arXiv subjects

Vilas Winstein

Publications and source records attributed to Vilas Winstein.

11 recordsLinked to original sources

arXiv2026

Correlation decay in area-tilted line ensembles

Random surfaces on a hard substrate often exhibit entropic repulsion, wherein the surface is propelled upwards to allow entropically preferable downward fluctuations. A particularly rich class of examples arises from the low-temperature 3D Ising model. A powerful approach to studying such surfaces is through their level curves, which form a family of non-intersecting random curves. In [CIW18, CIW19], an ensemble of Brownian lines with geometrically increasing area tilts was proposed as a putative limiting model in this case. This model falls outside the scope of techniques based on integrable or SDE structures, which have been key ingredients in the study of the Airy line ensemble. A particularly intriguing question about such line ensembles concerns their mixing properties when viewed as a Markov process, and in particular the rate of decay of correlations in time. For the Airy line ensemble, this decay is known to be inverse quadratic. The first quantitative bound on the decay of correlations in the area-tilted model, established in [CG25], was slower than polynomial in time. An earlier result [DLZ24] had established positivity of the spectral gap for the finite-line version of the ensemble, without quantitative bounds. This left open the important question of the true decay rate of correlations for the infinite ensemble. Settling this question for sufficiently large area-tilt strength, corresponding to sufficiently low temperature for the 3D Ising model, we prove exponential decay of correlations for the infinite ensemble and a uniform (in the number of lines) positive spectral gap for the finite ensemble. Our proof is based on establishing a precise form of separation of scales between curves of different indices, using a novel probabilistic approach involving embedding supercritical branching processes in the line ensemble.

math.PR↗

arXiv2026

The maximum-entropy median-martingale

This short note explores the maximum-entropy walk on the unit interval that is a median-martingale. That is, the median of its next state is equal to its current state. The stationary distribution of this walk is the arcsine distribution, and we provide a proof that elucidates the connection to two classical arcsine laws for Brownian motion. The notion of a martingale is further generalized, and a larger class of walks is considered and similarly characterized.

math.PR↗

arXiv2026

Wasserstein distances between ERGMs and Erd\H{o}s-R\'enyi models

Ferromagnetic exponential random graph models (ERGMs) are random graph models under which the presence of certain small structures (such as triangles) is encouraged; they can be constructed by tilting an Erd\H{o}s--R\'enyi model by the exponential of a particular nonlinear Hamiltonian. These models are mixtures of metastable wells which each behave macroscopically like an Erd\H{o}s--R\'enyi model, exhibiting the same laws of large numbers for subgraph counts [CD13]. However, on the microscopic scale these metastable wells are very different from Erd\H{o}s--R\'enyi models, with the total variation distance between the two measures tending to 1 [MX23]. In this article we clarify this situation by providing a sharp (up to constants) bound on the Hamming-Wasserstein distance between the two models, which is the average number of edges at which they differ, under the coupling which minimizes this average. In particular, we show that this distance is $\Theta(n^{3/2})$, quantifying exactly how these models differ. An upper bound of this form has appeared in the past [RR19], but this was restricted to the subcritical (high-temperature) regime of parameters. We extend this bound, using a new proof technique, to the supercritical (low-temperature) regime, and prove a matching lower bound which has only previously appeared in the subcritical regime of special cases of ERGMs satisfying a "triangle-free" condition [DF25]. To prove the lower bound in the presence of triangles, we introduce an approximation of the discrete derivative of the Hamiltonian, which controls the dynamical properties of the ERGM, in terms of local counts of triangles and wedges (two-stars) near an edge. This approximation is the main technical and conceptual contribution of the article, and we expect it will be useful in a variety of other contexts as well. Along the way, we also prove a bound on the marginal edge probability under the ERGM via a new bootstrapping argument. Such a bound has already appeared [FLSW25], but again only in the subcritical regime and using a different proof strategy.

math.PR↗

arXiv2026

Approximate FKG inequalities for phase-bound spin systems, with applications to central limit theorems for exponential random graphs

The Fortuin-Kasteleyn-Ginibre (FKG) inequality is an invaluable tool in monotone spin systems satisfying the FKG lattice condition, which provides positive correlations for all coordinate-wise increasing functions of spins. This inequality has numerous applications and plays an integral role in the proof of various central limit theorems (CLTs), including recent work on ferromagnetic exponential random graph models (ERGMs) wherein a Hamiltonian tilt promotes the presence of small subgraphs like triangles. However, the FKG lattice condition fails to hold when confining a spin system to a particular phase in the low-temperature regime of parameters. Thus it is not a priori clear if each phase internally has positive correlations for increasing functions, or if the positive correlations in the overall model (which is a mixture of phases) arise primarily from the global choice of phase. In this article, we show that the individual phases in ERGMs do indeed satisfy an approximate form of the FKG inequality internally. We use this to finish the proof of various CLTs within each individual phase in the phase-coexistence regime, answering a question posed by Bianchi, Collet, and Magnanini. We present the FKG inequality for ERGMs as a consequence of a more general result which holds under certain inputs related to metastable mixing; we expect this general result to be widely applicable, and we devote a section to spelling out the details of its application to a class of generalized higher-order ferromagnetic Curie-Weiss models where the necessary inputs are relatively transparent.

math.PR↗

arXiv2025

Decoupling of clusters in independent sets in a percolated hypercube

Independent sets in graphs are sets of vertices containing no neighbors, and they represent a canonical spin system with hardcore constraints. Of particular interest is the setting of the boolean hypercube, where counting independent sets was the original motivator for Sapozhenko's famous graph container method. A modern perspective on such problems is to consider the effect of disorder, and the study of independent sets in random subgraphs of the hypercube obtained via bond percolation with parameter $p$ was initiated by Kronenberg and Spinka. They employed tools from statistical mechanics to obtain detailed information about the moments of the number of independent sets (now a random variable), and posed many interesting questions. Previous work by the authors addressed many of these questions in the regime $p \geq \frac{2}{3}$, where the behavior is relatively simple and can be modeled well by a related family of independent particles. As $p$ decreases, though, typical independent sets become larger and feature more intricate clustering behavior. In the present article we overcome many of the challenges presented by this phenomenon and analyze the model for all $p> 0.465$. We obtain a sharp in-probability approximation for the number of independent sets in the percolated hypercube in terms of explicit random variables, as well as provide a sampling algorithm. Note that this shows, curiously, that $p = \frac{1}{2}$ is not a natural barrier for this problem unlike in many other problems where it appears as a point of a phase transition. A key contribution of this work is the introduction of a new probabilistic framework to handle the clustering behavior for these low values of $p$. Although our analysis is restricted to $p > 0.465$, our arguments are expected to be helpful for studying this model at even lower values of $p$, and possibly for other related problems.

math.PR↗

arXiv2025

Quantitative central limit theorems for exponential random graphs

Ferromagnetic exponential random graph models (ERGMs) are nonlinear exponential tilts of Erd\H{o}s-R\'enyi models, under which the presence of certain subgraphs such as triangles may be emphasized. These models are mixtures of metastable wells which each behave macroscopically like new Erd\H{o}s-R\'enyi models themselves, exhibiting the same laws of large numbers for the overall edge count as well as all subgraph counts. However, the microscopic fluctuations of these quantities remained elusive for some time. Building on a recent breakthrough by Fang, Liu, Shao and Zhao [FLSZ24] driven by Stein's method, we prove quantitative central limit theorems (CLTs) for these quantities and more in metastable wells under ferromagnetic ERGMs. One main novelty of our results is that they apply also in the supercritical (low temperature) regime of parameters, which has previously been relatively unexplored. To accomplish this, we develop a novel probabilistic technique based on the careful analysis of the evolution of relevant quantities under the ERGM Glauber dynamics. Our technique allows us to deliver the main input to the method developed by [FLSZ24], which is the fact that the fluctuations of subgraph counts are driven by those of the overall edge count. This was first shown for the triangle count by Sambale and Sinulis [SS20] in the Dobrushin (very high temperature) regime via functional-analytic methods. We feel our technique clarifies the underlying mechanisms at play, and it also supplies improved bounds on the Wasserstein and Kolmogorov distances between the observables at hand and the limiting Gaussians, as compared to the results of [FLSZ24] in the subcritical (high temperature) regime beyond the Dobrushin regime. Moreover, our technique is flexible enough to also yield quantitative CLTs for vertex degrees and local subgraph counts, which have not appeared before in any parameter regime.

math.PR↗

arXiv2025

Concentration via metastable mixing, with applications to the supercritical exponential random graph model

Folklore belief holds that metastable wells in low-temperature statistical mechanics models exhibit high-temperature behavior. We make this rigorous in the exponential random graph model (ERGM) through the lens of concentration of measure. We make use of the supercritical (low-temperature) metastable mixing which was recently proven by Bresler, Nagaraj, and Nichani, and obtain a novel concentration inequality for Lipschitz observables of the ERGM in a large metastable well, answering a question posed by those authors. To achieve this, we prove a new connectivity property for metastable mixing in the ERGM and introduce a new general result yielding concentration inequalities, which extends a result of Chatterjee. We also use a result of Barbour, Brightwell, and Luczak to cover all cases of interest. Our work extends a result of Ganguly and Nam from the subcritical (high-temperature) regime to metastable wells, and we also extend applications of this concentration, namely a central limit theorem for small subcollections of edges and a bound on the Wasserstein distance between the ERGM and the Erd\H{o}s-R\'enyi random graph. Finally, to supplement the mathematical content of the article, we present a simulation study of metastable wells in the supercritical ERGM.

math.PR↗

arXiv2024

Gaussian to log-normal transition for independent sets in a percolated hypercube

Independent sets in graphs, i.e., subsets of vertices where no two are adjacent, have long been studied, for instance as a model of hard-core gas. The $d$-dimensional hypercube, $\{0,1\}^d$, with the nearest neighbor structure, has been a particularly appealing choice for the base graph, owing in part to its many symmetries. Results go back to the work of Korshunov and Sapozhenko who proved sharp results on the count of such sets as well as structure theorems for random samples drawn uniformly. Of much interest is the behavior of such Gibbs measures in the presence of disorder. In this direction, Kronenberg and Spinka [KS] initiated the study of independent sets in a random subgraph of the hypercube obtained by considering an instance of bond percolation with probability $p$. Relying on tools from statistical mechanics they obtained a detailed understanding of the moments of the partition function, say $\mathcal{Z}$, of the hard-core model on such random graphs and consequently deduced certain fluctuation information, as well as posed a series of interesting questions. In particular, they showed in the uniform case that there is a natural phase transition at $p=2/3$ where $\mathcal{Z}$ transitions from being concentrated for $p>2/3$ to not concentrated at $p=2/3$. In this article, developing a probabilistic framework, as well as relying on certain cluster expansion inputs from [KS], we present a detailed picture of both the fluctuations of $\mathcal{Z}$ as well as the geometry of a randomly sampled independent set. In particular, we establish that $\mathcal{Z}$, properly centered and scaled, converges to a standard Gaussian for $p>2/3$, and to a sum of two i.i.d. log-normals at $p=2/3$. A particular step in the proof which could be of independent interest involves a non-uniform birthday problem for which collisions emerge at $p=2/3$.

math.PR↗

arXiv2022

Categorifying Biquandle Brackets

In their paper entitled "Quantum Enhancements and Biquandle Brackets," Nelson, Orrison, and Rivera introduced biquandle brackets, which are customized skein invariants for biquandle-colored links. These invariants generalize the Jones polynomial, which is categorified by Khovanov homology. At the end of their paper, Nelson, Orrison, and Rivera asked if the methods of Khovanov homology could be extended to obtain a categorification of biquandle brackets. We outline herein a Khovanov homology-style construction that is an attempt to obtain such a categorification of biquandle brackets. The resulting knot invariant generalizes Khovanov homology, but the biquandle bracket is not always recoverable, meaning the construction is not a true categorification of biquandle brackets. However, the construction does lead to a definition that gives a "canonical" biquandle 2-cocycle associated to a biquandle bracket, which, to the authors' knowledge, was not previously known.

math.QA↗

arXiv2022

Regularity based spectral clustering and mapping the Fiedler-carpet

Spectral clustering is discussed from many perspectives, by extending it to rectangular arrays and discrepancy minimization too. Near optimal clusters are obtained with singular value decomposition and with the weighted $k$-means algorithm. In case of rectangular arrays, this means enhancing the method of correspondence analysis with clustering, and in case of edge-weighted graphs, a normalized Laplacian based clustering. In the latter case it is proved that a spectral gap between the $(k-1)$th and $k$th smallest positive eigenvalues of the normalized Laplacian matrix gives rise to a sudden decrease of the inner cluster variances when the number of clusters of the vertex representatives is $2^{k-1}$, but only the first $k-1$ eigenvectors, constituting the so-called Fiedler-carpet, are used in the representation. Application to directed migration graphs is also discussed.

math.CO↗

arXiv2020

The Structure of Biquandle Brackets

In their paper entitled "Quantum Enhancements and Biquandle Brackets," Nelson, Orrison, and Rivera introduced biquandle brackets, which are customized skein invariants for biquandle-colored links. We prove herein that if a biquandle bracket is the pointwise product of another biquandle bracket with some function $ϕ$, then $ϕ$ is a a biquandle 2-cocycle (up to a constant multiple). As an application, we show that a new invariant introduced by Yang factors in this way, which allows us to show that the new invariant is in fact equivalent to the Jones polynomial on knots. Additionally, we provide a few new results about the structure of biquandle brackets and their relationship with biquandle 2-cocycles.

math.GT↗