Related papers: Uniqueness and Mixing in the Low-Temperature Rando…

We give a new rapid mixing result for a natural random walk on the independent sets of a graph $G$. We show that when $G$ has bounded treewidth, this random walk -- known as the Glauber dynamics for the hardcore model -- mixes rapidly for…

The approximate uniform sampling of graph realizations with a given degree sequence is an everyday task in several social science, computer science, engineering etc. projects. One approach is using Markov chains. The best available current…

We study Bernoulli bond percolation on a random recursive tree of size $n$ with percolation parameter $p(n)$ converging to $1$ as $n$ tends to infinity. The sizes of the percolation clusters are naturally stored in a tree. We prove…

We develop a new framework to prove the mixing or relaxation time for the Glauber dynamics on spin systems with unbounded degree. It works for general spin systems including both $2$-spin and multi-spin systems. As applications for this…

The mixing time of the Glauber dynamics for spin systems on trees is closely related to reconstruction problem. Martinelli, Sinclair and Weitz established this correspondence for a class of spin systems with soft constraints bounding the…

We introduce a correlated static model and investigate a percolation transition. The model is a modification of the static model and is characterized by assortative degree-degree correlation. As one varies the edge density, the network…

In this note, we prove that on any graph of maximal degree $d$ the mixing time of the Glauber Dynamics for the Ising Model at $\beta_c=\tanh^{-1}(\frac1{d-1})$, the uniqueness threshold on the infinite $d$-regular tree, is at most…

Using a maximum entropy principle to assign a statistical weight to any graph, we introduce a model of random graphs with arbitrary degree distribution in the framework of standard statistical mechanics. We compute the free energy and the…

We analyse the metastable behaviour of the dilute Curie-Weiss model subject to a Glauber dynamics. The model is a random version of a mean-field Ising model, where the coupling coefficients are Bernoulli random variables with mean $p\in…

Consider Bernoulli bond percolation on a locally finite, connected graph $G$ and let $p_{\mathrm{cut}}$ be the threshold corresponding to a "first-moment method" lower bound. Kahn (\textit{Electron.\ Comm.\ Probab.\ Volume 8, 184-187.}…

This paper is studying the critical regime of the planar random-cluster model on $\mathbb Z^2$ with cluster-weight $q\in[1,4)$. More precisely, we prove crossing estimates in quads which are uniform in their boundary conditions and depend…

We study conditional independence relationships for random networks and their interplay with exchangeability. We show that, for finitely exchangeable network models, the empirical subgraph densities are maximum likelihood estimates of their…

We provide a simple proof that graphs in a general class of self-similar networks have zero percolation threshold. The considered self-similar networks include random scale-free graphs with given expected node degrees and zero clustering,…

The Arboreal gas model on a finite graph $G$ is the Bernoulli bond percolation on $G$ conditioned on the event that the sampled subgraph is a forest. In this short note we study the arboreal gas on a regular tree wired at the leaves and…

We prove that any Markov chain that performs local, reversible updates on randomly chosen vertices of a bounded-degree graph necessarily has mixing time at least $\Omega(n\log n)$, where $n$ is the number of vertices. Our bound applies to…

Let $G$ be a finitely generated group and $X$ its Cayley graph with respect to a finite, symmetric generating set $S$. Furthermore, let $H$ be a finite group and $H \wr G$ the lamplighter group (wreath product) over $G$ with group of…

We analyze critical phenomena on networks generated as the union of hidden variables models (networks with any desired degree sequence) with arbitrary graphs. The resulting networks are general small-worlds similar to those a` la Watts and…

In this paper we investigate the eigenvalue statistics of exponentially weighted ensembles of full binary trees and $p$-branching star graphs. We show that spectral densities of corresponding adjacency matrices demonstrate peculiar…

We study random subgraphs of an arbitrary finite connected transitive graph $\mathbb G$ obtained by independently deleting edges with probability $1-p$. Let $V$ be the number of vertices in $\mathbb G$, and let $\Omega$ be their degree. We…

The random-cluster model is a unifying framework for studying random graphs, spin systems in physics and random spanning trees. The model is closely related to, though much more general than the classical Ising and Potts models, but its…