Critical Exponents on Fortuin–Kasteleyn Weighted Planar Maps

Critical Exponents on Fortuin–Kasteleyn Weighted Planar Maps In this paper we consider random planar maps weighted by the self-dual Fortuin–Kasteleyn model with parameter $${q \in (0,4)}$$ q ∈ ( 0 , 4 ) . Using a bijection due to Sheffield and a connection to planar Brownian motion in a cone we obtain rigorously the value of the annealed critical exponent associated with the length of cluster interfaces, which is shown to be $$\frac{4}{\pi} \arccos \left( \frac{\sqrt{2 - \sqrt{q}}}{2} \right)=\frac{\kappa'}{8},$$ 4 π arccos 2 - q 2 = κ ′ 8 , where $${\kappa' }$$ κ ′ is the SLE parameter associated with this model. We also derive the exponent corresponding to the area enclosed by a loop, which is shown to be 1 for all values of $${q \in (0,4)}$$ q ∈ ( 0 , 4 ) . Applying the KPZ formula we find that this value is consistent with the dimension of SLE curves and SLE duality. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Communications in Mathematical Physics Springer Journals

Critical Exponents on Fortuin–Kasteleyn Weighted Planar Maps

Loading next page...
 
/lp/springer_journal/critical-exponents-on-fortuin-kasteleyn-weighted-planar-maps-G5X0asxB5O
Publisher
Springer Berlin Heidelberg
Copyright
Copyright © 2017 by The Author(s)
Subject
Physics; Theoretical, Mathematical and Computational Physics; Mathematical Physics; Quantum Physics; Complex Systems; Classical and Quantum Gravitation, Relativity Theory
ISSN
0010-3616
eISSN
1432-0916
D.O.I.
10.1007/s00220-017-2933-7
Publisher site
See Article on Publisher Site

Abstract

In this paper we consider random planar maps weighted by the self-dual Fortuin–Kasteleyn model with parameter $${q \in (0,4)}$$ q ∈ ( 0 , 4 ) . Using a bijection due to Sheffield and a connection to planar Brownian motion in a cone we obtain rigorously the value of the annealed critical exponent associated with the length of cluster interfaces, which is shown to be $$\frac{4}{\pi} \arccos \left( \frac{\sqrt{2 - \sqrt{q}}}{2} \right)=\frac{\kappa'}{8},$$ 4 π arccos 2 - q 2 = κ ′ 8 , where $${\kappa' }$$ κ ′ is the SLE parameter associated with this model. We also derive the exponent corresponding to the area enclosed by a loop, which is shown to be 1 for all values of $${q \in (0,4)}$$ q ∈ ( 0 , 4 ) . Applying the KPZ formula we find that this value is consistent with the dimension of SLE curves and SLE duality.

Journal

Communications in Mathematical PhysicsSpringer Journals

Published: Jul 22, 2017

References

You’re reading a free preview. Subscribe to read the entire article.


DeepDyve is your
personal research library

It’s your single place to instantly
discover and read the research
that matters to you.

Enjoy affordable access to
over 18 million articles from more than
15,000 peer-reviewed journals.

All for just $49/month

Explore the DeepDyve Library

Search

Query the DeepDyve database, plus search all of PubMed and Google Scholar seamlessly

Organize

Save any article or search result from DeepDyve, PubMed, and Google Scholar... all in one place.

Access

Get unlimited, online access to over 18 million full-text articles from more than 15,000 scientific journals.

Your journals are on DeepDyve

Read from thousands of the leading scholarly journals from SpringerNature, Elsevier, Wiley-Blackwell, Oxford University Press and more.

All the latest content is available, no embargo periods.

See the journals in your area

DeepDyve

Freelancer

DeepDyve

Pro

Price

FREE

$49/month
$360/year

Save searches from
Google Scholar,
PubMed

Create lists to
organize your research

Export lists, citations

Read DeepDyve articles

Abstract access only

Unlimited access to over
18 million full-text articles

Print

20 pages / month

PDF Discount

20% off