Journal of Statistical Physics

Fyodorov, Y.; Nock, A.

doi: 10.1007/s10955-015-1209-xpmid: N/A

Correlation functions involving products and ratios of half-integer powers of characteristic polynomials of random matrices from the Gaussian orthogonal ensemble (GOE) frequently arise in applications of random matrix theory (RMT) to physics of quantum chaotic systems, and beyond. We provide an explicit evaluation of the large- $$N$$ N limits of a few non-trivial objects of that sort within a variant of the supersymmetry formalism, and via a related but different method. As one of the applications we derive the distribution of an off-diagonal entry $$K_{ab}$$ K a b of the resolvent (or Wigner $$K$$ K -matrix) of GOE matrices which, among other things, is of relevance for experiments on chaotic wave scattering in electromagnetic resonators.

Szehr, Oleg; Wolf, Michael

doi: 10.1007/s10955-015-1204-2pmid: N/A

This article investigates the stability of the ground state subspace of a canonical parent Hamiltonian of a Matrix product state against local perturbations. We prove that the spectral gap of such a Hamiltonian remains stable under weak local perturbations even in the thermodynamic limit, where the entire perturbation might not be bounded. Our discussion is based on preceding work by Yarotsky that develops a perturbation theory for relatively bounded quantum perturbations of classical Hamiltonians. We exploit a renormalization procedure, which on large scale transforms the parent Hamiltonian of a Matrix product state into a classical Hamiltonian plus some perturbation. We can thus extend Yarotsky’s results to provide a perturbation theory for parent Hamiltonians of Matrix product states and recover some of the findings of the independent contributions (Cirac et al in Phys Rev B 8(11):115108, 2013) and (Michalakis and Pytel in Comm Math Phys 322(2):277–302, 2013).

Lardizabal, Carlos; Souza, Rafael

doi: 10.1007/s10955-015-1217-xpmid: N/A

We study a particular class of trace-preserving completely positive maps, called PQ-channels, for which classical and quantum evolutions are isolated in a certain sense. By combining open quantum random walks with a notion of recurrence, we are able to describe criteria for recurrence of the walk related to this class of channels. Positive recurrence for open walks is also discussed in this context.

Lopes, Artur; Neumann, Adriana

doi: 10.1007/s10955-015-1205-1pmid: N/A

In the present paper, we consider a family of continuous time symmetric random walks indexed by $$k\in \mathbb {N}$$ k ∈ N , $$\{X_k(t),\,t\ge 0\}$$ { X k ( t ) , t ≥ 0 } . For each $$k\in \mathbb {N}$$ k ∈ N the matching random walk take values in the finite set of states $$\Gamma _k=\frac{1}{k}(\mathbb {Z}/k\mathbb {Z})$$ Γ k = 1 k ( Z / k Z ) ; notice that $$\Gamma _k$$ Γ k is a subset of $$\mathbb {S}^1$$ S 1 , where $$\mathbb {S}^1$$ S 1 is the unitary circle. The infinitesimal generator of such chain is denoted by $$L_k$$ L k . The stationary probability for such process converges to the uniform distribution on the circle, when $$k\rightarrow \infty $$ k → ∞ . Here we want to study other natural measures, obtained via a limit on $$k\rightarrow \infty $$ k → ∞ , that are concentrated on some points of $$\mathbb {S}^1$$ S 1 . We will disturb this process by a potential and study for each $$k$$ k the perturbed stationary measures of this new process when $$k\rightarrow \infty $$ k → ∞ . We disturb the system considering a fixed $$C^2$$ C 2 potential $$V: \mathbb {S}^1 \rightarrow \mathbb {R}$$ V : S 1 → R and we will denote by $$V_k$$ V k the restriction of $$V$$ V to $$\Gamma _k$$ Γ k . Then, we define a non-stochastic semigroup generated by the matrix $$k\,\, L_k + k\,\, V_k$$ k L k + k V k , where $$k\,\, L_k $$ k L k is the infinifesimal generator of $$\{X_k(t),\,t\ge 0\}$$ { X k ( t ) , t ≥ 0 } . From the continuous time Perron’s Theorem one can normalized such semigroup, and, then we get another stochastic semigroup which generates a continuous time Markov Chain taking values on $$\Gamma _k$$ Γ k . This new chain is called the continuous time Gibbs state associated to the potential $$k\,V_k$$ k V k , see (Lopes et al. in J Stat Phys 152:894–933, 2013). The stationary probability vector for such Markov Chain is denoted by $$\pi _{k,V}$$ π k , V . We assume that the maximum of $$V$$ V is attained in a unique point $$x_0$$ x 0 of $$\mathbb {S}^1$$ S 1 , and from this will follow that $$\pi _{k,V}\rightarrow \delta _{x_0}$$ π k , V → δ x 0 . Thus, here, our main goal is to analyze the large deviation principle for the family $$\pi _{k,V}$$ π k , V , when $$k \rightarrow \infty $$ k → ∞ . The deviation function $$I^V$$ I V , which is defined on $$ \mathbb {S}^1$$ S 1 , will be obtained from a procedure based on fixed points of the Lax–Oleinik operator and Aubry–Mather theory. In order to obtain the associated Lax–Oleinik operator we use the Varadhan’s Lemma for the process $$\{X_k(t),\,t\ge 0\}$$ { X k ( t ) , t ≥ 0 } . For a careful analysis of the problem we present full details of the proof of the Large Deviation Principle, in the Skorohod space, for such family of Markov Chains, when $$k\rightarrow \infty $$ k → ∞ . Finally, we compute the entropy of the invariant probabilities on the Skorohod space associated to the Markov Chains we analyze.

Perversi, Eleonora; Regazzini, Eugenio

doi: 10.1007/s10955-015-1200-6pmid: N/A

For a general inelastic Kac-like equation recently proposed, this paper studies the long-time behaviour of its probability-valued solution. In particular, the paper provides necessary and sufficient conditions for the initial datum in order that the corresponding solution converges to equilibrium. The proofs rest on the general CLT for independent summands applied to a suitable Skorokhod representation of the original solution evaluated at an increasing and divergent sequence of times. It turns out that, roughly speaking, the initial datum must belong to the standard domain of attraction of a stable law, while the equilibrium is presentable as a mixture of stable laws.

Taggi, Lorenzo

doi: 10.1007/s10955-015-1199-8pmid: N/A

This paper considers a class of probabilistic cellular automata undergoing a phase transition with an absorbing state. Denoting by $${\mathcal {U}}(x)$$ U ( x ) the neighbourhood of site $$x$$ x , the transition probability is $$T(\eta _x = 1 | \eta _{{\mathcal {U}}(x)}) = 0$$ T ( η x = 1 | η U ( x ) ) = 0 if $$\eta _{{\mathcal {U}}(x)}= \mathbf {0}$$ η U ( x ) = 0 or $$p$$ p otherwise, $$\forall x \in \mathbb {Z}$$ ∀ x ∈ Z . For any $$\mathcal {U}$$ U there exists a non-trivial critical probability $$p_c( {\mathcal {U}})$$ p c ( U ) that separates a phase with an absorbing state from a fluctuating phase. This paper studies how the neighbourhood affects the value of $$p_c( {\mathcal {U}})$$ p c ( U ) and provides lower bounds for $$p_c( {\mathcal {U}})$$ p c ( U ) . Furthermore, by using dynamic renormalization techniques, we prove that the expected convergence time of the processes on a finite space with periodic boundaries grows exponentially (resp. logarithmically) with the system size if $$p > p_c$$ p > p c (resp. $$p<p_c$$ p < p c ). This provides a partial answer to an open problem in Toom et al. (Stochastic Cellular Systems: Ergodicity, Memory, Morphogenesis, pp. 1–182. Manchester University Press, Manchester, 1990; Topics in Contemporary Probability and its Applications, pp. 117–157. CRC Press, Boca Raton, 1995).

Roy, Parna; Sen, Parongama

doi: 10.1007/s10955-015-1201-5pmid: N/A

In this paper we study generalised Ising Glauber models with inflow of information in one dimension and derive expressions for the exit probability using well established analytical methods. The analytical expressions agree very well with the results obtained from numerical simulation only when the interaction is restricted to the nearest neighbor. But as the range of interaction is increased the analytical results deviate from simulation results systematically. The reasons for the deviation as well as some related open questions are discussed.

Elsey, Matt

doi: 10.1007/s10955-015-1203-3pmid: N/A

In this work, we present a family of exact self-similar solutions for the distribution of enclosed areas under mean curvature flow on the union of disjoint closed curves in the plane. To our knowledge, this is the first example of closed-form exact self-similar solutions for the mean curvature flow. We perform numerical simulations for large sets of initial data obtained as the zero level set of Gaussian random fields and observe numerical convergence to one particular member of this family.

Baake, Michael; Kösters, Holger; Moody, Robert

doi: 10.1007/s10955-014-1178-5pmid: N/A

We discuss several examples of point processes [all taken from Hough et al. (Zeros of Gaussian analytic functions and determinantal point processes, 2009)] for which the autocorrelation and diffraction measures can be calculated explicitly. These include certain classes of determinantal and permanental point processes, as well as an isometry-invariant point process that arises as the zero set of a Gaussian random analytic function.

Holmes, Mark; Mohylevskyy, Yevhen; Newman, Charles

doi: 10.1007/s10955-015-1198-9pmid: N/A

Consider the voter model on a box of side length $$L$$ L (in the triangular lattice) with boundary votes fixed forever as type 0 or type 1 on two different halves of the boundary. Motivated by analogous questions in percolation, we study several geometric objects at stationarity, as $$L\rightarrow \infty $$ L → ∞ . One is the interface between the (large—i.e., boundary connected) 0-cluster and 1-cluster. Another is the set of large “coalescing classes” determined by the coalescing walk process dual to the voter model.