On the critical points of random matrix characteristic polynomials and of the Riemann ξ-function

On the critical points of random matrix characteristic polynomials and of the Riemann ξ-function Abstract A one-parameter family of point processes describing the distribution of the critical points of the characteristic polynomial of large random Hermitian matrices on the scale of mean spacing is investigated. Conditionally on the Riemann hypothesis and the multiple correlation conjecture, we show that one of these limiting processes also describes the distribution of the critical points of the Riemann ξ-function on the critical line. We prove that each of these processes boasts stronger level repulsion than the sine process describing the limiting statistics of the eigenvalues: the probability to find k critical points in a short interval is comparable to the probability to find k + 1 eigenvalues there. We also prove a similar property for the critical points and zeros of the Riemann ξ-function, conditionally on the Riemann hypothesis, but not on the multiple correlation conjecture. 1. Introduction 1.1. Statement of results Let Si be the sine point process, that is a random locally finite subset of R the distribution of which is determined by   E∑x1,…,xk∈Sipairwisedistinctf(x1,…,xk)=∫dkxf(x1,…,xk)det(sinπ(xj−xl)π(xj−xl))j,l=1k. (1.1) The sine process describes the eigenvalue distribution of random Hermitian matrices on the scale of mean eigenvalue spacing. For complex Wigner matrices (a class of high-dimensional Hermitian random matrices with independent entries above the main diagonal, cf. Section 2.1), this is expressed by the following relation, which is part of a series of results obtained by Erdős–Yau, Tao–Vu and coworkers, see [18, 62]: if HN is a sequence of random matrices of growing dimension satisfying the assumptions listed in Section 2.1, and λj,N are the eigenvalues of HN/N, then for E∈(−2,2)  {(λj,N−E)N4−E22π}j=1N⟶Siindistribution (1.2)with respect to the topology defined by continuous test functions of compact support. The factor 2πN4−E2 by which the eigenvalues are scaled is the (approximate) mean spacing between eigenvalues near E. Similar results are available for other random matrix ensembles, see the monographs [2, 53] and references therein. The correlation conjecture of Montgomery [50] in the extended version of Rudnick–Sarnak [57] and Bogomolny–Keating [7, 8] states that a similar relation holds for the zeros of the Riemann ζ-function on the critical line: if t is chosen uniformly at random in [0,T], then   {(γ−t)logT2π∣ζ(12+iγ)=0}⟶??Siindistribution (1.3)(the question marks are put to emphasize that this relation is still conjectural). The scaling factor 2πlogT is the approximate mean spacing between the zeros with imaginary part near T. The results of [37, 50, 57] imply that, conditionally on the Riemann hypothesis, convergence holds for a restricted family of test functions. Following Aizenman and Warzel [1] and Chhaibi et al. [12], consider the random entire function   Φ(z)=limR→∞∏x∈Si∩(−R,R)(1−z/x). (1.4) We study the one-parameter family of point processes   Sia′={z∈C∣Φ′(z)=aΦ(z)}⊂R,a∈R. For any a, the points of Sia′ interlace with those of Si. Therefore, the statistical properties of Sia′ on long scales are very close to those of Si. Also, Sia′→Si as a→∞. On the other hand, on short scales, the processes Sia′ are much more rigid. To quantify this, introduce the events   Ωk(Si,ϵ)={#[Si∩(−ϵ,ϵ)]≥k},Ωk(Sia′,ϵ)={#[Sia′∩(−ϵ,ϵ)]≥k}. From the special case   E(#[Si∩(−ϵ,ϵ)])!(#[Si∩(−ϵ,ϵ)]−k)!=∫(−ϵ,ϵ)kdkxdet(sinπ(xj−xl)π(xj−xl))j,l=1kof (1.1), the sine process boasts the following repulsion property: for any k≥1,   P(Ωk(Si,ϵ))=ckϵk2+o(ϵk2),ϵ→+0,where0<ck<∞. (1.5) For comparison, the probability of the corresponding event in the standard Poisson process decays as ϵk. Our first result is Theorem 1.1 For any a∈R, k≥2and 0<ϵ<1  P(Ωk(Sia′,ϵ)⧹Ωk+1(Si,5ϵ))≤Ck(ϵlog1ϵ)(k+2)2. (1.6) That is, k-tuples of critical points in a short interval (for a fixed value of k) are mostly due to (k+1)-tuples of zeros in a slightly larger interval. From (1.6) and (1.5), we have   (ck+1+o(1))ϵ(k+1)2≤P(Ωk(Sia′,ϵ))≤(5(k+1)2ck+1+o(1))ϵ(k+1)2,ϵ→+0. A slightly more careful argument shows that Corollary 1.2 For any k≥2, there exists a limit  ck′=limϵ→+0P(Ωk(Sia′,ϵ))ϵ(k+1)2∈[ck+1,(1+4k−1)(k+1)2ck+1],independent of a∈R. Combining Corollary 1.2 with a result of Aizenman–Warzel [1] which is stated as Proposition 2.1 below, we obtain Corollary 1.3 Let (HN)be a sequence of complex Wigner matrices satisfying the assumptions listed in Section2.1, and let (λj,N′)j=1N−1be the critical points of the characteristic polynomial PN(λ)=det(HN/N−λ). For E∈(−2,2)and k≥2,   limN→∞P{#[∣λj,N′−E∣<2πϵN4−E2]≥k}=(ck′+o(1))ϵ(k+1)2.The stronger repulsion between the critical points can be seen in Figs 1 and 2. Figure 1. View largeDownload slide The critical points of the characteristic polynomial of GUE40, the eigenvalues, and the eigenvalues of a principal submatrix of dimension 39. Figure 1. View largeDownload slide The critical points of the characteristic polynomial of GUE40, the eigenvalues, and the eigenvalues of a principal submatrix of dimension 39. Figure 2. View largeDownload slide A histogram of the absolute values of the eigenvalue (left) and critical value (right) second closest to zero for GUE50, multiplied by 50π. Figure 2. View largeDownload slide A histogram of the absolute values of the eigenvalue (left) and critical value (right) second closest to zero for GUE50, multiplied by 50π. In the number-theoretic setting, we consider the Riemann ξ-function   ξ(s)=s(s−1)2πs/2Γ(s2)ζ(s). This is an entire function which is real on the critical line; its zeros coincide with the non-trivial zeros of the ζ-function. Conditionally on the Riemann hypothesis, the zeros of ξ′ lie on the critical line Rs=12 and interlace with the zeros of ξ (cf. Section 2.3). Assuming the Riemann hypothesis together with the multiple correlation conjecture, we prove (see Corollary 2.3) that   {(γ′−t)logT2π∣ξ′(12+iγ′)=0}⟶??Si0′indistribution. (1.7)See Fig. 3. Figure 3. View largeDownload slide A histogram of spacings between the critical points for GUE300 near E=0 (line) and for the ξ-function near 12+232i (bars). Data courtesy of Dave Platt. Figure 3. View largeDownload slide A histogram of spacings between the critical points for GUE300 near E=0 (line) and for the ξ-function near 12+232i (bars). Data courtesy of Dave Platt. Denote   Ωk(ξ,T,ϵ)={0≤t≤T∣#[γ∈(t−2πϵlogT,t+2πϵlogT),ξ(12+iγ)=0]≥k}Ωk(ξ′,T,ϵ)={0≤t≤T∣#[γ′∈(t−2πϵlogT,t+2πϵlogT),ξ′(12+iγ′)=0]≥k}.From Corollary 1.2 and (1.7), we obtain Corollary 1.4 Assume the Riemann hypothesis and the multiple correlation conjecture (1.3). Then  limT→∞1Tmes(Ωk(ξ′,T,ϵ))=(ck′+o(1))ϵ(k+1)2. We also prove the following less conditional result with a similar message: k-tuples of critical points of the ξ-function crowding short intervals are mostly a consequence of (k+1)-tuples of zeros crowding slightly larger intervals. Theorem 1.5 Assume the Riemann hypothesis. For any k≥2, 0<ϵ<1, R≥5  1Tmes(Ωk(ξ′,T,ϵ)⧹(Ωk+1(ξ,T,5ϵ)∪Ωk+2(ξ,T,Rϵ)))≤CeckR. 1.2. Motivation Let us discuss the motivation for these results. The traditional object of study in random matrix theory is the joint distribution of the eigenvalues. The local eigenvalue statistics, that is the study of eigenvalues on the scale of mean eigenvalue spacing, is of particular interest due to the robust (universal) nature of the limiting objects. Recently, the value distribution of the characteristic polynomial of a random matrix also received significant attention. While the characteristic polynomial is determined by the eigenvalues, its restriction to an interval depends both on the eigenvalues inside the interval and those outside it. Therefore, the statistical properties of the characteristic polynomial on the scale of mean eigenvalue spacing are not necessarily determined by the local eigenvalue statistics. As one varies the argument over an interval containing many eigenvalues for a given realization of the random matrix, the value of the polynomial shows huge variations by the orders of magnitude. We refer to the works [26, 28] for a discussion and references, in particular, for a statistical mechanics perspective on the absolute value of the characteristic polynomial as a disordered landscape (a Boltzmann weight with a log-correlated potential). It was found that characteristic polynomials of random matrices can be used to model the value distribution of the Riemann zeta function on the critical line [35, 38, 39]. In particular, the statistics related to the global maximum of the modulus of characteristic polynomial has been studied for matrices drawn from the circular ensemble, and these results were used to study (on the physical and mathematical levels of rigour) the properties of the global maximum of ∣ζ(1/2+it)∣ in various intervals [3–5, 11, 12, 27, 28, 51, 52]. Parallel questions for the characteristic polynomial of Hermitian random matrices were investigated in [29–31]. The sequence of positions of the local maxima (and minima) of the characteristic polynomial which we study in this paper is one of the natural characteristics of the random landscape. Second, the zeros and the critical points of the characteristic polynomial form an interlacing pair of sequences. As put forth by Kerov [40, 41], such pairs naturally appear in numerous problems of analysis, probability theory and representation theory, and their limiting properties in various asymptotics regimes are of particular importance. In the recent work [59], we studied the statistical properties of the zeros and the critical points from the point of view of the global regime, namely, the fluctuations of linear statistics. In particular, the fluctuations differ from those of another natural interlacing pair, formed by the eigenvalues of a random matrix and those of a principal submatrix; see Erdős and Schröder [17]. Here we study the joint distribution of the zeros and the critical points in the local regime, that is on the scale of the mean spacing. See further Corollary 2.2 and Section 3.4 (‘Critical points versus submatrix eigenvalues’), and also Fig. 1. Finally, the strong repulsion between the critical points is an instance of a general phenomenon: the zeros of the derivative of a polynomial with real zeros (or of an analytic function in the Laguerre–Pólya class) are more evenly spaced than the zeros of the original polynomial. In the deterministic setting, this phenomenon goes back to the work of Stoyanoff and Riesz [61], see [21] for a historical discussion. Theorem 1.1 and its corollaries provide additional examples in the random setting. Here we remark that, under repeated differentiation, the zeros become more and more rigid and approach an arithmetic progression (after proper rescaling and with the right order of limits). This was established in various settings in [22, 42, 54] (of which [42] is applicable to the ξ-function), and is probably true for Φ(z) of (1.4) as well. 1.3. Some more notation The second source of motivation comes from number theory (disclaimer: there are no new number-theoretic ingredients in our arguments). The study of the zeros of ξ′ goes back to the work of Levinson [46] and Conrey [13, 14], who obtained unconditional lower bounds on the fraction of zeros lying on the critical line (see [45, 58], the more recent [23] and references therein for the corresponding results pertaining to the zeros of ξ). More recently, Farmer et al. [19] and further Bian [6] and Bui [10] studied the correlations between the zeros of ξ′, arguing that a detailed understanding of the joint statistics of the zeros of ξ and ξ′ may allow us to rule out the so-called Alternative Hypothesis, according to which the spacings between nearest high-lying zeros of ξ are close to half-integer multiples of the mean spacing. The main result in [19] asserts that, conditionally on the Riemann hypothesis,   1N(T)∑0<γ′,γ˜′<Tξ′(12+iγ′)=ξ′(12+iγ˜′)=04eiα(γ′−γ˜′)logT4+(γ′−γ˜′)2→∣α∣−4∣α∣2+∑k=1∞(k−1)!(2k)!(2∣α∣)2k+1 (1.8)as T→∞, for 0<α<1, where   N(T)=#{0≤t≤T∣ξ(1/2+it)=0}=T2πlogT(1+o(1)). (1.9) For comparison, Montgomery showed [50] that (conditionally on the Riemann hypothesis) for 0<α<1  1N(T)∑0<γ,γ˜<Tξ(12+iγ)=ξ(12+iγ˜)=04eiα(γ−γ˜)logT4+(γ−γ˜)2→∣α∣,T→∞. (1.10) The α→0 asymptotics of form factors on the left-hand side of (1.8) and (1.10) capture the behaviour of the spacings between zeros on long scales. The short scale behaviour roughly corresponds to the α→∞ asymptotics of the form factor. The pair correlation conjecture of Montgomery [50] states that for α≥1 the limit of the left-hand side of (1.10) is equal to 1, similar to the form factor of the sine process. Further, Hejhal [37] and Rudnick and Sarnak [57] extended the result (1.10) to higher correlations, which, together with the arguments of Bogomolny–Keating [7, 8], led to the conjecture [7, 8, 57] that the full asymptotic distribution of the zeros of ξ on the scale of mean spacing is described by the sine process of random matrix theory (1.3). On the other hand, a conjectural description of the full limiting distribution of the critical points seems to have been missing. Our results (Corollary 2.4 below) provide such a conjectural description: it turns out that (1.3) formally implies that   {(γ′−t)logT2π∣ξ′(12+iγ′)=0}⟶??Si0′indistribution. (1.11) Then, Corollary 1.4 provides a conditional description of the left tail of the spacing distribution, whereas Theorem 1.5 provides some less conditional information. Here we also mention the works devoted to the zeros of ζ′, particularly [15, 43, 47, 48] (and references therein). The zeros of ζ′ do not lie on the critical line, and are, therefore, thematically more distant from the current study; their counterparts in random matrix setting (in a sense made precise in the aforementioned works) are the critical points of the characteristic polynomial of a circular ensemble. 1.4. On a condition of Aizenman and Warzel Let us briefly discuss the derivation of Corollaries 1.3 and 1.4 from Theorem 1.1; see Sections 2.2 and 2.3 for the precise definition and proofs. The collection of all the critical points of the characteristic polynomial is determined by the collection of all the eigenvalues. However, it is not a priori clear whether this relation persists in the local limit regime: that is, whether the conditional distribution of the critical points in an interval of length, say, (5×meanspacing), conditioned on the eigenvalues outside a concentric interval of length (R×meanspacing) degenerates in the limit R→∞, uniformly in the matrix size. Technically, (1.2) amounts to the convergence in distribution of linear statistics of the form   ∑j=1Nf((λj,N−E)N4−E22π),f∈C0(R) (1.12)to the corresponding statistics of the sine process. It is possible to extend this to other integrable test functions satisfying mild regularity conditions. On the other hand, the critical points are controlled by linear statistics corresponding to functions of the form f(λ)=1λ−z, z∈C⧹R; the asymptotics of such linear statistics is not a formal consequence of (1.2). Recently, Aizenman and Warzel [1] put forth a general condition which ensures that (1.2) can be upgraded to the convergence of such linear statistics. In the setting of Wigner matrices, they verified the condition using the local semicircle law of Erdős–Schlein–Yau [16] and the universality results of Erdős–Yau, Tao–Vu and coworkers [18, 62], and obtained: Proposition 2.1′ (Aizenman–Warzel) Let PN(z)=det(HN/N−z), where HNis a sequence of complex Wigner matrices satisfying the assumptions listed in Section2.1. Then for E∈(−2,2)  2πN4−E2PN′(E+2πzN4−E2)PN(E+2πzN4−E2)⟶distrΦ′(z)Φ(z)+πEz4−E2,N→∞ (1.13)  PN(E+2πzN4−E2)PN(E)⟶distrΦ(z)exp[πEz4−E2],N→∞ (1.14)with respect to the topology of locally uniform convergence on C⧹R(in the first relation) and of locally uniform convergence on C(in the second relation). The second relation follows from the first one by (careful) integration (some care is required to upgrade uniform convergence on compact subsets of C⧹R to uniform convergence on compact subsets of C not containing the poles of the limiting meromorphic function). We mention that Chhaibi et al. established a counterpart of (1.13)–(1.14) for the Circular Unitary Ensemble; see further Section 3.4 (‘Other random matrix ensembles’). We show that a similar result holds for the Riemann ξ-function, conditionally on the Riemann hypothesis; see Proposition 2.3. Somewhat similar statements have been proved for the ζ-function, cf. [20, 34, 55]. Here we quote the following corollary (the non-trivial statement is ⟹): Corollary 2.4′ Conditionally on the Riemann hypothesis, the multiple correlation conjecture (1.3) is equivalent to each of the following relations:   2πilogTξ′(12+i(t+2πzlogT))ξ(12+i(t+2πzlogT))⟶distrΦ′(z)Φ(z),T→∞ (1.15)  ξ(12+i(t+2πzlogT))ξ(12+it)⟶distrΦ(z),T→∞ (1.16)when t is chosen uniformly at random in [0,T]. The combination of Theorem 1.1 with Proposition 2.1 and Corollary 2.4 implies Corollaries 1.3 and 1.4. Roughly speaking, the asymptotics of the statistics which depend on ratios of the characteristic polynomial [or the ξ-function], for example, the joint distribution of the zeros of the first k derivatives, are determined by the sine process. Here we remark that the fluctuations of linear statistics corresponding to functions such as f(λ)=log(λ−z) contain a component that depends on the eigenvalues [or zeros] outside the microscopic window. This component is not universal, and, for the ξ-function, contains an arithmetic piece; see Gonek et al. [35] and references therein. Therefore, the limiting value distribution of the characteristic polynomial differs from that of the ξ-function (for any deterministic regularization). 2. Convergence of random analytic functions 2.1. Local statistics Let HN=(H(i,j))i,j=1N be a complex Wigner matrix, which for us is a Hermitian random matrix such that {(RH(i,j))i<j,(IH(i,j))i<j,(H(i,i))i} are independent random variables; (RH(i,j))i<j,(IH(i,j))i<j are identically distributed, and   EH(i,j)=0,E∣H(i,j)∣2=1,Eexp(δ∣H(i,j)∣2)<∞for some δ>0; (H(i,i))i are identically distributed, and   EH(i,i)=0,EH(i,i)2<∞,Eexp(δ∣H(i,i)∣2)<∞. The main example is the Gaussian Unitary Ensemble (GUE), in which the joint probability density of the matrix elements of H is proportional to exp(−12trH2). Denote by (λj,N)j=1N the eigenvalues of H/N. The global statistics of the eigenvalues is described by Wigner’s law:   1N∑j=1Nδ(E−λj,N)⟶ρ(E)dE (2.1)weakly in distribution, where ρ(E)=12π(4−E2)+ is the semicircular density. Due to the interlacing between the critical points λj,N′ and the zeros λj,N of PN(λ)=det(HN/N−λ), one also has   1N−1∑j=1N−1δ(E−λj,N′)⟶ρ(E)dE. The local statistics of λj,N are described by the sine point process Si defined in (1.1) (see for example [60] for general properties of determinantal point processes): for any E∈(−2,2), one has the convergence in distribution:   ∑jδ(u−(λj,N−E)Nρ(E))⟶∑x∈Siδ(u−x),N→∞. (2.2) This result was proved in the 1960s for the Gaussian Unitary Ensemble (the eigenvalues of which form a determinantal point process); see [49]. In a series of works by Erdős–Yau, Tao–Vu, and coworkers, (2.2) was generalized to Wigner matrices satisfying assumptions such as (1)–(3) above; see [18, 62] and references therein. The correlation conjecture of Montgomery [50] in the extended version of Rudnick–Sarnak [57] and Bogomolny–Keating [7, 8] asserts that a similar statement holds for the non-trivial zeros of the ζ-function:   ∑ξ(12+iγ)=0δ(u−(γ−t)logT2π)⟶??∑x∈Siδ(u−x),T→∞ (2.3)in distribution, when t is uniformly chosen from [0,T]. The relations (2.2) and (2.3) mean that   ∑jf((λj,N−E)Nρ(E))⟶∑x∈Sif(x) (2.4)  ∑ξ(12+iγ)=0f((γ−t)logT2π)⟶??∑x∈Sif(x) (2.5)in distribution for continuous test functions f of compact support. It is possible to extend this to integrable test functions satisfying mild regularity conditions. On the other hand, going beyond integrable functions requires additional information about the zeros lying far away from the microscopic window. It turns out that the second relation (2.5) is (conditionally) valid for test functions f(x)=1/(x−z), if the sums are properly regularized, whereas the first relation (2.4) requires a deterministic correction depending on E; see Sections 2.3 and 2.2 (relying on the works [1, 34], respectively). These properties imply that the critical points depend quasi-locally, so to speak, on the zeros/eigenvalues. 2.2. Random functions of Nevanlinna class We recall the construction of Aizenman–Warzel [1]. Recall that a function w:C⧹R→C belongs to the Nevanlinna [= Herglotz = Pick] class ( w∈R) if it is analytic and   w(z)¯=w(z¯),Iw(z)Iz>0. The class R is equipped with the topology of pointwise convergence on compact subsets of C⧹R. Denote   W(z)=limR→∞∑x∈Si[1x−z−1x−iR]+iπ=limR→∞∑x∈Si∩(−R,R)1x−z. (2.6) The two limits exist and coincide according to a general criterion of [1], and W(z) is a random element of the Nevanlinna class. Also note that −W(z) is the logarithmic derivative of the function Φ(z) from (1.4). As before, let PN(z)=det(HN/N−z) be the characteristic polynomial of HN/N, and denote   WN(z;E)=−1Nρ(E)pN′(E+zNρ(E))pN(E+zNρ(E))=∑j1(λj,N−E)Nρ(E)−z. Proposition 2.1 (Aizenman–Warzel) For ∣E∣<2,   WN(z;E)⟶distrW(z)−πE4−E2 (2.7)  PN(E+2πzN4−E2)PN(E)⟶distrΦ(z)exp[πEz4−E2]. (2.8) Proof The first statement is proved in [1, Corollary 6.5], their argument relies on the results obtained in the works [16, 18, 62] on the local eigenvalue statistics of Wigner matrices, and on the general theory of random Nevanlinna functions which was developed in [1]. The second statement follows from the first one by (carefully) integrating from 0 to z.□ Denote by w−1(a) the collection of solutions of w(z)=a. Observe that the map w↦w−1(a) from R∩{meromorphicfunctions} to locally finite (multi-)subsets of R is continuous. From this observation and (2.7), we deduce: Corollary 2.2 Let (HN)be a sequence of random matrices satisfying the assumptions listed in Section2.1. Then for any E∈(−2,2)  (∑jδ(u−(λj,N−E)Nρ(E)),∑jδ(u−(λj,N′−E)Nρ(E)))⟶(W−1(∞),W−1(−a))=(Si,Sia′) (2.9)in distribution, where a=−πE4−E2. To recapitulate, the non-obvious part of the statement is that the zeros λj,N which are not in an O(1/N)-neighbourhood of E influence the critical points λj,N′ near E only via the deterministic quantity πE(4−E2)−1/2. Colloquially, the conditional distribution of the critical points in (E−r/N,E+r/N) given the eigenvalues in (E−R/N,E+R/N) degenerates in the limit R→∞. Proof of Corollary 1.3 Follows from Theorem 1.1 and Corollary 2.2.□ 2.3. A counterpart for the ξ-function Denote   Wt,T(z)=−2πilogTξ′(12+i[t+2πzlogT])ξ(12+i[t+2πzlogT]),0≤t≤T.Assuming the Riemann hypothesis, Wt,T belongs to the Nevanlinna class (the property Wt,T∈R is independent of t and T, and is in fact equivalent to the Riemann hypothesis). We shall treat t as a random variable uniformly chosen in [0,T], and denote the corresponding random function by WT. The next proposition is close to the results for the ζ-function which were proved in [20, 34]. Proposition 2.3 Assume the Riemann hypothesis. Let Tn→∞be a sequence and P—a point process such that, for t uniformly chosen in [0,Tn],   ∑ξ(12+iγ)=0δ(u−(γ−t)logTn2π)→Pindistribution,n→∞.Then WTn→WPin distribution, where WPis the unique random Nevanlinna function such that  WP−1(∞)=P,WP(i∞)=iπ,and in particular,   (∑ξ(12+iγ)=0δ(u−(γ−t)logTn2π),∑ξ′(12+iγ)=0δ(u−(γ′−t)logTn2π))→(P,P0′),where P0′=WP−1(0). Remark Assuming the Riemann hypothesis, it follows from the results of Fujii [24, 25] that   1T∫0T(N(t+2πRlogT)−N(t)−R)2dt≤Clog(e+R), (2.10)therefore, the family of point processes WT−1(∞) is precompact (that is for any finite interval I the family of random variables #(WT−1(∞)∩I) is tight), and, moreover, any limit point P satisfies   E#[P∩[−R,R]]=2R,E∣#[P∩[0,R]]−R∣2≤Clog(e+R). In particular, the conditions of [1, Theorem 4.1] are satisfied, and, therefore, the function WP (uniquely) exists. Proposition 2.3 implies Corollary 2.4 Assume the Riemann hypothesis and the multiple correlation conjecture (1.3). Then  WT⟶distrW,ξ(12+i(t+2πzlogT))ξ(12+it)⟶distrΦ(z),T→∞and consequently  (∑ξ(12+iγ)=0δ(u−(γ−t)logT2π),∑ξ′(12+iγ′)=0δ(u−(γ′−t)logT2π))⟶distr(Si,Si0′). Proof of Corollary 1.4 Follows from Theorem 1.1 and Corollary 2.4.□ The proof of Proposition 2.3 relies on the following lemma. Related results go back to the work of Selberg [58]. We essentially follow the argument in [34], relying on the work of Montgomery [50]. Lemma 2.5 Assuming the Riemann hypothesis, one has for any R>0:   limT→∞∫0TdtTWt,T(iR)=iπ (2.11)  limsupT→∞∫0TdtT∣Wt,T(iR)−iπ∣2≤CR2. (2.12) Proof of Proposition 2.3 The convergence of WT to Φ follows from the lemma, in combination with the general criterion of Aizenman and Warzel [1, Theorem 6.1]; it implies the other two statements.□ Proof of Lemma 2.5 To prove (2.11), consider the integral of ξ′/ξ along the closed contour Γ composed of the segments connecting the points   12−RlogT,12+RlogT,12+RlogT+Ti,12+RlogT+T′i,12−RlogT+Ticounterclockwise (see Fig. 4), where T′ is the real number closest to T such that there are no zeros of the ξ-function in the 1/(100logT)-neighbourhood of 12+iT. By the residue theorem and the asymptotics (1.9) of N(T), the integral is equal to   2πi#{zerosofξwithimaginarypartin[0,T′]}=2πiTlogT2π(1+o(1)). On the other hand, from the functional equation ξ(1−z)=ξ(z), the integral along the left vertical line is equal to the integral along the right vertical line, and the integral along the bottom horizontal line is zero; the integral along the two segments on the top is negligible (as one can see, for example, from (2.14) below). Therefore   ∫0T(ξ′/ξ)(12+i(t+iR/logT))dt=−πTlogT2π(1+o(1))which is equivalent to (2.11). We note for the sequel that (2.11) implies the following smoothened version:   limT→∞∫−∞∞Tdtπ(t2+T2)Wt,T(iR)=iπ. (2.13) To prove (2.12), we use the Hadamard product representation (cf. [63, 2.12])   ξ(12+iz)=ξ(12)ebˆz∏ξ(12+iγ)=0(1−z/γ)ez/γ,which implies that   Wt,T(iR)=∑γ[1(γ−t)logT2π−iR−1γlogT2π]−2πbˆlogT. (2.14)Integrating with the weight T/(π(t2+T2)) and using (2.13) and the Cauchy theorem, we obtain   limT→∞∑γ[1(γ−iT)logT−iR−1γlogT]=i2. (2.15)Note that (2.15) holds for any real R (positive or negative). Now we compute   I(R,T)=∫−∞∞Tdtπ(t2+T2)∣Wt,T(iR)2π+bˆlogT∣2=∑γ,γ˜Iγ,γ˜(R,T),where   Iγ,γ˜(R,T)=∫−∞∞Tdtπ(t2+T2)[1(γ−t)logT−iR−1γlogT]×[1(γ˜−t)logT+iR−1γ˜logT].By the Cauchy theorem,   Iγ,γ˜(R,T)=Iγ,γ˜′(R,T)+Iγ,γ˜″(R,T),where   Iγ,γ˜′(R,T)=[1(γ−iT)logT−iR−1γlogT][1(γ˜−iT)logT+iR−1γ˜logT]Iγ,γ˜″(R,T)=−2iTT2+(γ˜+iRlogT)2[1(γ−γ˜)logT−2iR−1γlogT]1logT.In view of (2.15),   limT→∞∑γ,γ˜Iγ,γ˜′(R,T)=(i/2)2=−1/4. (2.16)To estimate the sum of Iγ,γ˜″(R,T), let   Jγ,γ˜(R,T)=2TT2+(γ˜)22R(γ−γ˜)2log2T+4R21logT,Jγ,γ˜′(R,T)=RIγ,γ˜″(R,T)−Jγ,γ˜(R,T). Using the estimates   ∣1T2+(γ˜+iRlogT)2−1T2+(γ˜)2∣≤3R(T2+(γ˜)2)2∣1(γ−γ˜)logT−2iR−1γlogT∣≤2(∣γ˜∣logT+2R)∣γ∣logT(∣γ−γ˜∣logT+2R), we deduce that   limT→∞∑γ,γ˜Jγ,γ˜′(R,T)=0. (2.17) Now we turn to ∑γ,γ˜Jγ,γ˜(R,T) and show that   limsupT→∞∣∑γ,γ˜Jγ,γ˜(R,T)−12∣≤ConstR2. (2.18) It will suffice to prove that   limsupT→∞∣2πTlogT∑0≤γ,γ˜≤T2R(γ−γ˜)2log2T+4R2−1∣≤ConstR2. (2.19) Let   F(α,T)=2πTlogT∑0≤γ,γ˜≤TTiα(γ−γ′)w(γ−γ˜),w(u)=44+u2. (2.20) Montgomery showed [50] that   F(α,T)=∣α∣+T−2∣α∣logT(1+o(1))+o(1),∣α∣<1 (2.21)(which implies (1.10)). Therefore,   supx∫xx+1F(α,T)dα≤C (2.22) (see [32, 33], where this is proved with C=83+ϵ and 2912+ϵ,respectively). From the definition (2.20) of F, we have (cf. [34, (2.11)]):   2πTlogT∑0≤γ,γ˜≤T2R(γ−γ˜)2log2T+4R2w(γ−γ˜)=∫0∞F(α,T)e−2R∣α∣dα, (2.23)and thus, from (2.21) and (2.22),   limsupT→∞∣2πTlogT∑0≤γ,γ˜≤T4R(γ−γ˜)2log2T+4R2w(γ−γ˜)−1∣≤CR2. (2.24) Observing that   limT→∞2πTlogT∑0≤γ,γ˜≤T4R(γ−γ˜)2log2T+4R2(1−w(γ−γ˜))=0,we obtain (2.19) and thus (2.18). The relations (2.16), (2.17) and (2.18) imply that   limR→∞limsupT→∞∣I(R,T)−1/4∣=0and hence (2.12) holds.□ Figure 4. View largeDownload slide The contour Γ from the proof of Lemma 2.5. Figure 4. View largeDownload slide The contour Γ from the proof of Lemma 2.5. 3. Repulsion In this section, we prove Theorems 1.1 and 1.5 and Corollary 1.2, the (re-)formulation of which we recall for the convenience of the reader. Let   Ωk(Si,ϵ)={#[Si∩(−ϵ,ϵ)]≥k},Ωk(Sia′,ϵ)={#[Sia′∩(−ϵ,ϵ)]≥k}. Theorem 1.1′ For any a∈R, k≥2, 0<ϵ<18max(∣a∣,2e)and R≥5  P(Ωk(Sia′,ϵ)\(Ωk+1(Sia,(1+4k−1)ϵ)∪Ωk+2(Sia,Rϵ)))≤2exp(−kR64). Taking R=1000klog1ϵ and using (1.5), we obtain the version of Theorem 1.1 stated in the introduction. Corollary 1.2 For any k≥2, there exists a limit  ck′=limϵ→+0P(Ωk(Sia′,ϵ))ϵ(k+1)2∈(ck+1,(1+4k−1)ck+1),independent of a∈R. Next, denote   Ωk(ξ,T,ϵ)={0≤t≤T∣#[γ∈(t−2πϵlogT,t+2πϵlogT),ξ(12+iγ)=0]≥k}.Ωk(ξ′,T,ϵ)={0≤t≤T∣#[γ′∈(t−2πϵlogT,t+2πϵlogT),ξ′(12+iγ′)=0]≥k}. Theorem 1.5′ Assume the Riemann hypothesis. For any k≥2, 0<ϵ<1, R≥1+4k−1  1Tmes(Ωk(ξ′,T,ϵ)⧹(Ωk+1(ξ,T,(1+4k−1)ϵ)∪Ωk+2(ξ,T,Rϵ)))≤CeckR. Remark The coefficient 1+4k−1 in these results can be further improved to 1+2k+o(1). 3.1. Proof of Theorem 1.1 Proof of Theorem 1.1 ′ By Corollary 2.2, the theorem is equivalent to Corollary 1.3, and, moreover, to its special case pertaining to one (arbitrary) ensemble of random matrices; we choose the Gaussian Unitary Ensemble. (We could equally work directly with Si; in that case we would need to regularize all the sums.) Denote   xj,N=(λj,N−E)Nρ(E),xj,N′=(λj,N′−E)Nρ(E),where a=−πE4−E2. Let us first show that for any R≥1+4/(k−1)  Ωk(Sia′,ϵ)\(Ωk+1(Sia,(1+4k−1)ϵ)∪Ωk+2(Sia,Rϵ))⊂{∑∣xj,N−ϵ∣≥(R−1)ϵ1xj,N−ϵ≥k−14ϵ}∪{∑∣xj,N+ϵ∣≥(R−1)ϵ1xj,N+ϵ≤−k−14ϵ}. (3.1)Indeed, assume that   #[∣xj,N′∣<ϵ]≥k,#[∣xj,N∣<(1+4k−1)ϵ]≤k. (3.2) Then we have by interlacing   #[xj,N∈(−ϵ,ϵ)]=k−1, (3.3)and, on the other hand, there are no xj,N at least in one of the intervals   (−(1+4k−1)ϵ,−ϵ),(+ϵ,+(1+4k−1)ϵ);for example, in the second one. Then   ∑j=1N1xj,N−ϵ≥0,whence by (3.3) and (3.2)   ∑∣xj,N−ϵ∣≥4k−1ϵ1xj,N−ϵ≥∑∣xj,N∣<ϵ−1xj,N−ϵ≥k−12ϵ. Every xj,N contributes at most (k−1)/(4ϵ) to the sum on the left-hand side, therefore, either #[(1+4k−1)ϵ≤xj,N<Rϵ]≥2 or   ∑∣xj,N−ϵ∣≥(R−1)ϵ1xj,N−ϵ≥k−14ϵ. This proves (3.1), and it remains to bound the probability of the two terms on the right-hand side. By symmetry, we can focus on the first term, for which we use the following lemma, proved below (see for example Breuer–Duits [9, Theorem 3.1] for more sophisticated bounds):□ Lemma For any determinantal process Dwith self-adjoint kernel of finite rank and any (bounded Borel measurable) test function f,   P{∑x∈Df(x)≥E∑x∈Df(x)+r}≤exp(−A*(r)),r>0,where  A*(r)=supt≤∥f∥∞−1(rt−A(t)),A(t)=et22E∑x∈Df(x)2. (3.4)We apply the lemma to   f(x)=1∣x−ϵ∣≥(R−1)ϵx−ϵ,FN=∑f(xj,N),then, denoting by ρN(E)=1NddEE#{λj,N≤E} the mean density of eigenvalues, we have   EFN=∫∣x∣≥(R−1)ϵρN(E+x+ϵNρ(E))xdx=∫[1∣x∣≥(R−1)ϵx−xx2+1]ρN(E+x+ϵNρ(E))dx+∫xx2+1ρN(E+x+ϵNρ(E))dx. The first addend tends to zero since ρN→ρ uniformly, whereas the second addend tends to −πE4−E2 by (2.7). Therefore   limN→∞EFN=−πE4−E2. Next, for t≤Rϵ  limN→∞A(t)=et22limN→∞E∑f(xj,N)2=et22limN→∞∫∣x∣≥Rϵdx∣x−ϵ∣2≤et2(R−1)ϵ,whence, taking t=(R−1)ϵ in the definition (3.4) of A* and assuming that ϵ<1/(16e),   limN→∞A*(k−18ϵ)≥k−18ϵ(R−1)ϵ−e(R−1)ϵ≥(k−1)(R−1)16. According to the lemma, we have for ϵ≤min(4−E28π∣E∣,116e):   P{FN≥k−14ϵ}≤exp{−(k−1)(R−1)16}≤exp{−kR64},as claimed.□ Proof of Lemma Let K denote the operator defining the determinantal process; then for t∥f∥∞≤1  logEexp{t∑x∈Df(x)}=log det(1+(etf−1)K)=trlog(1+(etf−1)K)≤tr(etf−1)K≤et22trf2K+ttrfK≤A(t)+tE∑x∈Df(x).Therefore, by the Chebyshev inequality   P{∑x∈Df(x)≥E∑x∈Df(x)+r}≤exp(A(t)−rt).□ 3.2. Proof of Corollary 1.2 Proof of Corollary 1.2 Let us show that the limit   ck′=limϵ→0P(Ωk(Sia′,ϵ))ϵ(k+1)2,k≥2 (3.5)exists and does not depend on a. Choose αk>αk′>0 sufficiently small to ensure that   (1−αk)(k+2)2>(k+1)2. (3.6) Denote by ϒ(ϵ) the event   ∃pairwisedistinctdistinctX=(x1,…,xk+1)∈(Si∩(−ϵ1−αk,ϵ1−αk))k+1,suchthatallthezerosofPX(x)=ddx∏j=1k+1(x−xj)liein(−ϵ,ϵ). (3.7) Let us show that   P(ϒ(ϵ−ϵ1+αk′)⧹Ωk(Sia′,ϵ)),P(Ωk(Sia′,ϵ)⧹ϒ(ϵ+ϵ1+αk′))=o(ϵ(k+1)2). (3.8) Indeed, on the event ϒ(ϵ−ϵ1+αk′)⧹(Ωk(Sia′,ϵ)∪Ωk+2(Si,ϵ1−αk)) at least one of the xj lies outside (−ϵ,ϵ); assume for example that xk+1>ϵ and decompose   0>W(ϵ)=∑j=1k+11xj−ϵ+limr→∞∑x∈Siϵ1−αk≤∣x∣≤r1xj−ϵ. Then the first term is bounded from below by   ∑j=1k+11xj−ϵ≥∑j=1k+11xj−ϵ−ϵ1+αk′+(k+1)ϵ1+αk′4ϵ2≥k+14ϵ1−αk′. The probability of the event   limr→+∞∑x∈Siϵ1−αk≤∣x∣≤r1xj−ϵ≤−k+14ϵ1−αk′is o(ϵ(k+1)2) by a tail estimate which follows from the lemma in Section 3.1 (similar to the proof of Theorem 1.1), whereas P(Ωk+2(Si,ϵ1−αk))=o(ϵ(k+1)2) by the condition (3.6) on αk and (1.5). This proves the first part of (3.8); the second part is proved in a similar way. From (1.1) and the asymptotics   det(sinπ(xj−xm)π(xj−xm))j,m=1k=(1+o(1))ck,1∏j<m(xj−xm)2x→0(where ck,1 as well as ck,2 and ck,3 below are numerical constants), we obtain   P(ϒ(ϵ))=(ck,2+o(1))ϵ∫∑xj=0dkX∏j<m(xj−xm)21{thezerosofPXliein(−ϵ,ϵ)}=(ck,2+o(1))ck,3ϵ(k+1)2,which implies, with (3.8), that (3.5) holds with ck′=ck,2ck,3. Note that the indicator under the integral is compactly supported. The bound   ck′∈[ck+1,(1+4k−1)(k+1)2ck+1]follows from Theorem 1.1 ′ and (1.5).□ 3.3. Proof of Theorem 1.5 The proof of Theorem 1.5 relies on a bound, proved by Rodgers [56], on the moments of the logarithmic derivative of the ζ-function; see (3.12) below. The results of Farmer et al. [20] imply a more precise bound under additional hypotheses. Proof of Theorem 1.5′ Let 12+iγj be the zeros of ξ(z); rescale them as follows: xj,t,T=(γj−t)logT2π. We start with the following counterpart of (3.1):   1Tmes(Ωk(ξ′,T,ϵ)⧹(Ωk+1(ξ,T,(1+4k−1)ϵ)∪Ωk+2(ξ,T,Rϵ)))≤1Tmes({Wt,T(ϵ)−∑∣xj,t,T−ϵ∣<(R−1)ϵ1xj,t,T−ϵ≥k−14ϵ}\Ωk+2(ξ,T,Rϵ))+1Tmes({Wt,T(−ϵ)−∑∣xj,t,T+ϵ∣<(R−1)ϵ1xj,t,T+ϵ≤−k−14ϵ}\Ωk+2(ξ,T,Rϵ)).To show that each of the two terms is bounded by CeckR, it will suffice to prove that   1Tmes({∣Wt,T(0)−∑∣xj,t,T∣<Rϵ1xj,t,T∣≥k−14ϵ}\Ωk+2(ξ,T,Rϵ))≤CeckR. (3.9)Decompose   Wt,T(0)−∑∣xj,t,T∣<Rϵ1xj,t,T=RWt,T(20iRϵ)−∑∣xj,t,T∣<Rϵxj,t,Txj,t,T2+400R2ϵ2+∑∣xj,t,T∣≥Rϵ400R2ϵ2xj,t,T(xj,t,T2+400R2ϵ2)and observe that on the complement of Ωk+2(ξ,T,Rϵ)  ∣∑∣xj,t,T∣<Rϵxj,t,Txj,T2+400R2ϵ2∣≤k+120Rϵ≤k−16ϵ (3.10)whereas   ∣RWt,T(20iRϵ)∣,120∣∑∣xj,t,T∣≥Rϵ400R2ϵ2xj,t,T(xj,t,T2+400R2ϵ2)∣≤∣Wt,T(20iRϵ)∣,whence we need to show that   1Tmes{0≤t≤T∣∣Wt,T(20iRϵ)∣≥k−1480ϵ}≤Cexp(−ckR). (3.11)By a result of Rodgers [56, Theorem 2.1],   ∫0T∣ζ′(12+i(t+2πiδlogT))ζ(12+i(t+2πiδlogT))∣mdt≤Cmlogmδ−mTlogmT, (3.12)hence   ∫0T∣ξ′(12+i(t+2πiδlogT))ξ(12+i(t+2πiδlogT))∣mdt≤Cmlogmδ−mTlogmT,and, finally,   1T∫0T∣Wt,T(20iRϵ)∣mdt≤Cmlogm(Rϵ)−m, (3.13)from which (3.11) follows by the Chebyshev inequality.□ 3.4. Some questions and comments Other random matrix ensembles The results of this paper have counterparts for other random matrix ensembles (and other ξ-functions). In particular, Chhaibi et al. [12] showed that the characteristic polynomial ZN(z)=det(1N−zUN*) of the Circular Unitary Ensemble satisfies   ZN(exp(2πiz/N))ZN(1)⟶distrexp(iπz)Φ(z). Hence the collection of critical points of ZN(z)z−N/2 converges, after rescaling by N, to the same process Si0′ as in Corollaries 2.2 and 2.3. This is consistent with the discussion in [19, Sections 2.3 and 6.2]. Critical points vs. submatrix eigenvalues Let   Wdec(z)=limR→∞∑x∈Si∩(−R,R)∣gx∣2x−z,where, conditionally on Si, the random variables gx are independent complex standard Gaussian. Then Corollary 2.2 has the following counterpart:   ({(λj,N−E)Nρ(E)},{(λj,N−1−E)Nρ(E)})⟶((Wdec)−1(−πE4−E2),W−1(∞)). (3.14)(Amusingly, the distribution of the first term does not depend on E.) We are not sure whether (3.14) has a number-theoretic analogue. Form factors Suppose a point process P is a limit point of the rescaled zeros of ξ, as in Proposition 2.3. Then the rescaled critical points converge, along the same subsequence, to the point process P′=WP−1(0), the distribution of which is uniquely determined by that of P. By a result of Montgomery [50] and its extension by Hejhal [37] and Rudnick and Sarnak [57], the (multiple) form factors of P coincide, in a restricted domain of momenta, with those of the sine process. In view of the research programme suggested by Farmer et al. [19], it is natural to ask which constraints do this impose on the form factors of P′, and to compare these with the results of [6, 19]. It would also be interesting to compute the form factor of Sia′ and to check whether it coincides with the right-hand side of (1.8) for a=0 and ∣α∣≤1. A possible starting point is the identity ek(λ1′,…,λN−1′)=(1−kN)ek(λ1,…,λN) relating the elementary symmetric functions in the critical points of a polynomial to the elementary symmetric functions in the zeros. Zeros of higher derivatives The result (1.8) of [19] was extended to higher derivatives of ξ in the Ph.D. thesis of Bian [6]. In this context, we mention that if the Riemann hypothesis and the multiple correlation conjecture hold, Corollary 2.3 implies that for any k≥1  ∑ξ(k)(12+iγ‴)=0δ(u−(γ‴−t)logT2π)⟶distrSi0(k)={Φ(k)=0}. Logarithmic derivative From Lemma 2.5 and [1, Theorems 2.3 and 6.2] of Aizenman–Warzel, it follows, conditionally on the Riemann hypothesis, that (for t chosen uniformly in [0,T]) the rescaled logarithmic derivatives   2ilogTξ′(12+it)ξ(12+it),2ilogTζ′(12+it)ζ(12+it)+i⟶T→∞distrCauchy, (3.15)where the right-hand side is a standard (real) Cauchy random variable. For comparison, it was proved by Lester [44], following earlier results by Guo [36], that the distribution of (ζ′/ζ)(σ(T)+it), where ∣σ(T)−12∣logT→∞ and ∣σ(T)−12∣→0, is approximately (complex) Gaussian for large T. Funding Supported in part by the European Research Council start-up Grant 639305 (SPECTRUM). Acknowledgement This work originated from a question of Yan Fyodorov, who asked how to describe the local statistics of critical points, and from subsequent discussions, particularly, of the strong repulsion; he also made helpful comments and suggestions on all the subsequent stages. 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On the critical points of random matrix characteristic polynomials and of the Riemann ξ-function

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Abstract

Abstract A one-parameter family of point processes describing the distribution of the critical points of the characteristic polynomial of large random Hermitian matrices on the scale of mean spacing is investigated. Conditionally on the Riemann hypothesis and the multiple correlation conjecture, we show that one of these limiting processes also describes the distribution of the critical points of the Riemann ξ-function on the critical line. We prove that each of these processes boasts stronger level repulsion than the sine process describing the limiting statistics of the eigenvalues: the probability to find k critical points in a short interval is comparable to the probability to find k + 1 eigenvalues there. We also prove a similar property for the critical points and zeros of the Riemann ξ-function, conditionally on the Riemann hypothesis, but not on the multiple correlation conjecture. 1. Introduction 1.1. Statement of results Let Si be the sine point process, that is a random locally finite subset of R the distribution of which is determined by   E∑x1,…,xk∈Sipairwisedistinctf(x1,…,xk)=∫dkxf(x1,…,xk)det(sinπ(xj−xl)π(xj−xl))j,l=1k. (1.1) The sine process describes the eigenvalue distribution of random Hermitian matrices on the scale of mean eigenvalue spacing. For complex Wigner matrices (a class of high-dimensional Hermitian random matrices with independent entries above the main diagonal, cf. Section 2.1), this is expressed by the following relation, which is part of a series of results obtained by Erdős–Yau, Tao–Vu and coworkers, see [18, 62]: if HN is a sequence of random matrices of growing dimension satisfying the assumptions listed in Section 2.1, and λj,N are the eigenvalues of HN/N, then for E∈(−2,2)  {(λj,N−E)N4−E22π}j=1N⟶Siindistribution (1.2)with respect to the topology defined by continuous test functions of compact support. The factor 2πN4−E2 by which the eigenvalues are scaled is the (approximate) mean spacing between eigenvalues near E. Similar results are available for other random matrix ensembles, see the monographs [2, 53] and references therein. The correlation conjecture of Montgomery [50] in the extended version of Rudnick–Sarnak [57] and Bogomolny–Keating [7, 8] states that a similar relation holds for the zeros of the Riemann ζ-function on the critical line: if t is chosen uniformly at random in [0,T], then   {(γ−t)logT2π∣ζ(12+iγ)=0}⟶??Siindistribution (1.3)(the question marks are put to emphasize that this relation is still conjectural). The scaling factor 2πlogT is the approximate mean spacing between the zeros with imaginary part near T. The results of [37, 50, 57] imply that, conditionally on the Riemann hypothesis, convergence holds for a restricted family of test functions. Following Aizenman and Warzel [1] and Chhaibi et al. [12], consider the random entire function   Φ(z)=limR→∞∏x∈Si∩(−R,R)(1−z/x). (1.4) We study the one-parameter family of point processes   Sia′={z∈C∣Φ′(z)=aΦ(z)}⊂R,a∈R. For any a, the points of Sia′ interlace with those of Si. Therefore, the statistical properties of Sia′ on long scales are very close to those of Si. Also, Sia′→Si as a→∞. On the other hand, on short scales, the processes Sia′ are much more rigid. To quantify this, introduce the events   Ωk(Si,ϵ)={#[Si∩(−ϵ,ϵ)]≥k},Ωk(Sia′,ϵ)={#[Sia′∩(−ϵ,ϵ)]≥k}. From the special case   E(#[Si∩(−ϵ,ϵ)])!(#[Si∩(−ϵ,ϵ)]−k)!=∫(−ϵ,ϵ)kdkxdet(sinπ(xj−xl)π(xj−xl))j,l=1kof (1.1), the sine process boasts the following repulsion property: for any k≥1,   P(Ωk(Si,ϵ))=ckϵk2+o(ϵk2),ϵ→+0,where0<ck<∞. (1.5) For comparison, the probability of the corresponding event in the standard Poisson process decays as ϵk. Our first result is Theorem 1.1 For any a∈R, k≥2and 0<ϵ<1  P(Ωk(Sia′,ϵ)⧹Ωk+1(Si,5ϵ))≤Ck(ϵlog1ϵ)(k+2)2. (1.6) That is, k-tuples of critical points in a short interval (for a fixed value of k) are mostly due to (k+1)-tuples of zeros in a slightly larger interval. From (1.6) and (1.5), we have   (ck+1+o(1))ϵ(k+1)2≤P(Ωk(Sia′,ϵ))≤(5(k+1)2ck+1+o(1))ϵ(k+1)2,ϵ→+0. A slightly more careful argument shows that Corollary 1.2 For any k≥2, there exists a limit  ck′=limϵ→+0P(Ωk(Sia′,ϵ))ϵ(k+1)2∈[ck+1,(1+4k−1)(k+1)2ck+1],independent of a∈R. Combining Corollary 1.2 with a result of Aizenman–Warzel [1] which is stated as Proposition 2.1 below, we obtain Corollary 1.3 Let (HN)be a sequence of complex Wigner matrices satisfying the assumptions listed in Section2.1, and let (λj,N′)j=1N−1be the critical points of the characteristic polynomial PN(λ)=det(HN/N−λ). For E∈(−2,2)and k≥2,   limN→∞P{#[∣λj,N′−E∣<2πϵN4−E2]≥k}=(ck′+o(1))ϵ(k+1)2.The stronger repulsion between the critical points can be seen in Figs 1 and 2. Figure 1. View largeDownload slide The critical points of the characteristic polynomial of GUE40, the eigenvalues, and the eigenvalues of a principal submatrix of dimension 39. Figure 1. View largeDownload slide The critical points of the characteristic polynomial of GUE40, the eigenvalues, and the eigenvalues of a principal submatrix of dimension 39. Figure 2. View largeDownload slide A histogram of the absolute values of the eigenvalue (left) and critical value (right) second closest to zero for GUE50, multiplied by 50π. Figure 2. View largeDownload slide A histogram of the absolute values of the eigenvalue (left) and critical value (right) second closest to zero for GUE50, multiplied by 50π. In the number-theoretic setting, we consider the Riemann ξ-function   ξ(s)=s(s−1)2πs/2Γ(s2)ζ(s). This is an entire function which is real on the critical line; its zeros coincide with the non-trivial zeros of the ζ-function. Conditionally on the Riemann hypothesis, the zeros of ξ′ lie on the critical line Rs=12 and interlace with the zeros of ξ (cf. Section 2.3). Assuming the Riemann hypothesis together with the multiple correlation conjecture, we prove (see Corollary 2.3) that   {(γ′−t)logT2π∣ξ′(12+iγ′)=0}⟶??Si0′indistribution. (1.7)See Fig. 3. Figure 3. View largeDownload slide A histogram of spacings between the critical points for GUE300 near E=0 (line) and for the ξ-function near 12+232i (bars). Data courtesy of Dave Platt. Figure 3. View largeDownload slide A histogram of spacings between the critical points for GUE300 near E=0 (line) and for the ξ-function near 12+232i (bars). Data courtesy of Dave Platt. Denote   Ωk(ξ,T,ϵ)={0≤t≤T∣#[γ∈(t−2πϵlogT,t+2πϵlogT),ξ(12+iγ)=0]≥k}Ωk(ξ′,T,ϵ)={0≤t≤T∣#[γ′∈(t−2πϵlogT,t+2πϵlogT),ξ′(12+iγ′)=0]≥k}.From Corollary 1.2 and (1.7), we obtain Corollary 1.4 Assume the Riemann hypothesis and the multiple correlation conjecture (1.3). Then  limT→∞1Tmes(Ωk(ξ′,T,ϵ))=(ck′+o(1))ϵ(k+1)2. We also prove the following less conditional result with a similar message: k-tuples of critical points of the ξ-function crowding short intervals are mostly a consequence of (k+1)-tuples of zeros crowding slightly larger intervals. Theorem 1.5 Assume the Riemann hypothesis. For any k≥2, 0<ϵ<1, R≥5  1Tmes(Ωk(ξ′,T,ϵ)⧹(Ωk+1(ξ,T,5ϵ)∪Ωk+2(ξ,T,Rϵ)))≤CeckR. 1.2. Motivation Let us discuss the motivation for these results. The traditional object of study in random matrix theory is the joint distribution of the eigenvalues. The local eigenvalue statistics, that is the study of eigenvalues on the scale of mean eigenvalue spacing, is of particular interest due to the robust (universal) nature of the limiting objects. Recently, the value distribution of the characteristic polynomial of a random matrix also received significant attention. While the characteristic polynomial is determined by the eigenvalues, its restriction to an interval depends both on the eigenvalues inside the interval and those outside it. Therefore, the statistical properties of the characteristic polynomial on the scale of mean eigenvalue spacing are not necessarily determined by the local eigenvalue statistics. As one varies the argument over an interval containing many eigenvalues for a given realization of the random matrix, the value of the polynomial shows huge variations by the orders of magnitude. We refer to the works [26, 28] for a discussion and references, in particular, for a statistical mechanics perspective on the absolute value of the characteristic polynomial as a disordered landscape (a Boltzmann weight with a log-correlated potential). It was found that characteristic polynomials of random matrices can be used to model the value distribution of the Riemann zeta function on the critical line [35, 38, 39]. In particular, the statistics related to the global maximum of the modulus of characteristic polynomial has been studied for matrices drawn from the circular ensemble, and these results were used to study (on the physical and mathematical levels of rigour) the properties of the global maximum of ∣ζ(1/2+it)∣ in various intervals [3–5, 11, 12, 27, 28, 51, 52]. Parallel questions for the characteristic polynomial of Hermitian random matrices were investigated in [29–31]. The sequence of positions of the local maxima (and minima) of the characteristic polynomial which we study in this paper is one of the natural characteristics of the random landscape. Second, the zeros and the critical points of the characteristic polynomial form an interlacing pair of sequences. As put forth by Kerov [40, 41], such pairs naturally appear in numerous problems of analysis, probability theory and representation theory, and their limiting properties in various asymptotics regimes are of particular importance. In the recent work [59], we studied the statistical properties of the zeros and the critical points from the point of view of the global regime, namely, the fluctuations of linear statistics. In particular, the fluctuations differ from those of another natural interlacing pair, formed by the eigenvalues of a random matrix and those of a principal submatrix; see Erdős and Schröder [17]. Here we study the joint distribution of the zeros and the critical points in the local regime, that is on the scale of the mean spacing. See further Corollary 2.2 and Section 3.4 (‘Critical points versus submatrix eigenvalues’), and also Fig. 1. Finally, the strong repulsion between the critical points is an instance of a general phenomenon: the zeros of the derivative of a polynomial with real zeros (or of an analytic function in the Laguerre–Pólya class) are more evenly spaced than the zeros of the original polynomial. In the deterministic setting, this phenomenon goes back to the work of Stoyanoff and Riesz [61], see [21] for a historical discussion. Theorem 1.1 and its corollaries provide additional examples in the random setting. Here we remark that, under repeated differentiation, the zeros become more and more rigid and approach an arithmetic progression (after proper rescaling and with the right order of limits). This was established in various settings in [22, 42, 54] (of which [42] is applicable to the ξ-function), and is probably true for Φ(z) of (1.4) as well. 1.3. Some more notation The second source of motivation comes from number theory (disclaimer: there are no new number-theoretic ingredients in our arguments). The study of the zeros of ξ′ goes back to the work of Levinson [46] and Conrey [13, 14], who obtained unconditional lower bounds on the fraction of zeros lying on the critical line (see [45, 58], the more recent [23] and references therein for the corresponding results pertaining to the zeros of ξ). More recently, Farmer et al. [19] and further Bian [6] and Bui [10] studied the correlations between the zeros of ξ′, arguing that a detailed understanding of the joint statistics of the zeros of ξ and ξ′ may allow us to rule out the so-called Alternative Hypothesis, according to which the spacings between nearest high-lying zeros of ξ are close to half-integer multiples of the mean spacing. The main result in [19] asserts that, conditionally on the Riemann hypothesis,   1N(T)∑0<γ′,γ˜′<Tξ′(12+iγ′)=ξ′(12+iγ˜′)=04eiα(γ′−γ˜′)logT4+(γ′−γ˜′)2→∣α∣−4∣α∣2+∑k=1∞(k−1)!(2k)!(2∣α∣)2k+1 (1.8)as T→∞, for 0<α<1, where   N(T)=#{0≤t≤T∣ξ(1/2+it)=0}=T2πlogT(1+o(1)). (1.9) For comparison, Montgomery showed [50] that (conditionally on the Riemann hypothesis) for 0<α<1  1N(T)∑0<γ,γ˜<Tξ(12+iγ)=ξ(12+iγ˜)=04eiα(γ−γ˜)logT4+(γ−γ˜)2→∣α∣,T→∞. (1.10) The α→0 asymptotics of form factors on the left-hand side of (1.8) and (1.10) capture the behaviour of the spacings between zeros on long scales. The short scale behaviour roughly corresponds to the α→∞ asymptotics of the form factor. The pair correlation conjecture of Montgomery [50] states that for α≥1 the limit of the left-hand side of (1.10) is equal to 1, similar to the form factor of the sine process. Further, Hejhal [37] and Rudnick and Sarnak [57] extended the result (1.10) to higher correlations, which, together with the arguments of Bogomolny–Keating [7, 8], led to the conjecture [7, 8, 57] that the full asymptotic distribution of the zeros of ξ on the scale of mean spacing is described by the sine process of random matrix theory (1.3). On the other hand, a conjectural description of the full limiting distribution of the critical points seems to have been missing. Our results (Corollary 2.4 below) provide such a conjectural description: it turns out that (1.3) formally implies that   {(γ′−t)logT2π∣ξ′(12+iγ′)=0}⟶??Si0′indistribution. (1.11) Then, Corollary 1.4 provides a conditional description of the left tail of the spacing distribution, whereas Theorem 1.5 provides some less conditional information. Here we also mention the works devoted to the zeros of ζ′, particularly [15, 43, 47, 48] (and references therein). The zeros of ζ′ do not lie on the critical line, and are, therefore, thematically more distant from the current study; their counterparts in random matrix setting (in a sense made precise in the aforementioned works) are the critical points of the characteristic polynomial of a circular ensemble. 1.4. On a condition of Aizenman and Warzel Let us briefly discuss the derivation of Corollaries 1.3 and 1.4 from Theorem 1.1; see Sections 2.2 and 2.3 for the precise definition and proofs. The collection of all the critical points of the characteristic polynomial is determined by the collection of all the eigenvalues. However, it is not a priori clear whether this relation persists in the local limit regime: that is, whether the conditional distribution of the critical points in an interval of length, say, (5×meanspacing), conditioned on the eigenvalues outside a concentric interval of length (R×meanspacing) degenerates in the limit R→∞, uniformly in the matrix size. Technically, (1.2) amounts to the convergence in distribution of linear statistics of the form   ∑j=1Nf((λj,N−E)N4−E22π),f∈C0(R) (1.12)to the corresponding statistics of the sine process. It is possible to extend this to other integrable test functions satisfying mild regularity conditions. On the other hand, the critical points are controlled by linear statistics corresponding to functions of the form f(λ)=1λ−z, z∈C⧹R; the asymptotics of such linear statistics is not a formal consequence of (1.2). Recently, Aizenman and Warzel [1] put forth a general condition which ensures that (1.2) can be upgraded to the convergence of such linear statistics. In the setting of Wigner matrices, they verified the condition using the local semicircle law of Erdős–Schlein–Yau [16] and the universality results of Erdős–Yau, Tao–Vu and coworkers [18, 62], and obtained: Proposition 2.1′ (Aizenman–Warzel) Let PN(z)=det(HN/N−z), where HNis a sequence of complex Wigner matrices satisfying the assumptions listed in Section2.1. Then for E∈(−2,2)  2πN4−E2PN′(E+2πzN4−E2)PN(E+2πzN4−E2)⟶distrΦ′(z)Φ(z)+πEz4−E2,N→∞ (1.13)  PN(E+2πzN4−E2)PN(E)⟶distrΦ(z)exp[πEz4−E2],N→∞ (1.14)with respect to the topology of locally uniform convergence on C⧹R(in the first relation) and of locally uniform convergence on C(in the second relation). The second relation follows from the first one by (careful) integration (some care is required to upgrade uniform convergence on compact subsets of C⧹R to uniform convergence on compact subsets of C not containing the poles of the limiting meromorphic function). We mention that Chhaibi et al. established a counterpart of (1.13)–(1.14) for the Circular Unitary Ensemble; see further Section 3.4 (‘Other random matrix ensembles’). We show that a similar result holds for the Riemann ξ-function, conditionally on the Riemann hypothesis; see Proposition 2.3. Somewhat similar statements have been proved for the ζ-function, cf. [20, 34, 55]. Here we quote the following corollary (the non-trivial statement is ⟹): Corollary 2.4′ Conditionally on the Riemann hypothesis, the multiple correlation conjecture (1.3) is equivalent to each of the following relations:   2πilogTξ′(12+i(t+2πzlogT))ξ(12+i(t+2πzlogT))⟶distrΦ′(z)Φ(z),T→∞ (1.15)  ξ(12+i(t+2πzlogT))ξ(12+it)⟶distrΦ(z),T→∞ (1.16)when t is chosen uniformly at random in [0,T]. The combination of Theorem 1.1 with Proposition 2.1 and Corollary 2.4 implies Corollaries 1.3 and 1.4. Roughly speaking, the asymptotics of the statistics which depend on ratios of the characteristic polynomial [or the ξ-function], for example, the joint distribution of the zeros of the first k derivatives, are determined by the sine process. Here we remark that the fluctuations of linear statistics corresponding to functions such as f(λ)=log(λ−z) contain a component that depends on the eigenvalues [or zeros] outside the microscopic window. This component is not universal, and, for the ξ-function, contains an arithmetic piece; see Gonek et al. [35] and references therein. Therefore, the limiting value distribution of the characteristic polynomial differs from that of the ξ-function (for any deterministic regularization). 2. Convergence of random analytic functions 2.1. Local statistics Let HN=(H(i,j))i,j=1N be a complex Wigner matrix, which for us is a Hermitian random matrix such that {(RH(i,j))i<j,(IH(i,j))i<j,(H(i,i))i} are independent random variables; (RH(i,j))i<j,(IH(i,j))i<j are identically distributed, and   EH(i,j)=0,E∣H(i,j)∣2=1,Eexp(δ∣H(i,j)∣2)<∞for some δ>0; (H(i,i))i are identically distributed, and   EH(i,i)=0,EH(i,i)2<∞,Eexp(δ∣H(i,i)∣2)<∞. The main example is the Gaussian Unitary Ensemble (GUE), in which the joint probability density of the matrix elements of H is proportional to exp(−12trH2). Denote by (λj,N)j=1N the eigenvalues of H/N. The global statistics of the eigenvalues is described by Wigner’s law:   1N∑j=1Nδ(E−λj,N)⟶ρ(E)dE (2.1)weakly in distribution, where ρ(E)=12π(4−E2)+ is the semicircular density. Due to the interlacing between the critical points λj,N′ and the zeros λj,N of PN(λ)=det(HN/N−λ), one also has   1N−1∑j=1N−1δ(E−λj,N′)⟶ρ(E)dE. The local statistics of λj,N are described by the sine point process Si defined in (1.1) (see for example [60] for general properties of determinantal point processes): for any E∈(−2,2), one has the convergence in distribution:   ∑jδ(u−(λj,N−E)Nρ(E))⟶∑x∈Siδ(u−x),N→∞. (2.2) This result was proved in the 1960s for the Gaussian Unitary Ensemble (the eigenvalues of which form a determinantal point process); see [49]. In a series of works by Erdős–Yau, Tao–Vu, and coworkers, (2.2) was generalized to Wigner matrices satisfying assumptions such as (1)–(3) above; see [18, 62] and references therein. The correlation conjecture of Montgomery [50] in the extended version of Rudnick–Sarnak [57] and Bogomolny–Keating [7, 8] asserts that a similar statement holds for the non-trivial zeros of the ζ-function:   ∑ξ(12+iγ)=0δ(u−(γ−t)logT2π)⟶??∑x∈Siδ(u−x),T→∞ (2.3)in distribution, when t is uniformly chosen from [0,T]. The relations (2.2) and (2.3) mean that   ∑jf((λj,N−E)Nρ(E))⟶∑x∈Sif(x) (2.4)  ∑ξ(12+iγ)=0f((γ−t)logT2π)⟶??∑x∈Sif(x) (2.5)in distribution for continuous test functions f of compact support. It is possible to extend this to integrable test functions satisfying mild regularity conditions. On the other hand, going beyond integrable functions requires additional information about the zeros lying far away from the microscopic window. It turns out that the second relation (2.5) is (conditionally) valid for test functions f(x)=1/(x−z), if the sums are properly regularized, whereas the first relation (2.4) requires a deterministic correction depending on E; see Sections 2.3 and 2.2 (relying on the works [1, 34], respectively). These properties imply that the critical points depend quasi-locally, so to speak, on the zeros/eigenvalues. 2.2. Random functions of Nevanlinna class We recall the construction of Aizenman–Warzel [1]. Recall that a function w:C⧹R→C belongs to the Nevanlinna [= Herglotz = Pick] class ( w∈R) if it is analytic and   w(z)¯=w(z¯),Iw(z)Iz>0. The class R is equipped with the topology of pointwise convergence on compact subsets of C⧹R. Denote   W(z)=limR→∞∑x∈Si[1x−z−1x−iR]+iπ=limR→∞∑x∈Si∩(−R,R)1x−z. (2.6) The two limits exist and coincide according to a general criterion of [1], and W(z) is a random element of the Nevanlinna class. Also note that −W(z) is the logarithmic derivative of the function Φ(z) from (1.4). As before, let PN(z)=det(HN/N−z) be the characteristic polynomial of HN/N, and denote   WN(z;E)=−1Nρ(E)pN′(E+zNρ(E))pN(E+zNρ(E))=∑j1(λj,N−E)Nρ(E)−z. Proposition 2.1 (Aizenman–Warzel) For ∣E∣<2,   WN(z;E)⟶distrW(z)−πE4−E2 (2.7)  PN(E+2πzN4−E2)PN(E)⟶distrΦ(z)exp[πEz4−E2]. (2.8) Proof The first statement is proved in [1, Corollary 6.5], their argument relies on the results obtained in the works [16, 18, 62] on the local eigenvalue statistics of Wigner matrices, and on the general theory of random Nevanlinna functions which was developed in [1]. The second statement follows from the first one by (carefully) integrating from 0 to z.□ Denote by w−1(a) the collection of solutions of w(z)=a. Observe that the map w↦w−1(a) from R∩{meromorphicfunctions} to locally finite (multi-)subsets of R is continuous. From this observation and (2.7), we deduce: Corollary 2.2 Let (HN)be a sequence of random matrices satisfying the assumptions listed in Section2.1. Then for any E∈(−2,2)  (∑jδ(u−(λj,N−E)Nρ(E)),∑jδ(u−(λj,N′−E)Nρ(E)))⟶(W−1(∞),W−1(−a))=(Si,Sia′) (2.9)in distribution, where a=−πE4−E2. To recapitulate, the non-obvious part of the statement is that the zeros λj,N which are not in an O(1/N)-neighbourhood of E influence the critical points λj,N′ near E only via the deterministic quantity πE(4−E2)−1/2. Colloquially, the conditional distribution of the critical points in (E−r/N,E+r/N) given the eigenvalues in (E−R/N,E+R/N) degenerates in the limit R→∞. Proof of Corollary 1.3 Follows from Theorem 1.1 and Corollary 2.2.□ 2.3. A counterpart for the ξ-function Denote   Wt,T(z)=−2πilogTξ′(12+i[t+2πzlogT])ξ(12+i[t+2πzlogT]),0≤t≤T.Assuming the Riemann hypothesis, Wt,T belongs to the Nevanlinna class (the property Wt,T∈R is independent of t and T, and is in fact equivalent to the Riemann hypothesis). We shall treat t as a random variable uniformly chosen in [0,T], and denote the corresponding random function by WT. The next proposition is close to the results for the ζ-function which were proved in [20, 34]. Proposition 2.3 Assume the Riemann hypothesis. Let Tn→∞be a sequence and P—a point process such that, for t uniformly chosen in [0,Tn],   ∑ξ(12+iγ)=0δ(u−(γ−t)logTn2π)→Pindistribution,n→∞.Then WTn→WPin distribution, where WPis the unique random Nevanlinna function such that  WP−1(∞)=P,WP(i∞)=iπ,and in particular,   (∑ξ(12+iγ)=0δ(u−(γ−t)logTn2π),∑ξ′(12+iγ)=0δ(u−(γ′−t)logTn2π))→(P,P0′),where P0′=WP−1(0). Remark Assuming the Riemann hypothesis, it follows from the results of Fujii [24, 25] that   1T∫0T(N(t+2πRlogT)−N(t)−R)2dt≤Clog(e+R), (2.10)therefore, the family of point processes WT−1(∞) is precompact (that is for any finite interval I the family of random variables #(WT−1(∞)∩I) is tight), and, moreover, any limit point P satisfies   E#[P∩[−R,R]]=2R,E∣#[P∩[0,R]]−R∣2≤Clog(e+R). In particular, the conditions of [1, Theorem 4.1] are satisfied, and, therefore, the function WP (uniquely) exists. Proposition 2.3 implies Corollary 2.4 Assume the Riemann hypothesis and the multiple correlation conjecture (1.3). Then  WT⟶distrW,ξ(12+i(t+2πzlogT))ξ(12+it)⟶distrΦ(z),T→∞and consequently  (∑ξ(12+iγ)=0δ(u−(γ−t)logT2π),∑ξ′(12+iγ′)=0δ(u−(γ′−t)logT2π))⟶distr(Si,Si0′). Proof of Corollary 1.4 Follows from Theorem 1.1 and Corollary 2.4.□ The proof of Proposition 2.3 relies on the following lemma. Related results go back to the work of Selberg [58]. We essentially follow the argument in [34], relying on the work of Montgomery [50]. Lemma 2.5 Assuming the Riemann hypothesis, one has for any R>0:   limT→∞∫0TdtTWt,T(iR)=iπ (2.11)  limsupT→∞∫0TdtT∣Wt,T(iR)−iπ∣2≤CR2. (2.12) Proof of Proposition 2.3 The convergence of WT to Φ follows from the lemma, in combination with the general criterion of Aizenman and Warzel [1, Theorem 6.1]; it implies the other two statements.□ Proof of Lemma 2.5 To prove (2.11), consider the integral of ξ′/ξ along the closed contour Γ composed of the segments connecting the points   12−RlogT,12+RlogT,12+RlogT+Ti,12+RlogT+T′i,12−RlogT+Ticounterclockwise (see Fig. 4), where T′ is the real number closest to T such that there are no zeros of the ξ-function in the 1/(100logT)-neighbourhood of 12+iT. By the residue theorem and the asymptotics (1.9) of N(T), the integral is equal to   2πi#{zerosofξwithimaginarypartin[0,T′]}=2πiTlogT2π(1+o(1)). On the other hand, from the functional equation ξ(1−z)=ξ(z), the integral along the left vertical line is equal to the integral along the right vertical line, and the integral along the bottom horizontal line is zero; the integral along the two segments on the top is negligible (as one can see, for example, from (2.14) below). Therefore   ∫0T(ξ′/ξ)(12+i(t+iR/logT))dt=−πTlogT2π(1+o(1))which is equivalent to (2.11). We note for the sequel that (2.11) implies the following smoothened version:   limT→∞∫−∞∞Tdtπ(t2+T2)Wt,T(iR)=iπ. (2.13) To prove (2.12), we use the Hadamard product representation (cf. [63, 2.12])   ξ(12+iz)=ξ(12)ebˆz∏ξ(12+iγ)=0(1−z/γ)ez/γ,which implies that   Wt,T(iR)=∑γ[1(γ−t)logT2π−iR−1γlogT2π]−2πbˆlogT. (2.14)Integrating with the weight T/(π(t2+T2)) and using (2.13) and the Cauchy theorem, we obtain   limT→∞∑γ[1(γ−iT)logT−iR−1γlogT]=i2. (2.15)Note that (2.15) holds for any real R (positive or negative). Now we compute   I(R,T)=∫−∞∞Tdtπ(t2+T2)∣Wt,T(iR)2π+bˆlogT∣2=∑γ,γ˜Iγ,γ˜(R,T),where   Iγ,γ˜(R,T)=∫−∞∞Tdtπ(t2+T2)[1(γ−t)logT−iR−1γlogT]×[1(γ˜−t)logT+iR−1γ˜logT].By the Cauchy theorem,   Iγ,γ˜(R,T)=Iγ,γ˜′(R,T)+Iγ,γ˜″(R,T),where   Iγ,γ˜′(R,T)=[1(γ−iT)logT−iR−1γlogT][1(γ˜−iT)logT+iR−1γ˜logT]Iγ,γ˜″(R,T)=−2iTT2+(γ˜+iRlogT)2[1(γ−γ˜)logT−2iR−1γlogT]1logT.In view of (2.15),   limT→∞∑γ,γ˜Iγ,γ˜′(R,T)=(i/2)2=−1/4. (2.16)To estimate the sum of Iγ,γ˜″(R,T), let   Jγ,γ˜(R,T)=2TT2+(γ˜)22R(γ−γ˜)2log2T+4R21logT,Jγ,γ˜′(R,T)=RIγ,γ˜″(R,T)−Jγ,γ˜(R,T). Using the estimates   ∣1T2+(γ˜+iRlogT)2−1T2+(γ˜)2∣≤3R(T2+(γ˜)2)2∣1(γ−γ˜)logT−2iR−1γlogT∣≤2(∣γ˜∣logT+2R)∣γ∣logT(∣γ−γ˜∣logT+2R), we deduce that   limT→∞∑γ,γ˜Jγ,γ˜′(R,T)=0. (2.17) Now we turn to ∑γ,γ˜Jγ,γ˜(R,T) and show that   limsupT→∞∣∑γ,γ˜Jγ,γ˜(R,T)−12∣≤ConstR2. (2.18) It will suffice to prove that   limsupT→∞∣2πTlogT∑0≤γ,γ˜≤T2R(γ−γ˜)2log2T+4R2−1∣≤ConstR2. (2.19) Let   F(α,T)=2πTlogT∑0≤γ,γ˜≤TTiα(γ−γ′)w(γ−γ˜),w(u)=44+u2. (2.20) Montgomery showed [50] that   F(α,T)=∣α∣+T−2∣α∣logT(1+o(1))+o(1),∣α∣<1 (2.21)(which implies (1.10)). Therefore,   supx∫xx+1F(α,T)dα≤C (2.22) (see [32, 33], where this is proved with C=83+ϵ and 2912+ϵ,respectively). From the definition (2.20) of F, we have (cf. [34, (2.11)]):   2πTlogT∑0≤γ,γ˜≤T2R(γ−γ˜)2log2T+4R2w(γ−γ˜)=∫0∞F(α,T)e−2R∣α∣dα, (2.23)and thus, from (2.21) and (2.22),   limsupT→∞∣2πTlogT∑0≤γ,γ˜≤T4R(γ−γ˜)2log2T+4R2w(γ−γ˜)−1∣≤CR2. (2.24) Observing that   limT→∞2πTlogT∑0≤γ,γ˜≤T4R(γ−γ˜)2log2T+4R2(1−w(γ−γ˜))=0,we obtain (2.19) and thus (2.18). The relations (2.16), (2.17) and (2.18) imply that   limR→∞limsupT→∞∣I(R,T)−1/4∣=0and hence (2.12) holds.□ Figure 4. View largeDownload slide The contour Γ from the proof of Lemma 2.5. Figure 4. View largeDownload slide The contour Γ from the proof of Lemma 2.5. 3. Repulsion In this section, we prove Theorems 1.1 and 1.5 and Corollary 1.2, the (re-)formulation of which we recall for the convenience of the reader. Let   Ωk(Si,ϵ)={#[Si∩(−ϵ,ϵ)]≥k},Ωk(Sia′,ϵ)={#[Sia′∩(−ϵ,ϵ)]≥k}. Theorem 1.1′ For any a∈R, k≥2, 0<ϵ<18max(∣a∣,2e)and R≥5  P(Ωk(Sia′,ϵ)\(Ωk+1(Sia,(1+4k−1)ϵ)∪Ωk+2(Sia,Rϵ)))≤2exp(−kR64). Taking R=1000klog1ϵ and using (1.5), we obtain the version of Theorem 1.1 stated in the introduction. Corollary 1.2 For any k≥2, there exists a limit  ck′=limϵ→+0P(Ωk(Sia′,ϵ))ϵ(k+1)2∈(ck+1,(1+4k−1)ck+1),independent of a∈R. Next, denote   Ωk(ξ,T,ϵ)={0≤t≤T∣#[γ∈(t−2πϵlogT,t+2πϵlogT),ξ(12+iγ)=0]≥k}.Ωk(ξ′,T,ϵ)={0≤t≤T∣#[γ′∈(t−2πϵlogT,t+2πϵlogT),ξ′(12+iγ′)=0]≥k}. Theorem 1.5′ Assume the Riemann hypothesis. For any k≥2, 0<ϵ<1, R≥1+4k−1  1Tmes(Ωk(ξ′,T,ϵ)⧹(Ωk+1(ξ,T,(1+4k−1)ϵ)∪Ωk+2(ξ,T,Rϵ)))≤CeckR. Remark The coefficient 1+4k−1 in these results can be further improved to 1+2k+o(1). 3.1. Proof of Theorem 1.1 Proof of Theorem 1.1 ′ By Corollary 2.2, the theorem is equivalent to Corollary 1.3, and, moreover, to its special case pertaining to one (arbitrary) ensemble of random matrices; we choose the Gaussian Unitary Ensemble. (We could equally work directly with Si; in that case we would need to regularize all the sums.) Denote   xj,N=(λj,N−E)Nρ(E),xj,N′=(λj,N′−E)Nρ(E),where a=−πE4−E2. Let us first show that for any R≥1+4/(k−1)  Ωk(Sia′,ϵ)\(Ωk+1(Sia,(1+4k−1)ϵ)∪Ωk+2(Sia,Rϵ))⊂{∑∣xj,N−ϵ∣≥(R−1)ϵ1xj,N−ϵ≥k−14ϵ}∪{∑∣xj,N+ϵ∣≥(R−1)ϵ1xj,N+ϵ≤−k−14ϵ}. (3.1)Indeed, assume that   #[∣xj,N′∣<ϵ]≥k,#[∣xj,N∣<(1+4k−1)ϵ]≤k. (3.2) Then we have by interlacing   #[xj,N∈(−ϵ,ϵ)]=k−1, (3.3)and, on the other hand, there are no xj,N at least in one of the intervals   (−(1+4k−1)ϵ,−ϵ),(+ϵ,+(1+4k−1)ϵ);for example, in the second one. Then   ∑j=1N1xj,N−ϵ≥0,whence by (3.3) and (3.2)   ∑∣xj,N−ϵ∣≥4k−1ϵ1xj,N−ϵ≥∑∣xj,N∣<ϵ−1xj,N−ϵ≥k−12ϵ. Every xj,N contributes at most (k−1)/(4ϵ) to the sum on the left-hand side, therefore, either #[(1+4k−1)ϵ≤xj,N<Rϵ]≥2 or   ∑∣xj,N−ϵ∣≥(R−1)ϵ1xj,N−ϵ≥k−14ϵ. This proves (3.1), and it remains to bound the probability of the two terms on the right-hand side. By symmetry, we can focus on the first term, for which we use the following lemma, proved below (see for example Breuer–Duits [9, Theorem 3.1] for more sophisticated bounds):□ Lemma For any determinantal process Dwith self-adjoint kernel of finite rank and any (bounded Borel measurable) test function f,   P{∑x∈Df(x)≥E∑x∈Df(x)+r}≤exp(−A*(r)),r>0,where  A*(r)=supt≤∥f∥∞−1(rt−A(t)),A(t)=et22E∑x∈Df(x)2. (3.4)We apply the lemma to   f(x)=1∣x−ϵ∣≥(R−1)ϵx−ϵ,FN=∑f(xj,N),then, denoting by ρN(E)=1NddEE#{λj,N≤E} the mean density of eigenvalues, we have   EFN=∫∣x∣≥(R−1)ϵρN(E+x+ϵNρ(E))xdx=∫[1∣x∣≥(R−1)ϵx−xx2+1]ρN(E+x+ϵNρ(E))dx+∫xx2+1ρN(E+x+ϵNρ(E))dx. The first addend tends to zero since ρN→ρ uniformly, whereas the second addend tends to −πE4−E2 by (2.7). Therefore   limN→∞EFN=−πE4−E2. Next, for t≤Rϵ  limN→∞A(t)=et22limN→∞E∑f(xj,N)2=et22limN→∞∫∣x∣≥Rϵdx∣x−ϵ∣2≤et2(R−1)ϵ,whence, taking t=(R−1)ϵ in the definition (3.4) of A* and assuming that ϵ<1/(16e),   limN→∞A*(k−18ϵ)≥k−18ϵ(R−1)ϵ−e(R−1)ϵ≥(k−1)(R−1)16. According to the lemma, we have for ϵ≤min(4−E28π∣E∣,116e):   P{FN≥k−14ϵ}≤exp{−(k−1)(R−1)16}≤exp{−kR64},as claimed.□ Proof of Lemma Let K denote the operator defining the determinantal process; then for t∥f∥∞≤1  logEexp{t∑x∈Df(x)}=log det(1+(etf−1)K)=trlog(1+(etf−1)K)≤tr(etf−1)K≤et22trf2K+ttrfK≤A(t)+tE∑x∈Df(x).Therefore, by the Chebyshev inequality   P{∑x∈Df(x)≥E∑x∈Df(x)+r}≤exp(A(t)−rt).□ 3.2. Proof of Corollary 1.2 Proof of Corollary 1.2 Let us show that the limit   ck′=limϵ→0P(Ωk(Sia′,ϵ))ϵ(k+1)2,k≥2 (3.5)exists and does not depend on a. Choose αk>αk′>0 sufficiently small to ensure that   (1−αk)(k+2)2>(k+1)2. (3.6) Denote by ϒ(ϵ) the event   ∃pairwisedistinctdistinctX=(x1,…,xk+1)∈(Si∩(−ϵ1−αk,ϵ1−αk))k+1,suchthatallthezerosofPX(x)=ddx∏j=1k+1(x−xj)liein(−ϵ,ϵ). (3.7) Let us show that   P(ϒ(ϵ−ϵ1+αk′)⧹Ωk(Sia′,ϵ)),P(Ωk(Sia′,ϵ)⧹ϒ(ϵ+ϵ1+αk′))=o(ϵ(k+1)2). (3.8) Indeed, on the event ϒ(ϵ−ϵ1+αk′)⧹(Ωk(Sia′,ϵ)∪Ωk+2(Si,ϵ1−αk)) at least one of the xj lies outside (−ϵ,ϵ); assume for example that xk+1>ϵ and decompose   0>W(ϵ)=∑j=1k+11xj−ϵ+limr→∞∑x∈Siϵ1−αk≤∣x∣≤r1xj−ϵ. Then the first term is bounded from below by   ∑j=1k+11xj−ϵ≥∑j=1k+11xj−ϵ−ϵ1+αk′+(k+1)ϵ1+αk′4ϵ2≥k+14ϵ1−αk′. The probability of the event   limr→+∞∑x∈Siϵ1−αk≤∣x∣≤r1xj−ϵ≤−k+14ϵ1−αk′is o(ϵ(k+1)2) by a tail estimate which follows from the lemma in Section 3.1 (similar to the proof of Theorem 1.1), whereas P(Ωk+2(Si,ϵ1−αk))=o(ϵ(k+1)2) by the condition (3.6) on αk and (1.5). This proves the first part of (3.8); the second part is proved in a similar way. From (1.1) and the asymptotics   det(sinπ(xj−xm)π(xj−xm))j,m=1k=(1+o(1))ck,1∏j<m(xj−xm)2x→0(where ck,1 as well as ck,2 and ck,3 below are numerical constants), we obtain   P(ϒ(ϵ))=(ck,2+o(1))ϵ∫∑xj=0dkX∏j<m(xj−xm)21{thezerosofPXliein(−ϵ,ϵ)}=(ck,2+o(1))ck,3ϵ(k+1)2,which implies, with (3.8), that (3.5) holds with ck′=ck,2ck,3. Note that the indicator under the integral is compactly supported. The bound   ck′∈[ck+1,(1+4k−1)(k+1)2ck+1]follows from Theorem 1.1 ′ and (1.5).□ 3.3. Proof of Theorem 1.5 The proof of Theorem 1.5 relies on a bound, proved by Rodgers [56], on the moments of the logarithmic derivative of the ζ-function; see (3.12) below. The results of Farmer et al. [20] imply a more precise bound under additional hypotheses. Proof of Theorem 1.5′ Let 12+iγj be the zeros of ξ(z); rescale them as follows: xj,t,T=(γj−t)logT2π. We start with the following counterpart of (3.1):   1Tmes(Ωk(ξ′,T,ϵ)⧹(Ωk+1(ξ,T,(1+4k−1)ϵ)∪Ωk+2(ξ,T,Rϵ)))≤1Tmes({Wt,T(ϵ)−∑∣xj,t,T−ϵ∣<(R−1)ϵ1xj,t,T−ϵ≥k−14ϵ}\Ωk+2(ξ,T,Rϵ))+1Tmes({Wt,T(−ϵ)−∑∣xj,t,T+ϵ∣<(R−1)ϵ1xj,t,T+ϵ≤−k−14ϵ}\Ωk+2(ξ,T,Rϵ)).To show that each of the two terms is bounded by CeckR, it will suffice to prove that   1Tmes({∣Wt,T(0)−∑∣xj,t,T∣<Rϵ1xj,t,T∣≥k−14ϵ}\Ωk+2(ξ,T,Rϵ))≤CeckR. (3.9)Decompose   Wt,T(0)−∑∣xj,t,T∣<Rϵ1xj,t,T=RWt,T(20iRϵ)−∑∣xj,t,T∣<Rϵxj,t,Txj,t,T2+400R2ϵ2+∑∣xj,t,T∣≥Rϵ400R2ϵ2xj,t,T(xj,t,T2+400R2ϵ2)and observe that on the complement of Ωk+2(ξ,T,Rϵ)  ∣∑∣xj,t,T∣<Rϵxj,t,Txj,T2+400R2ϵ2∣≤k+120Rϵ≤k−16ϵ (3.10)whereas   ∣RWt,T(20iRϵ)∣,120∣∑∣xj,t,T∣≥Rϵ400R2ϵ2xj,t,T(xj,t,T2+400R2ϵ2)∣≤∣Wt,T(20iRϵ)∣,whence we need to show that   1Tmes{0≤t≤T∣∣Wt,T(20iRϵ)∣≥k−1480ϵ}≤Cexp(−ckR). (3.11)By a result of Rodgers [56, Theorem 2.1],   ∫0T∣ζ′(12+i(t+2πiδlogT))ζ(12+i(t+2πiδlogT))∣mdt≤Cmlogmδ−mTlogmT, (3.12)hence   ∫0T∣ξ′(12+i(t+2πiδlogT))ξ(12+i(t+2πiδlogT))∣mdt≤Cmlogmδ−mTlogmT,and, finally,   1T∫0T∣Wt,T(20iRϵ)∣mdt≤Cmlogm(Rϵ)−m, (3.13)from which (3.11) follows by the Chebyshev inequality.□ 3.4. Some questions and comments Other random matrix ensembles The results of this paper have counterparts for other random matrix ensembles (and other ξ-functions). In particular, Chhaibi et al. [12] showed that the characteristic polynomial ZN(z)=det(1N−zUN*) of the Circular Unitary Ensemble satisfies   ZN(exp(2πiz/N))ZN(1)⟶distrexp(iπz)Φ(z). Hence the collection of critical points of ZN(z)z−N/2 converges, after rescaling by N, to the same process Si0′ as in Corollaries 2.2 and 2.3. This is consistent with the discussion in [19, Sections 2.3 and 6.2]. Critical points vs. submatrix eigenvalues Let   Wdec(z)=limR→∞∑x∈Si∩(−R,R)∣gx∣2x−z,where, conditionally on Si, the random variables gx are independent complex standard Gaussian. Then Corollary 2.2 has the following counterpart:   ({(λj,N−E)Nρ(E)},{(λj,N−1−E)Nρ(E)})⟶((Wdec)−1(−πE4−E2),W−1(∞)). (3.14)(Amusingly, the distribution of the first term does not depend on E.) We are not sure whether (3.14) has a number-theoretic analogue. Form factors Suppose a point process P is a limit point of the rescaled zeros of ξ, as in Proposition 2.3. Then the rescaled critical points converge, along the same subsequence, to the point process P′=WP−1(0), the distribution of which is uniquely determined by that of P. By a result of Montgomery [50] and its extension by Hejhal [37] and Rudnick and Sarnak [57], the (multiple) form factors of P coincide, in a restricted domain of momenta, with those of the sine process. In view of the research programme suggested by Farmer et al. [19], it is natural to ask which constraints do this impose on the form factors of P′, and to compare these with the results of [6, 19]. It would also be interesting to compute the form factor of Sia′ and to check whether it coincides with the right-hand side of (1.8) for a=0 and ∣α∣≤1. A possible starting point is the identity ek(λ1′,…,λN−1′)=(1−kN)ek(λ1,…,λN) relating the elementary symmetric functions in the critical points of a polynomial to the elementary symmetric functions in the zeros. Zeros of higher derivatives The result (1.8) of [19] was extended to higher derivatives of ξ in the Ph.D. thesis of Bian [6]. In this context, we mention that if the Riemann hypothesis and the multiple correlation conjecture hold, Corollary 2.3 implies that for any k≥1  ∑ξ(k)(12+iγ‴)=0δ(u−(γ‴−t)logT2π)⟶distrSi0(k)={Φ(k)=0}. Logarithmic derivative From Lemma 2.5 and [1, Theorems 2.3 and 6.2] of Aizenman–Warzel, it follows, conditionally on the Riemann hypothesis, that (for t chosen uniformly in [0,T]) the rescaled logarithmic derivatives   2ilogTξ′(12+it)ξ(12+it),2ilogTζ′(12+it)ζ(12+it)+i⟶T→∞distrCauchy, (3.15)where the right-hand side is a standard (real) Cauchy random variable. For comparison, it was proved by Lester [44], following earlier results by Guo [36], that the distribution of (ζ′/ζ)(σ(T)+it), where ∣σ(T)−12∣logT→∞ and ∣σ(T)−12∣→0, is approximately (complex) Gaussian for large T. Funding Supported in part by the European Research Council start-up Grant 639305 (SPECTRUM). Acknowledgement This work originated from a question of Yan Fyodorov, who asked how to describe the local statistics of critical points, and from subsequent discussions, particularly, of the strong repulsion; he also made helpful comments and suggestions on all the subsequent stages. 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