Related papers: Potts and random cluster measures on locally regul…

An important experimental effort has been accomplished in recent years in the measurement of rates, polarization and CP observables in $B$ decays into two light vector mesons. On the theoretical side, after a very consistent effort done…

The $q$-state Potts model is an archetypical model for various types of phase transitions. We consider it on the square grid and focus on the regime where it undergoes a discontinuous transition, that is $q>4$. At the transition point…

Using the theory of negative association for measures and the notion of random weak limits of sparse graphs, we establish the validity of the cavity method for counting spanning subgraphs subject to local constraints in asymptotically…

We consider the Ising model with inverse temperature beta and without external field on sequences of graphs G_n which converge locally to the k-regular tree. We show that for such graphs the Ising measure locally weak converges to the…

In arXiv:1609.05666v1 [math.PR] a functional limit theorem was proved. It states that symmetric processes associated with resistance metric measure spaces converge when the underlying spaces converge with respect to the…

We study configuration spaces $C(n; p, q)$ of $n$ ordered unit squares in a $p$ by $q$ rectangle. Our goal is to estimate the Betti numbers for large $n$, $j$, $p$, and $q$. We consider sequences of area-normalized coordinates, where…

For a family of random-cluster models with cluster weights $q\geq 1$, we prove that the probability that $0$ is connected to $x$ is asymptotically equal to $\tfrac{1}{q}\chi(\beta)^{2}\beta J_{0,x}$. The method developed in this article can…

The arboreal gas is the random (unrooted) spanning forest of a graph in which each forest is sampled with probability proportional to $\beta^{\# \text{edges}}$ for some $\beta\geq 0$, which arises as the $q\to 0$ limit of the…

It is known for many algorithmic problems that if a tree decomposition of width $t$ is given in the input, then the problem can be solved with exponential dependence on $t$. A line of research by Lokshtanov, Marx, and Saurabh [SODA 2011]…

We consider dynamical graphs, namely graphs that evolve over time, and investigate a notion of local weak convergence that extends naturally the usual Benjamini-Schramm local weak convergence for static graphs. One of the well-known results…

We study the random-cluster model on trees and treelike graphs at low temperatures. This is a model of dependent percolation parametrized by an edge probability $p\in (0,1)$ and a clustering weight $q\in [1,\infty)$, generalizing…

We study numerically the fractal dimensions and the bulk three-point connectivity for the spin clusters of the Q-state Potts model in two dimensions with $1\leq Q\leq 4$. We check that the usually invoked correspondence between FK clusters…

We study the agglomerative percolation (AP) models on the Bethe lattice and the triangular cactus to establish the exact mean-field theory for AP. Using the self-consistent simulation method, based on the exact self-consistent equation, we…

Work is a continuation of work TMPh, 175(2), 2013. We study the Potts model with zero external field on the Cayley tree. It is shown that for any values of the parameter all periodic Gibbs measures are translation-invariant for the q-state…

Recently, Vontobel showed the relationship between Bethe free energy and annealed free energy for protograph factor graph ensembles. In this paper, annealed free energy of any random regular, irregular and Poisson factor graph ensembles are…

We study a variant of the ferromagnetic Potts model, recently introduced by Tamura, Tanaka and Kawashima, consisting of a ferromagnetic interaction among $q$ "visible" colours along with the presence of $r$ non-interacting "invisible"…

In this paper, we study a class of multilinear Gibbs measures with Hamiltonian given by a generalized $\mathrm{U}$-statistic and with a general base measure. Expressing the asymptotic free energy as an optimization problem over a space of…

Let $(Z^{(n)}_k)_{1 \leq k \leq n}$ be a random set of points and let $\mu_n$ be its \emph{empirical measure}: $$\mu_n = \frac{1}{n} \sum_{k=1}^n \delta_{Z^{(n)}_k}. $$ Let $$P_n(z) := (z - Z^{(n)}_1)\cdots (z - Z^{(n)}_n)\quad…

We discuss recent results on ground state entropy in Potts antiferromagnets and connections with chromatic polynomials. These include rigorous lower and upper bounds, Monte Carlo measurements, large--$q$ series, exact solutions, and studies…

We introduce an exact replica method for the study of critical systems with quenched bond randomness in two dimensions. For the $q$-state Potts model we show that a line of renormalization group fixed points interpolates from weak to strong…