arXiv subjects

Tyler Helmuth

Publications and source records attributed to Tyler Helmuth.

At least 19 recordsLinked to original sources

arXiv2026

Imry-Ma phenomenon for the hard-core model on $\mathbb{Z}^{2}$

The \emph{Imry-Ma phenomenon} refers to the dramatic effect that disorder can have on first-order phase transitions for two-dimensional spin systems. The most famous example is the absence of a phase transition for the two-dimensional random-field Ising model. This paper establishes that a similar phenomena takes place for the hard-core model, a discrete model of crystallization: arbitrarily weak disorder prevents the formation of a crystal. Our proof of this behaviour is an adaptation of the Aizenman-Wehr argument for the Imry-Ma phenomenon, with the use of internal (spin space) symmetries for spin systems being replaced by the use spatial symmetries.

math.PR↗

arXiv2025

Simple analyticity criteria for repulsive multi-body potentials

We prove a simple, explicit lower bound on the radius of a zero-free disk for Gibbs point processes defined by finite-range, repulsive multi-body interactions. Our lower bound improves on those previously known, and we demonstrate that it is essentially sharp in the generality with which our arguments apply. The key ingredient is a multi-body generalization of integral identities for point densities of Gibbs point processes in the spirit of earlier work of Michelen and Perkins.

math-ph↗

arXiv2024

Pirogov--Sinai Theory Beyond Lattices

Pirogov--Sinai theory is a well-developed method for understanding the low-temperature phase diagram of statistical mechanics models on lattices. Motivated by physical and algorithmic questions beyond the setting of lattices, we develop a combinatorially flexible version of Pirogov--Sinai theory for the hard-core model of independent sets. Our results illustrate that the main conclusions of Pirogov--Sinai theory can be obtained in significantly greater generality than that of $\mathbb Z^{d}$. The main ingredients in our generalization are combinatorial and involve developing appropriate definitions of contours based on the notion of cycle basis connectivity. This is inspired by works of Tim\'{a}r and Georgakopoulos--Panagiotis.

math.PR↗

arXiv2023

Percolation on hypergraphs and the hard-core model

We prove tight bounds on the site percolation threshold for $k$-uniform hypergraphs of maximum degree $\Delta$ and for $k$-uniform hypergraphs of maximum degree $\Delta$ in which any pair of edges overlaps in at most $r$ vertices. The hypergraphs that achieve these bounds are hypertrees, but unlike in the case of graphs, there are many different $k$-uniform, $\Delta$-regular hypertrees. Determining the extremal tree for a given $k, \Delta, r$ involves an optimization problem, and our bounds arise from a convex relaxation of this problem. By combining our percolation bounds with the method of disagreement percolation we obtain improved bounds on the uniqueness threshold for the hard-core model on hypergraphs satisfying the same constraints. Our uniqueness conditions imply exponential weak spatial mixing, and go beyond the known bounds for rapid mixing of local Markov chains and existence of efficient approximate counting and sampling algorithms. Our results lead to natural conjectures regarding the aforementioned algorithmic tasks, based on the intuition that uniqueness thresholds for the extremal hypertrees for percolation determine computational thresholds.

math.PR↗

arXiv2023

Directed Spatial Permutations on Asymmetric Tori

We investigate a model of random spatial permutations on two-dimensional tori, and establish that the joint distribution of large cycles is asymptotically given by the Poisson--Dirichlet distribution with parameter one. The asymmetry of the tori we consider leads to a spatial bias in the permutations, and this allows for a simple argument to deduce the existence of mesoscopic cycles. The main challenge is to leverage this mesoscopic structure to establish the existence and distribution of macroscopic cycles. We achieve this by a dynamical resampling argument in conjunction with a method developed by Schramm for the study of random transpositions on the complete graph. Our dynamical analysis implements generic heuristics for the occurrence of the Poisson--Dirichlet distribution in random spatial permutations, and hence may be of more general interest.

math.PR↗

arXiv2022

Efficient Algorithms for Approximating Quantum Partition Functions at Low Temperature

We establish an efficient approximation algorithm for the partition functions of a class of quantum spin systems at low temperature, which can be viewed as stable quantum perturbations of classical spin systems. Our algorithm is based on combining the contour representation of quantum spin systems of this type due to Borgs, Koteck\'y, and Ueltschi with the algorithmic framework developed by Helmuth, Perkins, and Regts, and Borgs et al.

quant-ph↗

arXiv2021

Spin systems with hyperbolic symmetry: a survey

Spin systems with hyperbolic symmetry originated as simplified models for the Anderson metal--insulator transition, and were subsequently found to exactly describe probabilistic models of linearly reinforced walks and random forests. In this survey we introduce these models, discuss their origins and main features, some existing tools available for their study, recent probabilistic results, and relations to other well-studied probabilistic models. Along the way we discuss some of the (many) open questions that remain.

math.PR↗

arXiv2021

Approximation algorithms for the random-field Ising model

Approximating the partition function of the ferromagnetic Ising model with general external fields is known to be #BIS-hard in the worst case, even for bounded-degree graphs, and it is widely believed that no polynomial-time approximation scheme exists. This motivates an average-case question: are there classes of instances for which polynomial-time approximation schemes exist? We investigate this question for the random field Ising model on graphs with maximum degree $\Delta$. We establish the existence of fully polynomial-time approximation schemes and samplers with high probability over the random fields if the external fields are IID Gaussians with variance larger than a constant depending only on the inverse temperature and $\Delta$. The main challenge comes from the positive density of vertices at which the external field is small. These regions, which may have connected components of size $\Theta(\log n)$, are a barrier to algorithms based on establishing a zero-free region, and cause worst-case analyses of Glauber dynamics to fail. The analysis of our algorithm is based on percolation on a self-avoiding walk tree.

cs.DS↗

arXiv2021

Percolation transition for random forests in $d\geq 3$

The arboreal gas is the probability measure on (unrooted spanning) forests of a graph in which each forest is weighted by a factor $\beta>0$ per edge. It arises as the $q\to 0$ limit of the $q$-state random cluster model with $p=\beta q$. We prove that in dimensions $d\geq 3$ the arboreal gas undergoes a percolation phase transition. This contrasts with the case of $d=2$ where no percolation transition occurs. The starting point for our analysis is an exact relationship between the arboreal gas and a non-linear sigma model with target space the fermionic hyperbolic plane $\mathbb{H}^{0|2}$. This latter model can be thought of as the $0$-state Potts model, with the arboreal gas being its random cluster representation. Unlike the standard Potts models, the $\mathbb{H}^{0|2}$ model has continuous symmetries. By combining a renormalisation group analysis with Ward identities we prove that this symmetry is spontaneously broken at low temperatures. In terms of the arboreal gas, this symmetry breaking translates into the existence of infinite trees in the thermodynamic limit. Our analysis also establishes massless free field correlations at low temperatures and the existence of a macroscopic tree on finite tori.

math.PR↗

arXiv2020

Finite-size scaling, phase coexistence, and algorithms for the random cluster model on random graphs

For $\Delta \ge 5$ and $q$ large as a function of $\Delta$, we give a detailed picture of the phase transition of the random cluster model on random $\Delta$-regular graphs. In particular, we determine the limiting distribution of the weights of the ordered and disordered phases at criticality and prove exponential decay of correlations and central limit theorems away from criticality. Our techniques are based on using polymer models and the cluster expansion to control deviations from the ordered and disordered ground states. These techniques also yield efficient approximate counting and sampling algorithms for the Potts and random cluster models on random $\Delta$-regular graphs at all temperatures when $q$ is large. This includes the critical temperature at which it is known the Glauber and Swendsen-Wang dynamics for the Potts model mix slowly. We further prove new slow-mixing results for Markov chains, most notably that the Swendsen-Wang dynamics mix exponentially slowly throughout an open interval containing the critical temperature. This was previously only known at the critical temperature. Many of our results apply more generally to $\Delta$-regular graphs satisfying a small-set expansion condition.

math.PR↗

arXiv2020

Efficient Algorithms for Approximating Quantum Partition Functions

We establish a polynomial-time approximation algorithm for partition functions of quantum spin models at high temperature. Our algorithm is based on the quantum cluster expansion of Neto\v{c}n\'y and Redig and the cluster expansion approach to designing algorithms due to Helmuth, Perkins, and Regts. Similar results have previously been obtained by related methods, and our main contribution is a simple and slightly sharper analysis for the case of pairwise interactions on bounded-degree graphs.

cs.DS↗

arXiv2020

Loop-erased random walk as a spin system observable

The determination of the Hausdorff dimension of the scaling limit of loop-erased random walk is closely related to the study of the one-point function of loop-erased random walk, i.e., the probability a loop-erased random walk passes through a given vertex. Recent work in the theoretical physics literature has investigated the Hausdorff dimension of loop-erased random walk in three dimensions by applying field theory techniques to study spin systems that heuristically encode the one-point function of loop-erased random walk. Inspired by this, we introduce two different spin systems whose correlation functions can be rigorously shown to encode the one-point function of loop-erased random walk.

math.PR↗

arXiv2020

Correlation decay for hard spheres via Markov chains

We improve upon all known lower bounds on the critical fugacity and critical density of the hard sphere model in dimensions two and higher. As the dimension tends to infinity our improvements are by factors of $2$ and $1.7$, respectively. We make these improvements by utilizing techniques from theoretical computer science to show that a certain Markov chain for sampling from the hard sphere model mixes rapidly at low enough fugacities. We then prove an equivalence between optimal spatial and temporal mixing for hard spheres to deduce our results.

math.PR↗

arXiv2019

Random spanning forests and hyperbolic symmetry

We study (unrooted) random forests on a graph where the probability of a forest is multiplicatively weighted by a parameter $\beta>0$ per edge. This is called the arboreal gas model, and the special case when $\beta=1$ is the uniform forest model. The arboreal gas can equivalently be defined to be Bernoulli bond percolation with parameter $p=\beta/(1+\beta)$ conditioned to be acyclic, or as the limit $q\to 0$ with $p=\beta q$ of the random cluster model. It is known that on the complete graph $K_{N}$ with $\beta=\alpha/N$ there is a phase transition similar to that of the Erd\H{o}s--R\'enyi random graph: a giant tree percolates for $\alpha > 1$ and all trees have bounded size for $\alpha<1$. In contrast to this, by exploiting an exact relationship between the arboreal gas and a supersymmetric sigma model with hyperbolic target space, we show that the forest constraint is significant in two dimensions: trees do not percolate on $\mathbb{Z}^2$ for any finite $\beta>0$. This result is a consequence of a Mermin--Wagner theorem associated to the hyperbolic symmetry of the sigma model. Our proof makes use of two main ingredients: techniques previously developed for hyperbolic sigma models related to linearly reinforced random walks and a version of the principle of dimensional reduction.

math.PR↗

arXiv2019

Efficient sampling and counting algorithms for the Potts model on $\mathbb Z^d$ at all temperatures

For $d \ge 2$ and all $q\geq q_{0}(d)$ we give an efficient algorithm to approximately sample from the $q$-state ferromagnetic Potts and random cluster models on finite tori $(\mathbb Z / n \mathbb Z )^d$ for any inverse temperature $\beta\geq 0$. This shows that the physical phase transition of the Potts model presents no algorithmic barrier to efficient sampling, and stands in contrast to Markov chain mixing time results: the Glauber dynamics mix slowly at and below the critical temperature, and the Swendsen--Wang dynamics mix slowly at the critical temperature. We also provide an efficient algorithm (an FPRAS) for approximating the partition functions of these models at all temperatures. Our algorithms are based on representing the random cluster model as a contour model using Pirogov--Sinai theory, and then computing an accurate approximation of the logarithm of the partition function by inductively truncating the resulting cluster expansion. The main innovation of our approach is an algorithmic treatment of unstable ground states, which is essential for our algorithms to apply to all inverse temperatures $\beta$. By treating unstable ground states our work gives a general template for converting probabilistic applications of Pirogov-Sinai theory to efficient algorithms.

math.PR↗

arXiv2019

The continuous-time lace expansion

We derive a continuous-time lace expansion for a broad class of self-interacting continuous-time random walks. Our expansion applies when the self-interaction is a sufficiently nice function of the local time of a continuous-time random walk. As a special case we obtain a continuous-time lace expansion for a class of spin systems that admit continuous-time random walk representations. We apply our lace expansion to the $n$-component $g|\varphi|^4$ model on $\mathbb{Z}^{d}$ when $n=1,2$, and prove that the critical Green's function $G_{\nu_{c}}(x)$ is asymptotically a multiple of $|x|^{2-d}$ when $d\geq 5$ at weak coupling. As another application of our method we establish the analogous result for the lattice Edwards model at weak coupling.

math.PR↗

arXiv2019

The geometry of random walk isomorphism theorems

The classical random walk isomorphism theorems relate the local times of a continuous-time random walk to the square of a Gaussian free field. A Gaussian free field is a spin system that takes values in Euclidean space, and this article generalises the classical isomorphism theorems to spin systems taking values in hyperbolic and spherical geometries. The corresponding random walks are no longer Markovian: they are the vertex-reinforced and vertex-diminished jump processes. We also investigate supersymmetric versions of these formulas. Our proofs are based on exploiting the continuous symmetries of the corresponding spin systems. The classical isomorphism theorems use the translation symmetry of Euclidean space, while in hyperbolic and spherical geometries the relevant symmetries are Lorentz boosts and rotations, respectively. These very short proofs are new even in the Euclidean case. Isomorphism theorems are useful tools, and to illustrate this we present several applications. These include simple proofs of exponential decay for spin system correlations, exact formulas for the resolvents of the joint processes of random walks together with their local times, and a new derivation of the Sabot--Tarr\`es formula for the limiting local time of the vertex-reinforced jump process.

math.PR↗

arXiv2018

Algorithmic Pirogov-Sinai theory

We develop an efficient algorithmic approach for approximate counting and sampling in the low-temperature regime of a broad class of statistical physics models on finite subsets of the lattice $\mathbb Z^d$ and on the torus $(\mathbb Z/n \mathbb Z)^d$. Our approach is based on combining contour representations from Pirogov-Sinai theory with Barvinok's approach to approximate counting using truncated Taylor series. Some consequences of our main results include an FPTAS for approximating the partition function of the hard-core model at sufficiently high fugacity on subsets of $\mathbb Z^d$ with appropriate boundary conditions and an efficient sampling algorithm for the ferromagnetic Potts model on the discrete torus $(\mathbb Z/n \mathbb Z)^d$ at sufficiently low temperature.

cs.DS↗