Related papers: The dimer and Ising models on Klein bottles

With a graph $G=(V,E)$ we associate a collection of non-negative real weights $\cup_{v\in V}{\lambda_{i,v}:1\leq i \leq m} \cup \cup_{uv \in E} {\lambda_{ij,uv}:1\leq i \leq j \leq m}$. We consider the probability distribution on…

The Ashkin-Teller model can be formulated as a pair of 2D Ising models, interacting via a four-spin interaction. I consider the case of weak anisotropy (slight a-symmetry between the two Ising layers) and weak coupling. I show that the…

We consider sampling in the so-called low-temperature regime, which is typically characterised by non-local behaviour and strong global correlations. Canonical examples include sampling independent sets on bipartite graphs and sampling from…

We study numerically the non-equilibrium critical properties of the Ising model defined on direct products of graphs, obtained from factor graphs without phase transition (Tc = 0). On this class of product graphs, the Ising model features a…

We consider the ferromagnetic Ising model on a sequence of graphs $G_n$ converging locally weakly to a rooted random tree. Generalizing [Montanari, Mossel, Sly '11], under an appropriate "continuity" property, we show that the Ising…

We investigate the finite size corrections to the equilibrium magnetization of an Ising model on a random graph with $N$ nodes and $N^{\gamma}$ edges, with $1 < \gamma \leq 2$. By conveniently rescaling the coupling constant, the free…

We establish necessary and sufficient conditions for convergence (in the sense of finite dimensional distributions) of multiplicative measures on the set of partitions. We show that this convergence is equivalent to asymptotic independence…

We apply dimer diagram techniques to uncover discrete global symmetries in the fields theories on D3-branes at singularities given by general orbifolds of general toric Calabi-Yau threefold singularities. The discrete symmetries are…

This paper is an endeavor to discuss some properties of zero-divisor graphs of the ring $\mathbb{Z}_n$, the ring of integers modulo $n$. The zero divisor graph of a commutative ring $R$, is an undirected graph whose vertices are the nonzero…

For a graph $G=(V,E)$, $k\in \mathbb{N}$, and a complex number $w$ the partition function of the univariate Potts model is defined as \[ {\bf Z}(G;k,w):=\sum_{\phi:V\to [k]}\prod_{\substack{uv\in E \\ \phi(u)=\phi(v)}}w, \] where…

A systematic investigation is given of finite size effects in $d=2$ quantum gravity or equivalently the theory of dynamically triangulated random surfaces. For Ising models coupled to random surfaces, finite size effects are studied on the…

We develop an asymptotic expansion of the spectral measures on a degenerating family of hyperbolic Riemann surfaces of finite volume. As an application of our results, we study the asymptotic behavior of weighted counting functions, which,…

In this paper we develop a general approach to dimer models analogous to Krichever's scheme in the theory of integrable systems. We start with a Riemann surface and the simplest generic meromorphic functions on it and demonstrate how to…

We consider the asymptotics of the partition function of the extended Gross-Witten-Wadia unitary matrix model by introducing an extra logarithmic term in the potential. The partition function can be written as a Toeplitz determinant with…

We show that the two dimensional Ising model is complete, in the sense that the partition function of any lattice model on any graph is equal to the partition function of the 2D Ising model with complex coupling. The latter model has all…

Non-invertible Kramers-Wannier (KW) duality symmetries are constructed for the transverse-field Ising model (TFIM) at the self-dual point under various boundary conditions (BCs), as long as the resultant Hamiltonian commutes with the ${\rm…

We present a new approach to a classical problem in statistical physics: estimating the partition function and other thermodynamic quantities of the ferromagnetic Ising model. Markov chain Monte Carlo methods for this problem have been…

We consider two different versions of the double dimer model on a planar domain, where we either fold a single dimer cover on a symmetric domain onto itself across the line of symmetry, or we superimpose two independent dimer covers on two,…

We study the stochastic Ising model on finite graphs with n vertices and bounded degree and analyze the effect of boundary conditions on the mixing time. We show that for all low enough temperatures, the spectral gap of the dynamics with…

In these lectures three different methods of computing the asymptotic expansion of a Hermitian matrix integral is presented. The first one is a combinatorial method using Feynman diagrams. This leads us to the generating function of the…