arXiv subjects

Alistair Sinclair

Publications and source records attributed to Alistair Sinclair.

At least 19 recordsLinked to original sources

arXiv2026

Nonlinear Exchange Dynamics for Independent Sets

In recent years, nonlinear dynamics derived from kinetic theory have gained attention in the context of sampling configurations of spin systems such as the Ising model. We focus on nonlinear dynamics for the hard-core model, a canonical spin system with hard constraints that specifies a distribution over independent sets in a graph, weighted by their sizes. We explore two distinct types of nonlinear dynamics: the mean-field dynamics, which preserves the density (or average size) of independent sets, and the single-site dynamics, which preserves the marginal vector (i.e., the occupancy probabilities of the vertices). These dynamics are natural stochastic processes for sampling from the hard-core model with a specified density or marginal vector, respectively, both of which are canonical instances of maximum entropy distributions that have been studied in various contexts. In contrast to linear Markov chains, there is a significant lack of a fundamental theoretical framework for nonlinear dynamics. We develop foundational theoretical tools for analyzing nonlinear dynamics within the context of the hard-core model. We establish almost linear convergence of both the mean-field and single-site dynamics at sufficiently low density through novel coupling arguments. We also establish exponential decay of relative entropy for the mean-field dynamics all the way up to the critical density. Additionally, we design new algorithms for sampling from the hard-core distribution with either a specified density or a specified marginal vector. These algorithms are based on a related linear Markov chain, called the particle-system dynamics and inspired by the so-called Kac's program, that approximates the associated nonlinear dynamics. As we demonstrate in the paper, they are comparable in time complexity, but simpler to implement, than traditional approaches based on learning parameter values.

cs.DS↗

arXiv2026

Mixing times and spectra of non-equilibrium symmetric exclusion processes on general graphs

The symmetric exclusion process (SEP) is a classical model of interacting particles on a graph, in which multiple particles execute random walks subject to the constraint that no two particles may occupy the same vertex. The version of the process in which the number of particles is fixed, is by now very well understood, through the study of reversible Markov chains. In this paper, we study the non-equilibrium version of the SEP, in which some vertices of the graph interact with external heat baths, held at different temperatures, that mediate transport through the graph. The resulting steady-state distribution is still the stationary distribution of a natural Markov chain, but the chain is no longer reversible. This means that even the steady-state distribution is very hard to describe, and indeed is the subject of a rich literature, which has focused almost exclusively on graphs which are finite subsets of the lattice $Z^d$. Very little is known about other important quantities, notably the mixing time. We derive various quantitative results about the non-equilibrium SEP for arbitrary graphs with arbitrary heat baths. Our main result is a bound on the mixing time in terms of the worst-case hitting time of a single particle to any of the heat baths; this bound is tight up to two logarithmic factors in the size of the graph. We then use this result to derive a simple algorithm, running in time roughly $n^{O(k)}$, that computes the steady-state joint occupation distribution for any set of $k$ vertices; since the steady-state distribution is very elusive, this provides a potentially useful tool to study its $k$-wise correlations for small values of $k$. Finally, we prove that the spectrum of the Markov chain associated with the non-reversible dynamics for the SEP is independent of the heat bath temperatures; thus in particular the spectrum is always real.

math.PR↗

arXiv2025

On quantum to classical comparison for Davies generators

Despite extensive study, our understanding of quantum Markov chains remains far less complete than that of their classical counterparts. [Temme'13] observed that the Davies Lindbladian, a well-studied model of quantum Markov dynamics, contains an embedded classical Markov generator, raising the natural question of how the convergence properties of the quantum and classical dynamics are related. While [Temme'13] showed that the spectral gap of the Davies Lindbladian can be much smaller than that of the embedded classical generator for certain highly structured Hamiltonians, we show that if the spectrum of the Hamiltonian does not contain long arithmetic progressions, then the two spectral gaps must be comparable. As a consequence, we prove that for a large class of Hamiltonians, including those obtained by perturbing a fixed Hamiltonian with a generic external field, the quantum spectral gap remains within a constant factor of the classical spectral gap. Our result aligns with physical intuition and enables the application of classical Markov chain techniques to the quantum setting. The proof is based on showing that any ``off-diagonal'' eigenvector of the Davies generator can be used to construct an observable which commutes with the Hamiltonian and has a Lindbladian Rayleigh quotient which can be upper bounded in terms of that of the original eigenvector's Lindbladian Rayleigh quotient. Thus, a spectral gap for such observables implies a spectral gap for the full Davies generator.

quant-ph↗

arXiv2024

Diversity in Evolutionary Dynamics

We consider the dynamics imposed by natural selection on the populations of two competing, sexually reproducing, haploid species. In this setting, the fitness of any genome varies over time due to the changing population mix of the competing species; crucially, this fitness variation arises naturally from the model itself, without the need for imposing it exogenously as is typically the case. Previous work on this model [14] showed that, in the special case where each of the two species exhibits just two phenotypes, genetic diversity is maintained at all times. This finding supported the tenet that sexual reproduction is advantageous because it promotes diversity, which increases the survivability of a species. In the present paper we consider the more realistic case where there are more than two phenotypes available to each species. The conclusions about diversity in general turn out to be very different from the two-phenotype case. Our first result is negative: namely, we show that sexual reproduction does not guarantee the maintenance of diversity at all times, i.e., the result of [14] does not generalize. Our counterexample consists of two competing species with just three phenotypes each. We show that, for any time~$t_0$ and any $\varepsilon>0$, there is a time $t\ge t_0$ at which the combined diversity of both species is smaller than~$\varepsilon$. Our main result is a complementary positive statement, which says that in any non-degenerate example, diversity is maintained in a weaker, "infinitely often" sense. Thus, our results refute the supposition that sexual reproduction ensures diversity at all times, but affirm a weaker assertion that extended periods of high diversity are necessarily a recurrent event.

q-bio.PE↗

arXiv2023

Spatial mixing and the random-cluster dynamics on lattices

An important paradigm in the understanding of mixing times of Glauber dynamics for spin systems is the correspondence between spatial mixing properties of the models and bounds on the mixing time of the dynamics. This includes, in particular, the classical notions of weak and strong spatial mixing, which have been used to show the best known mixing time bounds in the high-temperature regime for the Glauber dynamics for the Ising and Potts models. Glauber dynamics for the random-cluster model does not naturally fit into this spin systems framework because its transition rules are not local. In this paper, we present various implications between weak spatial mixing, strong spatial mixing, and the newer notion of spatial mixing within a phase, and mixing time bounds for the random-cluster dynamics in finite subsets of $\mathbb Z^d$ for general $d\ge 2$. These imply a host of new results, including optimal $O(N\log N)$ mixing for the random cluster dynamics on torii and boxes on $N$ vertices in $\mathbb Z^d$ at all high temperatures and at sufficiently low temperatures, and for large values of $q$ quasi-polynomial (or quasi-linear when $d=2$) mixing time bounds from random phase initializations on torii at the critical point (where by contrast the mixing time from worst-case initializations is exponentially large). In the same parameter regimes, these results translate to fast sampling algorithms for the Potts model on $\mathbb Z^d$ for general $d$.

math.PR↗

arXiv2023

Nonlinear dynamics for the Ising model

We introduce and analyze a natural class of nonlinear dynamics for spin systems such as the Ising model. This class of dynamics is based on the framework of mass action kinetics, which models the evolution of systems of entities under pairwise interactions, and captures a number of important nonlinear models from various fields, including chemical reaction networks, Boltzmann's model of an ideal gas, recombination in population genetics and genetic algorithms. In the context of spin systems, it is a natural generalization of linear dynamics based on Markov chains, such as Glauber dynamics and block dynamics, which are by now well understood. However, the inherent nonlinearity makes the dynamics much harder to analyze, and rigorous quantitative results so far are limited to processes which converge to essentially trivial stationary distributions that are product measures. In this paper we provide the first quantitative convergence analysis for natural nonlinear dynamics in a combinatorial setting where the stationary distribution contains non-trivial correlations, namely the Ising model at high temperatures. We prove that nonlinear versions of both the Glauber dynamics and the block dynamics converge to the Gibbs distribution of the Ising model (with given external fields) in times $O(n\log n)$ and $O(\log n)$ respectively, where $n$ is the size of the underlying graph (number of spins). Given the lack of general analytical methods for such nonlinear systems, our analysis is unconventional, and combines tools such as information percolation (due in the linear setting to Lubetzky and Sly), a novel coupling of the Ising model with Erd\H{o}s-R\'enyi random graphs, and non-traditional branching processes augmented by a fragmentation process.

math.PR↗

arXiv2022

Random-Cluster Dynamics in $\mathbb{Z}^2$

The random-cluster model has been widely studied as a unifying framework for random graphs, spin systems and electrical networks, but its dynamics have so far largely resisted analysis. In this paper we analyze the Glauber dynamics of the random-cluster model in the canonical case where the underlying graph is an $n \times n$ box in the Cartesian lattice $\mathbb{Z}^2$. Our main result is a $O(n^2\log n)$ upper bound for the mixing time at all values of the model parameter $p$ except the critical point $p=p_c(q)$, and for all values of the second model parameter $q\ge 1$. We also provide a matching lower bound proving that our result is tight. Our analysis takes as its starting point the recent breakthrough by Beffara and Duminil-Copin on the location of the random-cluster phase transition in $\mathbb{Z}^2$. It is reminiscent of similar results for spin systems such as the Ising and Potts models, but requires the reworking of several standard tools in the context of the random-cluster model, which is not a spin system in the usual sense.

cs.DM↗

arXiv2022

The Critical Mean-field Chayes-Machta Dynamics

The random-cluster model is a unifying framework for studying random graphs, spin systems and electrical networks that plays a fundamental role in designing efficient Markov Chain Monte Carlo (MCMC) sampling algorithms for the classical ferromagnetic Ising and Potts models. In this paper, we study a natural non-local Markov chain known as the Chayes-Machta dynamics for the mean-field case of the random-cluster model, where the underlying graph is the complete graph on $n$ vertices. The random-cluster model is parametrized by an edge probability $p$ and a cluster weight $q$. Our focus is on the critical regime: $p = p_c(q)$ and $q \in (1,2)$, where $p_c(q)$ is the threshold corresponding to the order-disorder phase transition of the model. We show that the mixing time of the Chayes-Machta dynamics is $O(\log n \cdot \log \log n)$ in this parameter regime, which reveals that the dynamics does not undergo an exponential slowdown at criticality, a surprising fact that had been predicted (but not proved) by statistical physicists. This also provides a nearly optimal bound (up to the $\log\log n$ factor) for the mixing time of the mean-field Chayes-Machta dynamics in the only regime of parameters where no non-trivial bound was previously known. Our proof consists of a multi-phased coupling argument that combines several key ingredients, including a new local limit theorem, a precise bound on the maximum of symmetric random walks with varying step sizes, and tailored estimates for critical random graphs. In addition, we derive an improved comparison inequality between the mixing time of the Chayes-Machta dynamics and that of the local Glauber dynamics on general graphs; this results in better mixing time bounds for the local dynamics in the mean-field setting.

math.PR↗

arXiv2022

Correlation decay and partition function zeros: Algorithms and phase transitions

We explore connections between the phenomenon of correlation decay and the location of Lee-Yang and Fisher zeros for various spin systems. In particular we show that, in many instances, proofs showing that weak spatial mixing on the Bethe lattice (infinite $Δ$-regular tree) implies strong spatial mixing on all graphs of maximum degree $Δ$ can be lifted to the complex plane, establishing the absence of zeros of the associated partition function in a complex neighborhood of the region in parameter space corresponding to strong spatial mixing. This allows us to give unified proofs of several recent results of this kind, including the resolution by Peters and Regts of the Sokal conjecture for the partition function of the hard core lattice gas. It also allows us to prove new results on the location of Lee-Yang zeros of the anti-ferromagnetic Ising model. We show further that our methods extend to the case when weak spatial mixing on the Bethe lattice is not known to be equivalent to strong spatial mixing on all graphs. In particular, we show that results on strong spatial mixing in the anti-ferromagnetic Potts model can be lifted to the complex plane to give new zero-freeness results for the associated partition function. This extension allows us to give the first deterministic FPTAS for counting the number of $q$-colorings of a graph of maximum degree $Δ$ provided only that $q\ge 2Δ$. This matches the natural bound for randomized algorithms obtained by a straightforward application of Markov chain Monte Carlo. We also give an improved version of this result for triangle-free graphs.

cs.CC↗

arXiv2021

Efficiently list-edge coloring multigraphs asymptotically optimally

We give polynomial time algorithms for the seminal results of Kahn, who showed that the Goldberg-Seymour and List-Coloring conjectures for (list-)edge coloring multigraphs hold asymptotically. Kahn's arguments are based on the probabilistic method and are non-constructive. Our key insight is to show that the main result of Achlioptas, Iliopoulos and Kolmogorov for analyzing local search algorithms can be used to make constructive applications of a powerful version of the so-called Lopsided Lovasz Local Lemma. In particular, we use it to design algorithms that exploit the fact that correlations in the probability spaces on matchings used by Kahn decay with distance.

cs.DM↗

arXiv2021

Low-temperature Ising dynamics with random initializations

It is well known that Glauber dynamics on spin systems typically suffer exponential slowdowns at low temperatures. This is due to the emergence of multiple metastable phases in the state space, separated by narrow bottlenecks that are hard for the dynamics to cross. It is a folklore belief that if the dynamics is initialized from an appropriate random mixture of ground states, one for each phase, then convergence to the Gibbs distribution should be much faster. However, such phenomena have largely evaded rigorous analysis, as most tools in the study of Markov chain mixing times are tailored to worst-case initializations. In this paper we develop a general framework towards establishing this conjectured behavior for the Ising model. In the classical setting of the Ising model on an $N$-vertex torus in $\mathbb Z^d$, our framework implies that the mixing time for the Glauber dynamics, initialized from a $\frac 12$-$\frac 12$ mixture of the all-plus and all-minus configurations, is $N^{1+o(1)}$ in dimension $d=2$, and at most quasi-polynomial in all dimensions $d\ge 3$, at all temperatures below the critical one. The key innovation in our analysis is the introduction of the notion of "weak spatial mixing within a phase", a low-temperature adaptation of the classical concept of weak spatial mixing. We show both that this new notion is strong enough to control the mixing time from the above random initialization (by relating it to the mixing time with plus boundary condition at $O(\log N)$ scales), and that it holds at all low temperatures in all dimensions. This framework naturally extends to much more general families of graphs. To illustrate this, we also use the same approach to establish optimal $O(N\log N)$ mixing for the Ising Glauber dynamics on random regular graphs at sufficiently low temperatures, when initialized from the same random mixture.

math.PR↗

arXiv2021

Entropy decay in the Swendsen-Wang dynamics on ${\mathbb Z}^d$

We study the mixing time of the Swendsen-Wang dynamics for the ferromagnetic Ising and Potts models on the integer lattice ${\mathbb Z}^d$. This dynamics is a widely used Markov chain that has largely resisted sharp analysis because it is non-local, i.e., it changes the entire configuration in one step. We prove that, whenever Strong Spatial Mixing (SSM) holds, the mixing time on any $n$-vertex cube in ${\mathbb Z}^d$ is $O(\log n)$, and we prove this is tight by establishing a matching lower bound on the mixing time. The previous best known bound was $O(n)$. SSM is a standard condition corresponding to exponential decay of correlations with distance between spins on the lattice and is known to hold in $d=2$ dimensions throughout the high-temperature (single phase) region. Our result follows from a Modified Log-Sobolev Inequality, which expresses the fact that the dynamics contracts relative entropy at a constant rate at each step. The proof of this fact utilizes a new factorization of the entropy in the joint probability space over spins and edges that underlies the Swendsen-Wang dynamics, which extends to general bipartite graphs of bounded degree. This factorization leads to several additional results, including mixing time bounds for a number of natural local and non-local Markov chains on the joint space, as well as for the standard random-cluster dynamics.

math.PR↗

arXiv2020

Beyond the Lovasz Local Lemma: Point to Set Correlations and Their Algorithmic Applications

Following the groundbreaking algorithm of Moser and Tardos for the Lovasz Local Lemma (LLL), there has been a plethora of results analyzing local search algorithms for various constraint satisfaction problems. The algorithms considered fall into two broad categories: resampling algorithms, analyzed via different algorithmic LLL conditions; and backtracking algorithms, analyzed via entropy compression arguments. This paper introduces a new convergence condition that seamlessly handles resampling, backtracking, and hybrid algorithms, i.e., algorithms that perform both resampling and backtracking steps. Unlike all past LLL work, our condition replaces the notion of a dependency or causality graph by quantifying point-to-set correlations between bad events. As a result, our condition simultaneously: (i)~captures the most general algorithmic LLL condition known as a special case; (ii)~significantly simplifies the analysis of entropy compression applications; (iii)~relates backtracking algorithms, which are conceptually very different from resampling algorithms, to the LLL; and most importantly (iv)~allows for the analysis of hybrid algorithms, which were outside the scope of previous techniques. We give several applications of our condition, including a new hybrid vertex coloring algorithm that extends the recent breakthrough result of Molloy for coloring triangle-free graphs to arbitrary graphs.

cs.DM↗

arXiv2018

Fisher zeros and correlation decay in the Ising model

We study the complex zeros of the partition function of the Ising model, viewed as a polynomial in the "interaction parameter"; these are known as Fisher zeros in light of their introduction by Fisher in 1965. While the zeros of the partition function as a polynomial in the "field" parameter have been extensively studied since the classical work of Lee and Yang, comparatively little is known about Fisher zeros for general graphs. Our main result shows that the zero-field Ising model has no Fisher zeros in a complex neighborhood of the entire region of parameters where the model exhibits correlation decay. In addition to shedding light on Fisher zeros themselves, this result also establishes a formal connection between two distinct notions of phase transition for the Ising model: the absence of complex zeros (analyticity of the free energy, or the logarithm of the partition function) and decay of correlations with distance. We also discuss the consequences of our result for efficient deterministic approximation of the partition function. Our proof relies heavily on algorithmic techniques, notably Weitz's self-avoiding walk tree, and as such belongs to a growing body of work that uses algorithmic methods to resolve classical questions in statistical physics.

math-ph↗

arXiv2018

The Ising Partition Function: Zeros and Deterministic Approximation

We study the problem of approximating the partition function of the ferromagnetic Ising model in graphs and hypergraphs. Our first result is a deterministic approximation scheme (an FPTAS) for the partition function in bounded degree graphs that is valid over the entire range of parameters $β$ (the interaction) and $λ$ (the external field), except for the case $\vertλ\vert=1$ (the "zero-field" case). A randomized algorithm (FPRAS) for all graphs, and all $β,λ$, has long been known. Unlike most other deterministic approximation algorithms for problems in statistical physics and counting, our algorithm does not rely on the "decay of correlations" property. Rather, we exploit and extend machinery developed recently by Barvinok, and Patel and Regts, based on the location of the complex zeros of the partition function, which can be seen as an algorithmic realization of the classical Lee-Yang approach to phase transitions. Our approach extends to the more general setting of the Ising model on hypergraphs of bounded degree and edge size, where no previous algorithms (even randomized) were known for a wide range of parameters. In order to achieve this extension, we establish a tight version of the Lee-Yang theorem for the Ising model on hypergraphs, improving a classical result of Suzuki and Fisher.

cs.DS↗

arXiv2017

Spatial Mixing and Systematic Scan Markov chains

We consider spin systems on the integer lattice graph $\mathbb{Z}^d$ with nearest-neighbor interactions. We develop a combinatorial framework for establishing that exponential decay with distance of spin correlations, specifically the strong spatial mixing condition (SSM), implies rapid mixing of a large class of Markov chains. As a first application of our method we prove that SSM implies $O(\log n)$ mixing of systematic scan dynamics (under mild conditions) on an $n$-vertex $d$-dimensional cube of the integer lattice graph $\mathbb{Z}^d$. Systematic scan dynamics are widely employed in practice but have proved hard to analyze. A second application of our technology concerns the Swendsen-Wang dynamics for the ferromagnetic Ising and Potts models. We show that SSM implies an $O(1)$ bound for the relaxation time (i.e., the inverse spectral gap). As a by-product of this implication we observe that the relaxation time of the Swendsen-Wang dynamics in square boxes of $\mathbb{Z}^2$ is $O(1)$ throughout the subcritical regime of the $q$-state Potts model, for all $q \ge 2$. We also use our combinatorial framework to give a simple coupling proof of the classical result that SSM entails optimal mixing time of the Glauber dynamics. Although our results in the paper focus on $d$-dimensional cubes in $\mathbb{Z}^d$, they generalize straightforwardly to arbitrary regions of $\mathbb{Z}^d$ and to graphs with subexponential growth.

cs.DM↗

arXiv2017

Spatial Mixing and Non-local Markov chains

We consider spin systems with nearest-neighbor interactions on an $n$-vertex $d$-dimensional cube of the integer lattice graph $\mathbb{Z}^d$. We study the effects that exponential decay with distance of spin correlations, specifically the strong spatial mixing condition (SSM), has on the rate of convergence to equilibrium distribution of non-local Markov chains. We prove that SSM implies $O(\log n)$ mixing of a block dynamics whose steps can be implemented efficiently. We then develop a methodology, consisting of several new comparison inequalities concerning various block dynamics, that allow us to extend this result to other non-local dynamics. As a first application of our method we prove that, if SSM holds, then the relaxation time (i.e., the inverse spectral gap) of general block dynamics is $O(r)$, where $r$ is the number of blocks. A second application of our technology concerns the Swendsen-Wang dynamics for the ferromagnetic Ising and Potts models. We show that SSM implies an $O(1)$ bound for the relaxation time. As a by-product of this implication we observe that the relaxation time of the Swendsen-Wang dynamics in square boxes of $\mathbb{Z}^2$ is $O(1)$ throughout the subcritical regime of the $q$-state Potts model, for all $q \ge 2$. We also prove that for monotone spin systems SSM implies that the mixing time of systematic scan dynamics is $O(\log n (\log \log n)^2)$. Systematic scan dynamics are widely employed in practice but have proved hard to analyze. Our proofs use a variety of techniques for the analysis of Markov chains including coupling, functional analysis and linear algebra.

cs.DM↗

arXiv2016

Entropy production in nonlinear recombination models

We study the convergence to equilibrium of a class of nonlinear recombination models. In analogy with Boltzmann's H theorem from kinetic theory, and in contrast with previous analysis of these models, convergence is measured in terms of relative entropy. The problem is formulated within a general framework that we refer to as Reversible Quadratic Systems. Our main result is a tight quantitative estimate for the entropy production functional. Along the way we establish some new entropy inequalities generalizing Shearer's and related inequalities.

math.PR↗