arXiv subjects

Will Perkins

Publications and source records attributed to Will Perkins.

At least 19 recordsLinked to original sources

arXiv2026

Satisfiability and freezing in random integer programs

We study a model of a random centrally symmetric polytope and ask about the existence of points in the polytope with non-zero, integer coordinates. Our main results are the location of a sharp threshold for the existence of these integer points, a characterization of the typical empirical distribution of their coordinates, and showing that near the feasibility threshold these integer points are almost always totally isolated --- at linear Hamming distance to the nearest other integer point in the polytope. The behavior of these random integer programs is related to that of the symmetric binary perceptron (the case when coordinates of points are $\pm 1$-valued). The lack of symmetry in the set of potential solutions leads to new questions and phenomena, such as the distribution of the coordinate sizes in typical solutions, and requires the development of new information-theoretic techniques to deal with higher-dimensional optimization problems arising in the method of moments.

math.PR↗

arXiv2026

The replica symmetric solution for hypergraph independent sets in the critical regime

We prove a variational formula for the logarithmic asymptotics of a non-existence probability in a broad class of combinatorial problems such as avoiding cliques in random graphs and $k$-term arithmetic progressions in random subsets of integers. These results follow from a formula for the probability that a binomial random subset of the vertices of a locally sparse hypergraph is an independent set. The formula holds throughout the critical regime, interpolating between the regimes in which Janson's inequality and the method of hypergraph containers give the respective asymptotics. The formula is the replica-symmetric Bethe free energy formula from statistical physics applied to a hypergraph hardcore model. The proof uses two new techniques. The first, for the upper bound, involves revealing a small `window' of a random independent set and then applying entropy methods. The second, for the lower bound, involves the analysis of a random greedy algorithm guided by a Belief Propagation fixed point. In the case of avoiding cliques in random graphs, the variational formula can be expressed as an optimization problem over graphons. Moreover, typical random graphs conditioned on not containing $K_r$, when suitably normalized, approach the set of optimizing graphons in cut distance.

math.CO↗

arXiv2026

Local large deviations for triangles in sparse random graphs

We revisit a classic topic in probabilistic combinatorics, the lower-tail large-deviation problem for triangles in the random graph $G(n,p)$. Here we aim for first-order asymptotics for the quantity $\mathbb{P}[X=k]$ with $X$ the number of triangles in $G(n,p)$ and $0 \le k \le (1-ε) \mathbb{E}X$, in the sparse regime in which the logarithmic asymptotics are Poissonian. When $k=0$ (the case of triangle-freeness) and $p = p(n)$ is sufficiently small, first-order asymptotics are known via Janson's inequality and results of Stark and Wormald; when $k$ is sufficiently close to $\mathbb{E}X$, first-order asymptotics are known via local central limit theorems. Our main result gives first-order asymptotics for all $k$ in the above range when $p = o(n^{-2/3})$, improving upon the result of Frieze that required $p = o(n^{-4/5})$. We also characterize, up to vanishing total variation distance, the distribution of the $k$ triangles in the corresponding conditional distribution and give an efficient algorithm to approximately sample from this conditional distribution. Notably, when $k = Θ(μ)$ there is a transition at $p = Θ(n^{-4/5})$ from an asymptotically uniform triangle distribution to a distribution asymptotically singular to uniform.

math.CO↗

arXiv2026

Structural Corrections to the Bethe Approximation of the Permanent

We study deterministic approximation algorithms for the permanent of a nonnegative matrix through the Bethe permanent, an approximation computable in polynomial time. The tight analysis of Anari and Rezaei gives a universal comparison between the permanent and the Bethe permanent within a factor $(\sqrt 2)^n$. The simple example of the unweighted $4$-cycle $C_4$ (or a union of disjoint $C_4$'s) shows that this bound is tight. We show that such $4$-cycle obstructions can be identified and exploited algorithmically. Given a Bethe optimizer, our algorithm identifies nearly isolated weighted $2\times2$ blocks and peels off a vertex-disjoint family of them. If the total weighted correction is large, we can improve the Bethe approximation; if it is small, we show that the Bethe permanent is within a factor of $(\sqrt2 - \varepsilon)^n$ of the truth. Combining these facts, we obtain a deterministic polynomial time $(\sqrt2-\varepsilon)^n$-approximation algorithm for the permanent of an arbitrary nonnegative $n\times n$ matrix, where $\varepsilon>0$ is some absolute constant.

cs.DS↗

arXiv2026

The Hard-Core Model on Bipartite Spectral Expanders: Counting and Sampling at All Fugacities

We study approximate counting and sampling algorithms for the hard-core model on $Δ$-regular bipartite graphs under a spectral expansion condition. Let $M_G$ be the biadjacency matrix of $G$. For every fixed $ξ\in(0,1)$, we give an FPRAS for the hard-core partition function and an efficient approximate sampler whenever \[ λ\leq \frac{1-ξ}{σ_2(M_G)}. \] The main idea is to introduce a family of quadratic tilts in the left-right occupation imbalance and show that each tilted measure can be sampled efficiently using Glauber dynamics. A discrete Gaussian identity expresses the original hard-core model as an exact positive mixture of these tilted measures; truncation and simulated annealing then yield efficient counting and sampling algorithms. For the complementary high-fugacity regime, we refine the polymer-model approach and show that the required phase-dominance and cluster expansion conditions follow from the singular-spectrum bound alone. Combining the two regimes, we obtain efficient approximate counting and sampling at every fugacity $λ>0$ whenever \[ σ_2(M_G)\leq c\left(\frac{Δ^2}{\log(\mathrm eΔ)}\right)^{1/3} \] for an absolute constant $c>0$. In particular, this recovers all-fugacity algorithms for random $Δ$-regular bipartite graphs for all sufficiently large $Δ$, while providing an efficiently verifiable certificate of their success on a given instance.

cs.DS↗

arXiv2026

Computational Thresholds for Balanced and Fixed-Slice Independent Sets in Bipartite Graphs

Motivated by recent work of Kocurek, Oveis Gharan, and Tjowasi, which gives an efficient sampling algorithm for the hard-core model on random regular bipartite graphs by decomposing into fixed-size slices, we study the worst-case tractability of approximate counting and sampling of fixed-size slices for bipartite independent set problems. Let $G=(L\sqcup R,E)$ be a bipartite graph with $|L|=|R|=n$ and maximum degree $Δ$. The fixed-slice problem asks to sample uniformly from independent sets satisfying $|I\cap L|=α_L n$ and $|I\cap R|=α_R n$. We show that if the overall density $α$ lies in the interval $(\frac{1}Δ, \tfrac{1}{2})$, and the densities on the two sides are more balanced than the typical phase densities of a random $Δ$-regular bipartite graph, then there is no FPRAS or efficient sampling scheme unless $\mathbf{NP}=\mathbf{RP}$. We then study a related fugacity model in which the densities are not fixed, but the independent set is required to be balanced between the two sides of the bipartition. For $λ>0$, the balanced hard-core model is the ordinary hard-core model with fugacity $λ$, conditioned on the event $|I\cap L|=|I\cap R|$. We prove that this model has the same computational threshold as the hard-core model on general bounded-degree graphs. That is, for every fixed $Δ\ge 3$, if $λ<λ_c(Δ)$, then the balanced partition function admits an FPTAS and the balanced hard-core distribution admits an efficient sampling scheme. Conversely, if $λ>λ_c(Δ)$, then no FPRAS or efficient sampler exists on this graph class unless $\mathbf{NP}=\mathbf{RP}$.

cs.DS↗

arXiv2026

A phase transition for the hard sphere model on the hyperbolic plane

The hard sphere model is a classical model from statistical physics in which particles are represented by equal-sized spheres. Longstanding predictions from the physics literature indicate that in $\mathbb{R}^2$ and $\mathbb{R}^3$ the system undergoes a phase transition, but it remains a major open problem to confirm this. We prove the existence of a phase transition for this model in the hyperbolic plane.

math-ph↗

arXiv2026

Uniqueness, analyticity and mixing for Gibbs point processes via spectral gaps

A Gibbs point process models particles interacting in the continuum through a potential. Among the most classical examples is the hard-sphere model, where given an activity parameter $λ$, a radius $r$, and a bounded set $Λ\subset \mathbb{R}^d$ one samples a Poisson process of intensity $λ$ in $Λ$ conditioned on the points forming the centers of an $r$-sphere packing. We prove uniqueness of infinite-volume Gibbs measure, analyticity of the pressure, and various notions of spatial and temporal mixing for activities up to what we define as the spectral threshold $λ_{spec}$ of the potential. For each fixed dimension $d \geq 2$, this improves the uniqueness and analyticity bounds for the hard-sphere model. As $d \to \infty$, our improvement over the classical bounds grows exponentially. We also prove an optimal mixing time bound for heat bath dynamics for the hard-sphere model up to an expected density of $Θ(d / 2^d)$, the first result that asymptotically matches the maximum density for rapid mixing predicted by Parisi and Zamponi. We also exhibit repulsive, radial pair potentials for which $λ_{spec} = + \infty$, showing that the corresponding Gibbs point processes have no phase transition at any activity $λ> 0$. Further, in dimensions $8$ and $24$ we exhibit such a potential with no phase transition for which the work of Cohn-Kumar-Miller-Radchenko-Viazovska proves that the unique ground state at any fixed density is given by the $E_8$ and Leech lattices, respectively. Our work builds upon a 2013 work of Kondratiev-Kuna-Ohlerich that implicitly defined $λ_{spec}$ and proved a spectral gap for a Glauber-like continuum birth-death dynamics. Our main work shows that such a spectral gap implies several strong notions of absence of phase transition and analyzes the behavior of $λ_{spec}$ for interesting potentials.

math-ph↗

arXiv2026

A simple proof of rapid mixing on random regular graphs beyond uniqueness

A recent breakthrough of Chen, Chen, Chen, Yin, and Zhang shows rapid mixing for Glauber dynamics for the hard-core model on random regular graphs beyond the tree uniqueness threshold. Their approach builds upon the literature of various local-to-global techniques and applies to a more general setting of discrete distributions supported on downward-closed set families. We give a short and self-contained proof via a Bochner--Bakry--Émery approach and directly show a Poincaré inequality by expanding the Dirichlet form in terms of the $L^2$-norm of the generator applied to a test function and eliminating a sum of squares term. Our proof is a streamlined version of an argument of Kondratiev, Kuna, and Ohlerich used to study spatial birth-and-death dynamics for Gibbs point processes in the continuum, which we adapt to the discrete setting.

math.PR↗

arXiv2026

Fast and perfect sampling of subgraphs and polymer systems

We give an efficient perfect sampling algorithm for weighted, connected induced subgraphs (or graphlets) of rooted, bounded degree graphs. Our algorithm utilizes a vertex-percolation process with a carefully chosen rejection filter and works under a percolation subcriticality condition. We show that this condition is optimal in the sense that the task of (approximately) sampling weighted rooted graphlets becomes impossible in finite expected time for infinite graphs and intractable for finite graphs when the condition does not hold. We apply our sampling algorithm as a subroutine to give near linear-time perfect sampling algorithms for polymer models and weighted non-rooted graphlets in finite graphs, two widely studied yet very different problems. This new perfect sampling algorithm for polymer models gives improved sampling algorithms for spin systems at low temperatures on expander graphs and unbalanced bipartite graphs, among other applications.

cs.DS↗

arXiv2026

Non-existence probabilities and lower tails in the critical regime via Belief Propagation

We compute the logarithmic asymptotics of the non-existence probability (and more generally the lower-tail probability) for a wide variety of combinatorial problems for a range of parameters in the `critical regime' between the regime amenable to hypergraph container methods and that amenable to Janson's inequality. Examples include lower tails and non-existence probabilities for subgraphs of random graphs and for $k$-term arithmetic progressions in random sets of integers. Our methods apply in the general framework of estimating the probability that a $p$-random subset of vertices in a $k$-uniform hypergraph induces significantly fewer hyperedges than expected. We show that under some simple structural conditions on the hypergraph and an upper bound on $p$ determined by a phase transition in the hard-core model on the infinite $k$-uniform, $Δ$-regular, linear hypertree, this probability can be accurately approximated by the Bethe free energy evaluated at the unique fixed point of a Belief Propagation operator on the hypergraph.

math.CO↗

arXiv2026

Sampling Colorings with Fixed Color Class Sizes

In 1970 Hajnal and Szemerédi proved a conjecture of Erdös that for a graph with maximum degree $Δ$, there exists an equitable $Δ+1$ coloring; that is a coloring where color class sizes differ by at most $1$. In 2007 Kierstand and Kostochka reproved their result and provided a polynomial-time algorithm which produces such a coloring. In this paper we study the problem of approximately sampling uniformly random equitable colorings. A series of works gives polynomial-time sampling algorithms for colorings without the color class constraint, the latest improvement being by Carlson and Vigoda for $q\geq 1.809 Δ$. In this paper we give a polynomial-time sampling algorithm for equitable colorings when $q> 2Δ$. Moreover, our results extend to colorings with small deviations from equitable (and as a corollary, establishing their existence). The proof uses the framework of the geometry of polynomials for multivariate polynomials, and as a consequence establishes a multivariate local Central Limit Theorem for color class sizes of uniform random colorings.

math.CO↗

arXiv2026

Optimal detection of planted stars via a random energy model

We study the problem of detecting a planted star in the Erd{ő}s--R{é}nyi random graph $G(n,m)$, formulated as a hypothesis test. We determine the scaling window for critical detection in $m$ in terms of the star size, and characterize the asymptotic total variation distance between the null and alternative hypotheses in this window. In the course of the proofs we show a condensation phase transition in the likelihood ratio that closely resembles that of the random energy model from spin glass theory.

math.ST↗

arXiv2025

Fast and Slow Mixing of the Kawasaki Dynamics on Bounded-Degree Graphs

We study the worst-case mixing time of the global Kawasaki dynamics for the fixed-magnetization Ising model on the class of graphs of maximum degree $Δ$. Proving a conjecture of Carlson, Davies, Kolla, and Perkins, we show that below the tree uniqueness threshold, the Kawasaki dynamics mix rapidly for all magnetizations. Disproving a conjecture of Carlson, Davies, Kolla, and Perkins, we show that the regime of fast mixing does not extend throughout the regime of tractability for this model: there is a range of parameters for which there exist efficient sampling algorithms for the fixed-magnetization Ising model on max-degree $Δ$ graphs, but the Kawasaki dynamics can take exponential time to mix. Our techniques involve showing spectral independence in the fixed-magnetization Ising model and proving a sharp threshold for the existence of multiple metastable states in the Ising model with external field on random regular graphs.

cs.DS↗

arXiv2025

Fixed-magnetization Ising on random graphs up to reconstruction

We study the fixed-magnetization ferromagnetic Ising model on random $d$-regular graphs for $d\ge 3$ and inverse temperature below the tree reconstruction threshold. Our main result is that for each magnetization $η$, the free energy density of the fixed-magnetization Ising model converges to the annealed free energy density, itself the Bethe free energy of an Ising measure on the infinite $d$-regular tree. Moreover, the fixed-magnetization Ising model exhibits local weak convergence to this tree measure. A key challenge to proving these results is that for magnetizations between the model's spinodal points, the limiting tree measure corresponds to an unstable fixed point of the belief propagation equations. As an application, we prove that the positive-temperature Zdeborová--Boettcher conjecture on max-cut and min-bisection holds up to the reconstruction threshold: on the random $d$-regular graph, the expected fraction of bichromatic edges in the anti-ferromagnetic Ising model plus the expected fraction of bichromatic edges in the zero-magnetization ferromagnetic Ising model equals $1+o(1)$. A second application is completely determining the large deviation rate function for the magnetization in the Ising model on the random regular graph up to reconstruction. Finally, we use the precise understanding of this rate function to show that the Glauber dynamics for the full Ising model on the random graph mixes in sub-exponential time from uniformly random initialization, well into the non-uniqueness regime where the worst-case initialization mixing time is exponentially slow.

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↗

arXiv2025

On the chromatic number of random triangle-free graphs

We study the chromatic number of typical triangle-free graphs with $Θ\left( n^{3/2} (\log n)^{1/2} \right)$ edges and establish the width of the scaling window for the transitions from $χ= 3$ to $χ= 4$ and from $χ= 4$ to $χ= 5$. The transition from $3$- to $4$-colorability has scaling window of width $Θ(n^{4/3} (\log n)^{-1/3})$. To prove this, we show a high probability equivalence of the $3$-colorability of a random triangle-free graph at this density and the satisfiability of an instance of bipartite random $2$-SAT, for which we establish the width of the scaling window following the techniques of Bollob{á}s, Borgs, Chayes, Kim, and Wilson. The transition from $4$- to $5$-colorability has scaling window of width $Θ(n^{3/2} (\log n)^{-1/2})$. To prove this, we show a high probability equivalence of the $4$-colorability of a random triangle-free graph at this density and the simultaneous $2$-colorability of two independent Erdős--Rényi random graphs. For this transition, we also establish the limiting probability of $4$-colorability inside the scaling window.

math.CO↗

arXiv2025

On the evolution of structure in triangle-free graphs

We study the typical structure and the number of triangle-free graphs with $n$ vertices and $m$ edges where $m$ is large enough so that a typical triangle-free graph has a cut containing nearly all of its edges, but may not be bipartite. Erdős, Kleitman, and Rothschild showed that almost every triangle-free graph is bipartite. Osthus, Prömel, and Taraz later showed that for $m \ge (1+ε)\frac{\sqrt{3}}{4}n^{3/2}\sqrt{\log n}$, almost every triangle-free graph on $n$ vertices and $m$ edges is bipartite. Here we give a precise characterization of the distribution of edges within each part of the max cut of a uniformly chosen triangle-free graph $G$ on $n$ vertices and $m$ edges, for a larger range of densities with $m=Θ(n^{3/2} \sqrt{\log n})$. Using this characterization, we describe the evolution of the structure of typical triangle-free graphs as the density changes. We show that as the number of edges decreases below $\frac{\sqrt{3}}{4} n^{3/2}\sqrt{\log n}$, the following structural changes occur in $G$: -Isolated edges, then trees, then more complex subgraphs emerge as `defect edges', edges within parts of a max cut of $G$. The distribution of defect edges is first that of independent Erdős-Rényi random graphs, then that of independent exponential random graphs, conditioned on a small maximum degree and no triangles. -There is a sharp threshold for $3$-colorability at $m \sim \frac{\sqrt{2}}{4} n^{3/2}\sqrt{\log n}$ and a sharp threshold between $4$-colorability and unbounded chromatic number at $m\sim\frac{1}{4}n^{3/2}\sqrt{\log n}$. -Giant components emerge in the defect edges at $m\sim\frac{1}{4} n^{3/2}\sqrt{\log n}$. We use these results to prove asymptotic formulas for the number of triangle-free graphs at these densities. We likewise prove analogous results for the random graph $G(n,p)$ conditioned on triangle-freeness.

math.CO↗