We give the first comprehensive analysis of the effect of boundary conditions on the mixing time of the Glauber dynamics in the so-called Bethe approximation. Specifically, we show that spectral gap and the log-Sobolev constant of the Glauber dynamics for the Ising model on an n-vertex regular tree with plus-boundary are bounded below by a constant independent of n at all temperatures and all external fields. This implies that the mixing time is O(log n) (in contrast to the free boundary case, where it is not bounded by any fixed polynomial at low temperatures). In addition, our methods yield simpler proofs and stronger results for the spectral gap and log-Sobolev constant in the regime where there are multiple phases but the mixing time is insensitive to the boundary condition. Our techniques also apply to a much wider class of models, including those with hard-core constraints like the antiferromagnetic Potts model at zero temperature (proper colorings) and the hard--core lattice gas (independent sets).
In this paper, we present a novel extension to the classical path coupling method to statistical mechanical models which we refer to as aggregate path coupling. In conjunction with large deviations estimates, we use this aggregate path coupling method to prove rapid mixing of Glauber dynamics for a large class of statistical mechanical models, including models that exhibit discontinuous phase transitions which have traditionally been more difficult to analyze rigorously. The parameter region for rapid mixing for the generalized Curie-Weiss-Potts model is derived as a new application of the agg…
In the study of Markov chain mixing times, analysis has centered on the performance from a worst-case starting state. Here, in the context of Glauber dynamics for the one-dimensional Ising model, we show how new ideas from information percolation can be used to establish mixing times from other starting states. At high temperatures we show that the alternating initial condition is asymptotically the fastest one, and, surprisingly, its mixing time is faster than at infinite temperature, accelerating as the inverse-temperature $β$ ranges from 0 to $β_0=\frac12\mathrm{arctanh}(\frac13)$. Moreover…
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 (+)-boundary condition on a class of nonamenable graphs, is strictly positive uniformly in n. This implies that the mixing time grows at most linearly in n. The class of graphs we consider includes hyperbolic graphs with sufficiently high degree, where the best upper bound on the mixing time of the free boundary dynamics is polynomial in n, with exponent growi…
We prove an optimal $Ω(n^{-1})$ lower bound on the spectral gap of Glauber dynamics for anti-ferromagnetic two-spin systems with $n$ vertices in the tree uniqueness regime. This spectral gap holds for all, including unbounded, maximum degree $Δ$. Consequently, we have the following mixing time bounds for the models satisfying the uniqueness condition with a slack $δ\in(0,1)$: $\bullet$ $C(δ) n^2\log n$ mixing time for the hardcore model with fugacity $λ\le (1-δ)λ_c(Δ)= (1-δ)\frac{(Δ- 1)^{Δ- 1}}{(Δ- 2)^Δ}$; $\bullet$ $C(δ) n^2$ mixing time for the Ising model with edge activity $β\in\left[\frac…
The six-vertex model in statistical physics is a weighted generalization of the ice model on $\mathbb{Z}^2$ (i.e., Eulerian orientations) and the zero-temperature three-state Potts model (i.e., proper three-colorings). The phase diagram of the model depicts its physical properties and suggests where local Markov chains will be efficient. In this paper, we analyze the mixing time of Glauber dynamics for the six-vertex model in the ordered phases. Specifically, we show that for all Boltzmann weights in the ferroelectric phase, there exist boundary conditions such that local Markov chains require…
We study Glauber dynamics for the mean-field (Curie-Weiss) Potts model with $q\geq 3$ states and show that it undergoes a critical slowdown at an inverse-temperature $β_s(q)$ strictly lower than the critical $β_c(q)$ for uniqueness of the thermodynamic limit. The dynamical critical $β_s(q)$ is the spinodal point marking the onset of metastability. We prove that when $β<β_s(q)$ the mixing time is asymptotically $C(β, q) n \log n$ and the dynamics exhibits the cutoff phenomena, a sharp transition in mixing, with a window of order $n$. At $β=β_s(q)$ the dynamics no longer exhibits cutoff and its …
We prove an optimal mixing time bound on the single-site update Markov chain known as the Glauber dynamics or Gibbs sampling in a variety of settings. Our work presents an improved version of the spectral independence approach of Anari et al. (2020) and shows $O(n\log{n})$ mixing time on any $n$-vertex graph of bounded degree when the maximum eigenvalue of an associated influence matrix is bounded. As an application of our results, for the hard-core model on independent sets weighted by a fugacity $λ$, we establish $O(n\log{n})$ mixing time for the Glauber dynamics on any $n$-vertex graph of c…
We show that for any attractive Glauber-Exclusion process on the one-dimensional lattice of size $N$ with periodic boundary condition, if the corresponding hydrodynamic limit equation has a reaction term with a strictly convex potential, then the total-variation mixing time is of order $O(\log N)$. In particular, the result covers the full high-temperature regime in the original model introduced by De Masi, Ferrari and Lebowitz (1985).
We provide a general bound on the Wasserstein distance between two arbitrary distributions of sequences of Bernoulli random variables. The bound is in terms of a mixing quantity for the Glauber dynamics of one of the sequences, and a simple expectation of the other. The result is applied to estimate, with explicit error, expectations of functions of random vectors for some Ising models and exponential random graphs in "high temperature" regimes.
Glauber dynamics is a powerful tool to generate randomized, approximate solutions to combinatorially difficult problems. Applications include Markov Chain Monte Carlo (MCMC) simulation and distributed scheduling for wireless networks. In this paper, we derive bounds on the mixing time of a generalization of Glauber dynamics where multiple vertices are allowed to update their states in parallel and the fugacity of each vertex can be different. The results can be used to obtain various conditions on the system parameters such as fugacities, vertex degrees and update probabilities, under which th…
The Gaussian Free Field (GFF) is a canonical random surface in probability theory generalizing Brownian motion to higher dimensions. In two dimensions, it is critical in several senses, and is expected to be the universal scaling limit of a host of random surface models in statistical physics. It also arises naturally as the stationary solution to the stochastic heat equation with additive noise. Focusing on the dynamical aspects of the corresponding universality class, we study the mixing time, i.e., the rate of convergence to stationarity, for the canonical prelimiting object, namely the dis…
For general antiferromagnetic 2-spin systems, including the hardcore model and the antiferromagnetic Ising model, there is an $\mathsf{FPTAS}$ for the partition function on graphs of maximum degree $Δ$ when the infinite regular tree lies in the uniqueness region by Li et al. (2013). Moreover, in the tree non-uniqueness region, Sly (2010) showed that there is no $\mathsf{FPRAS}$ to estimate the partition function unless $\mathsf{NP}=\mathsf{RP}$. The algorithmic results follow from the correlation decay approach due to Weitz (2006) or the polynomial interpolation approach developed by Barvinok …
We extend results of R. Holley beyond the integer lattice to a large class of groups which includes free groups. In particular we show that a shift-invariant measure is Gibbs if and only if it is Glauber-invariant. Moreover, any shift-invariant measure converges weakly to the set of Gibbs measures when evolved under Glauber dynamics. These results are proven using a new notion of free energy density relative to a sofic approximation by homomorphisms. Any measure which minimizes free energy density is Gibbs.
A well-known conjecture in computer science and statistical physics is that Glauber dynamics on the set of $k$-colorings of a graph $G$ on $n$ vertices with maximum degree $Δ$ is rapidly mixing for $k \geq Δ+2$. In FOCS 1999, Vigoda showed rapid mixing of flip dynamics with certain flip parameters on the set of proper $k$-colorings for $k > \frac{11}{6}Δ$, implying rapid mixing for Glauber dynamics. In this paper, we obtain the first improvement beyond the $\frac{11}{6}Δ$ barrier for general graphs by showing rapid mixing for $k > (\frac{11}{6} - η)Δ$ for some positive constant $η$. The key to…
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 log-Sobolev constant by a comparison with the block dynamics. However, when there are hard constraints, the block dynamics may be reducible. We introduce a variant of the block dynamics extending these results to a wide class of spin systems with hard constraints. This applies for essentially any spin system that has non-reconstruction provided that on average…
In this work we prove sufficient conditions for the Glauber dynamics corresponding to a sequence of (non-product) measures on finite product spaces to be rapidly mixing, i.e. that the mixing time with respect to the total variation distance satisfies $t_{mix} = O(N \log N)$, where $N$ is the system size. The proofs do not rely on coupling arguments, but instead use functional inequalities. As a byproduct, we obtain exponential decay of the relative entropy along the Glauber semigroup. These conditions can be checked in various examples, which include the exponential random graph models with su…
Let $T$ be a tree on $n$ vertices and with maximum degree $Δ$. We show that for $k\geq Δ+1$ the Glauber dynamics for $k$-edge-colourings of $T$ mixes in polynomial time in $n$. The bound on the number of colours is best possible as the chain is not even ergodic for $k \leq Δ$. Our proof uses a recursive decomposition of the tree into subtrees; we bound the relaxation time of the original tree in terms of the relaxation time of its subtrees using block dynamics and chain comparison techniques. Of independent interest, we also introduce a monotonicity result for Glauber dynamics that simplifies …
We study the single-site Glauber dynamics for the fugacity $λ$, Hard-core model on the random graph $G(n, d/n)$. We show that for the typical instances of the random graph $G(n,d/n)$ and for fugacity $λ< \frac{d^d}{(d-1)^{d+1}}$, the mixing time of Glauber dynamics is $n^{1 + O(1/\log \log n)}$. Our result improves on the recent elegant algorithm in [Bezakova, Galanis, Goldberg Stefankovic; ICALP'22]. The algorithm there is a MCMC based sampling algorithm, but it is not the Glauber dynamics. Our algorithm here is simpler, as we use the classic Glauber dynamics. Furthermore, the bounds on mixin…
We prove an optimal $O(n \log n)$ mixing time of the Glauber dynamics for the Ising models with edge activity $β\in \left(\frac{Δ-2}Δ, \fracΔ{Δ-2}\right)$. This mixing time bound holds even if the maximum degree $Δ$ is unbounded. We refine the boosting technique developed in [CFYZ21], and prove a new boosting theorem by utilizing the entropic independence defined in [AJK+21]. The theorem relates the modified log-Sobolev (MLS) constant of the Glauber dynamics for a near-critical Ising model to that for an Ising model in a sub-critical regime.
We study continuous time Glauber dynamics for random configurations with local constraints (e.g. proper coloring, Ising and Potts models) on finite graphs with $n$ vertices and of bounded degree. We show that the relaxation time (defined as the reciprocal of the spectral gap $|λ_1-λ_2|$) for the dynamics on trees and on planar hyperbolic graphs, is polynomial in $n$. For these hyperbolic graphs, this yields a general polynomial sampling algorithm for random configurations. We then show that if the relaxation time $τ_2$ satisfies $τ_2=O(1)$, then the correlation coefficient, and the mutual info…
We study Glauber dynamics for the Ising model on the complete graph on $n$ vertices, known as the Curie-Weiss Model. It is well known that at high temperature ($β< 1$) the mixing time is $Θ(n\log n)$, whereas at low temperature ($β> 1$) it is $\exp(Θ(n))$. Recently, Levin, Luczak and Peres considered a censored version of this dynamics, which is restricted to non-negative magnetization. They proved that for fixed $β> 1$, the mixing-time of this model is $Θ(n\log n)$, analogous to the high-temperature regime of the original dynamics. Furthermore, they showed \emph{cutoff} for the original dynam…
We prove that Ising models on the hypercube with general quadratic interactions satisfy a Poincaré inequality with respect to the natural Dirichlet form corresponding to Glauber dynamics, as soon as the operator norm of the interaction matrix is smaller than $1$. The inequality implies a control on the mixing time of the Glauber dynamics. Our techniques rely on a localization procedure which establishes a structural result, stating that Ising measures may be decomposed into a mixture of measures with quadratic potentials of rank one, and provides a framework for proving concentration bounds fo…
We study the Glauber dynamics for the Ising model on the complete graph, also known as the Curie-Weiss Model. For beta < 1, we prove that the dynamics exhibits a cut-off: the distance to stationarity drops from near 1 to near 0 in a window of order n centered at [2(1-beta)]^{-1} n log n. For beta = 1, we prove that the mixing time is of order n^{3/2}. For beta > 1, we study metastability. In particular, we show that the Glauber dynamics restricted to states of non-negative magnetization has mixing time O(n log n).
We establish rapid mixing of the random-cluster Glauber dynamics on random $Δ$-regular graphs for all $q\ge 1$ and $p<p_u(q,Δ)$, where the threshold $p_u(q,Δ)$ corresponds to a uniqueness/non-uniqueness phase transition for the random-cluster model on the (infinite) $Δ$-regular tree. It is expected that this threshold is sharp, and for $q>2$ the Glauber dynamics on random $Δ$-regular graphs undergoes an exponential slowdown at $p_u(q,Δ)$. More precisely, we show that for every $q\ge 1$, $Δ\ge 3$, and $p<p_u(q,Δ)$, with probability $1-o(1)$ over the choice of a random $Δ$-regular graph on $n$ v…
Motivated by a recent use of Glauber dynamics for Monte-Carlo simulations of path integral representation of quantum spin models [Krzakala, Rosso, Semerjian, and Zamponi, Phys. Rev. B (2008)], we analyse a natural Glauber dynamics for the quantum Ising model with a transverse field on a finite graph $G$. We establish strict monotonicity properties of the equilibrium distribution and we extend (and improve) the censoring inequality of Peres and Winkler to the quantum setting. Then we consider the case when $G$ is a regular $b$-ary tree and prove the same fast mixing results established in [Mart…
We prove that the mixing time of the Glauber dynamics for random k-colorings of the complete tree with branching factor b undergoes a phase transition at $k=b(1+o_b(1))/\ln{b}$. Our main result shows nearly sharp bounds on the mixing time of the dynamics on the complete tree with n vertices for $k=Cb/\ln{b}$ colors with constant C. For $C\geq1$ we prove the mixing time is $O(n^{1+o_b(1)}\ln{n})$. On the other side, for $C<1$ the mixing time experiences a slowing down; in particular, we prove it is $O(n^{1/C+o_b(1)}\ln{n})$ and $Ω(n^{1/C-o_b(1)})$. The critical point C=1 is interesting since it…
We consider Glauber dynamics for the Ising model on the complete graph on $n$ vertices, known as the Curie-Weiss model. It is well-known that the mixing-time in the high temperature regime ($β< 1$) has order $n\log n$, whereas the mixing-time in the case $β> 1$ is exponential in $n$. Recently, Levin, Luczak and Peres proved that for any fixed $β< 1$ there is cutoff at time $[2(1-β)]^{-1} n\log n$ with a window of order $n$, whereas the mixing-time at the critical temperature $β=1$ is $Θ(n^{3/2})$. It is natural to ask how the mixing-time transitions from $Θ(n\log n)$ to $Θ(n^{3/2})$ and finall…
We study dynamical aspects of the $q$-state Potts model on an $n\times n$ box at its critical $β_c(q)$. Heat-bath Glauber dynamics and cluster dynamics such as Swendsen--Wang (that circumvent low-temperature bottlenecks) are all expected to undergo "critical slowdowns" in the presence of periodic boundary conditions: the inverse spectral gap, which in the subcritical regime is $O(1)$, should at criticality be polynomial in $n$ for $1< q \leq 4$, and exponential in $n$ for $q>4$ in accordance with the predicted discontinuous phase transition. This was confirmed for $q=2$ (the Ising model) by th…
The Glauber dynamics on the colourings of a graph is a random process which consists in recolouring at each step a random vertex of a graph with a new colour chosen uniformly at random among the colours not already present in its neighbourhood. It is known that when the total number of colours available is at least $Δ+2$, where $Δ$ is the maximum degree of the graph, this process converges to a uniform distribution on the set of all the colourings. Moreover, a well known conjecture is that the time it takes for the convergence to happen, called the mixing time, is polynomial in the size of the…
We study the effect of boundary conditions on the relaxation time of the Glauber dynamics for the hard-core model on the tree. The hard-core model is defined on the set of independent sets weighted by a parameter $λ$, called the activity. The Glauber dynamics is the Markov chain that updates a randomly chosen vertex in each step. On the infinite tree with branching factor $b$, the hard-core model can be equivalently defined as a broadcasting process with a parameter $ω$ which is the positive solution to $λ=ω(1+ω)^b$, and vertices are occupied with probability $ω/(1+ω)$ when their parent is uno…
Exponential random graph models have become increasingly important in the study of modern networks ranging from social networks, economic networks, to biological networks. They seek to capture a wide variety of common network tendencies such as connectivity and reciprocity through local graph properties. Sampling from these exponential distributions is crucial for parameter estimation, hypothesis testing, as well as understanding the features of the network in question. We inspect the efficiency of a popular sampling technique, the Glauber dynamics, for vertex-weighted exponential random graph…
The random-cluster model with parameters $(p,q)$ is a random graph model that generalizes bond percolation ($q=1$) and the Ising and Potts models ($q\geq 2$). We study its Glauber dynamics on $n\times n$ boxes $Λ_{n}$ of the integer lattice graph $\mathbb Z^2$, where the model exhibits a sharp phase transition at $p=p_c(q)$. Unlike traditional spin systems like the Ising and Potts models, the random-cluster model has non-local interactions. Long-range interactions can be imposed as external connections in the boundary of $Λ_n$, known as boundary conditions. For select boundary conditions that …
We give a systematic development of the application of matrix norms to rapid mixing in spin systems. We show that rapid mixing of both random update Glauber dynamics and systematic scan Glauber dynamics occurs if any matrix norm of the associated dependency matrix is less than 1. We give improved analysis for the case in which the diagonal of the dependency matrix is $\mathbf{0}$ (as in heat bath dynamics). We apply the matrix norm methods to random update and systematic scan Glauber dynamics for coloring various classes of graphs. We give a general method for estimating a norm of a symmetric …
Peres and Winkler proved a "censoring" inequality for Glauber dynamics on monotone spins systems such as the Ising model. Specifically, if, starting from a constant-spin configuration, the spins are updated at some sequence of sites, then inserting another site into this sequence brings the resulting configuration closer in total variation to the stationary distribution. We show by means of simple counterexamples that the analogous statements fail for Glauber dynamics on proper colorings of a graph, and for lazy transpositions on permutations, answering two questions of Peres. It is not known …
Consider Glauber dynamics for the Ising model on a graph of $n$ vertices. Hayes and Sinclair showed that the mixing time for this dynamics is at least $n\log n/f(Δ)$, where $Δ$ is the maximum degree and $f(Δ) = Θ(Δ\log^2 Δ)$. Their result applies to more general spin systems, and in that generality, they showed that some dependence on $Δ$ is necessary. In this paper, we focus on the ferromagnetic Ising model and prove that the mixing time of Glauber dynamics on any $n$-vertex graph is at least $(1/4+o(1))n \log n$.
In zero-temperature Glauber dynamics, vertices of a graph are given i.i.d.~initial spins $σ_x(0)$ from $\{-1,+1\}$ with $\mathbb{P}_p(σ_x(0) = +1)=p$, and they update their spins at the arrival times of i.i.d. Poisson processes to agree with a majority of their neighbors. We study this process on the 3-regular tree $\mathbb{T}_3$, where it is known that the critical threshold $p_c$, below which $\mathbb{P}_p$-a.s. all spins fixate to $-1$, is strictly less than $1/2$. Defining $θ(p)$ to be the $\mathbb{P}_p$-probability that a vertex fixates to $+1$, we show that $θ$ is a continuous function o…
In this paper we investigate the relationship between the mixing times of the Glauber dynamics of a statistical mechanical system with its thermodynamic equilibrium structure. For this we consider the mean-field Blume-Capel model, one of the simplest statistical mechanical models that exhibits the following intricate phase transition structure: within a two dimensional parameter space there exists a curve at which the model undergoes a second-order, continuous phase transition, a curve where the model undergoes a first-order, discontinuous phase transition, and a tricritical point which separa…
We extend the notion of spectral independence (introduced by Anari, Liu, and Oveis Gharan [ALO20]) from the Boolean domain to general discrete domains. This property characterises distributions with limited correlations, and implies that the corresponding Glauber dynamics is rapidly mixing. As a concrete application, we show that Glauber dynamics for sampling proper $q$-colourings mixes in polynomial-time for the family of triangle-free graphs with maximum degree $Δ$ provided $q\ge (α^*+δ)Δ$ where $α^*\approx 1.763$ is the unique solution to $α^*=\exp(1/α^*)$ and $δ>0$ is any constant. This is…
