arXiv subjects

Benjamin Schweinhart

Publications and source records attributed to Benjamin Schweinhart.

At least 19 recordsLinked to original sources

arXiv2026

A Topological Formula for Potts Lattice Gauge Theory Correlations

We exhibit a formula relating the correlation between Wilson loop variables in Potts lattice gauge theory to a topological quantity in the plaquette random cluster model. As applications we show that the correlation length of the model on $\mathbb{Z}^4$ with free boundary conditions equals that of the dual model with constant boundary conditions, we prove exponential decay of correlations between slowly growing Wilson loop variables for Ising lattice gauge theory on $\mathbb{Z}^3$ at all but the critical temperature, and we demonstrate that the correlation length is finite at sufficiently high or low temperatures in any dimension.

math.PR↗

arXiv2026

Voronoi Percolation: Topological Stability and Giant Cycles

We study the topological stability of Voronoi percolation in higher dimensions. We show that slightly increasing p allows a discretization that preserves increasing topological properties with high probability. This strengthens a theorem of Bollobás and Riordan and generalizes it to higher dimensions. As a consequence, we prove a sharp phase transition for the emergence of i-dimensional giant cycles in Voronoi percolation on the 2i-dimensional torus.

math.PR↗

arXiv2026

Generalized cluster algorithms for Potts lattice gauge theory

Monte Carlo algorithms, like the Swendsen-Wang and invaded-cluster, sample the Ising and Potts models asymptotically faster than single-spin Glauber dynamics do. Here, we generalize both algorithms to sample Potts lattice gauge theory by way of a $2$-dimensional cellular representation called the plaquette random-cluster model. The invaded-cluster algorithm targets Potts lattice gauge theory at criticality by implementing a stopping condition defined in terms of homological percolation, the emergence of spanning surfaces on the torus. Simulations for $\mathbb Z(2)$ and $\mathbb Z(3)$ lattice gauge theories on the cubical $4$-dimensional torus indicate that both generalized algorithms exhibit much faster autocorrelation decay than single-spin dynamics and allow for efficient sampling on $4$-dimensional tori of linear scale at least $40$.

cond-mat.stat-mech↗

arXiv2026

Classification of Histopathology Slides with Persistent Homology Convolutions

Convolutional neural networks (CNNs) are a standard tool for computer vision tasks such as image classification. However, typical model architectures may result in the loss of topological information. In specific domains such as histopathology, topology is an important descriptor that can be used to distinguish between disease-indicating tissue by analyzing the shape characteristics of cells. Current literature suggests that reintroducing topological information using persistent homology can improve medical diagnostics; however, previous methods utilize global topological summaries which do not contain information about the locality of topological features. To address this gap, we present a novel method that generates local persistent homology-based data using a modified version of the convolution operator called \textit{Persistent Homology Convolutions}. This method captures information about the locality and translation equivariance of topological features. We perform a comparative study using various representations of histopathology slides and find that models trained with persistent homology convolutions outperform conventionally trained models and are less sensitive to hyperparameters. These results indicate that persistent homology convolutions extract meaningful geometric information from the histopathology slides.

eess.IV↗

arXiv2026

A Cellular Representation of the Potts Lattice Higgs Model

The $i$-dimensional Potts lattice Higgs model is a random assignment of spins in $\mathbb{Z}_q$ to the $i$-dimensional cells of a cell complex induced by a Hamiltonian with a Potts interaction on the $(i+1)$-cells and an additional term playing the role of an external field. We develop a representation of this model as a pair of dependent plaquette percolations, and prove that Wilson line expectations can be expressed in terms of the probability of a topological event. As an application, we prove the existence of a phase transition for the Marcu--Fredenhagen ratio in the Potts lattice Higgs model on $\mathbb{Z}^d$ when $i=1.$

math.PR↗

arXiv2026

Hadwiger Models: Low-Temperature Behavior in a Natural Extension of the Ising Model

All isometrically invariant Markov (strictly local) fields on binary assignments are induced by energy functions that can be represented as linear combinations of area, perimeter, and Euler characteristic. This class of model includes the Ising model, both ferro- and antiferro-magnetic, with and without a field, as well as the "triplet" Ising model We determine the low-temperature behavior for this class of model, and construct a phase diagram of that behavior. In particular, we identify regions with three geometric phases, regions with a single unique phase, and coexistence lines between them.

math-ph↗

arXiv2025

Dependence of Microstructure Classification Accuracy on Crystallographic Data Representation

Convolutional neural networks are increasingly being used to analyze and classify material microstructures, motivated by the possibility that they will be able to identify relevant microstructural features more efficiently and impartially than human experts. While up to now convolutional neural networks have mostly been applied to light optimal microscopy and scanning electron microscope micrographs, application to EBSD micrographs will be increasingly common as rational design generates materials with unknown textures and phase compositions. This raises the question of how crystallographic orientation should be represented in such a convolutional neural network, and whether this choice has a significant effect on the network's analysis and classification accuracy. Four representations of orientation information are examined and are used with convolutional neural networks to classify five synthetic microstructures with varying textures and grain geometries. Of these, a spectral embedding of crystallographic orientations in a space that respects the crystallographic symmetries performs by far the best, even when the network is trained on small volumes of data such as could be accessible by practical experiments.

physics.comp-ph↗

arXiv2025

Disks, Surfaces, and Entanglement Percolation

We study the probability that a loop is null-homotopic -- that is, bounded by the continuous image of a disk -- in plaquette percolation on $\mathbb{Z}^3.$ Locally, the event that there is a ``horizontal disk crossing'' of a rectangular prism is dual to the event that there is a vertical crossing in entanglement percolation (with wired boundary conditions). However, the analysis of analogous events on the full lattice is complicated by the long-range nature of entanglement percolation. We show that the probability that a rectangular loop is contractible exhibits a phase transition from area law to perimeter law dual to the entanglement percolation threshold, conditional on a conjecture concerning the continuity of entanglement percolation thresholds with respect to truncation. We also show the continuity of a truncated entanglement percolation threshold in slabs and apply that to identify a regime where large plaquette surfaces exist but typically have many handles.

math.PR↗

arXiv2025

On the Continuity of Enhancement Percolation

We study bond percolation in $\mathbb{Z}^d$ with an unbounded family of enhancements that enable additional bonds to act as open. A natural question is whether percolation occurs in this model if and only if percolation also occurs in the system with a finite subcollection of enhancements. We give an affirmative answer in dimension $d=2$ for symmetric families of connected enhancements, and in dimensions $d\ge 3$ we prove a partial result.

math.PR↗

arXiv2024

A Metric on the Polycrystalline Microstructure State Space

Material microstructures are traditionally compared using sets of statistical measures that are incomplete, e.g., two visually distinct microstructures can have identical grain size distributions and phase fractions. While this is not a severe concern for materials fabricated by traditional means, the microstructures produced by advanced manufacturing methods can depend sensitively and unpredictably on the processing conditions. Moreover, the advent of computational materials design has increased the frequency of synthetic microstructure generation, and there is not yet a standard approach in the literature to validate the generated microstructures with experimental ones. This work proposes an idealized distance on the space of single-phase polycrystalline microstructures such that two microstructures that are close with respect to the distance exhibit statistically similar grain geometries in all respects below a user-specified length scale. Given a pair of micrographs, the distance is approximated by sampling windows from the micrographs, defining a distance between pairs of windows, and finding a window matching that minimizes the sum of pairwise window distances. The approach is used to compare a variety of synthetic microstructures and to develop a procedure to query a proof-of-concept database suitable for general single-phase polycrystalline microstructures.

physics.comp-ph↗

arXiv2024

Some Properties of the Plaquette Random-Cluster Model

We show that the $i$-dimensional plaqutte random-cluster model with coefficients in $\mathbb{Z}_q$ is dual to a $(d-i)$-dimensional plaquette random cluster model. In addition, we explore boundary conditions, infinite volume limits, and uniqueness for these models. For previously known results, we provide new proofs that rely more on the tools of algebraic topology.

math.PR↗

arXiv2024

A Sharp Deconfinement Transition for Potts Lattice Gauge Theory in Codimension Two

In 1983, Aizenman, Chayes, Chayes, Fröhlich, and Russo proved that $2$-dimensional Bernoulli plaquette percolation in $\mathbb{Z}^3$ exhibits a sharp phase transition for the event that a large rectangular loop is "bounded by a surface of plaquettes.'' We extend this result both to $(d-1)$-dimensional plaquette percolation in $\mathbb{Z}^d,$ and to a dependent model of plaquette percolation called the plaquette random-cluster model. As a consequence, we obtain a sharp phase transition for Wilson loop expectations in $(d-2)$-dimensional $q$-state Potts hyperlattice gauge theory on $\mathbb{Z}^d$ dual to that of the Potts model. Our proof is unconditional for Ising lattice gauge theory, but relies on a regularity conjecture for the random-cluster model in slabs when $q>2.$ We also further develop the general theory of the $i$-plaquette random cluster model and its relationship with $(i-1)$-dimensional Potts lattice gauge

math.PR↗

arXiv2023

Homological percolation on a torus: plaquettes and permutohedra

We study higher-dimensional homological analogues of bond percolation on a square lattice and site percolation on a triangular lattice. By taking a quotient of certain infinite cell complexes by growing sublattices, we obtain finite cell complexes with a high degree of symmetry and with the topology of the torus $\mathbb{T}^d$. When random subcomplexes induce nontrivial $i$-dimensional cycles in the homology of the ambient torus, we call such cycles \emph{giant}. We show that for every $i$ and $d$ there is a sharp transition from nonexistence of giant cycles to giant cycles spanning the homology of the torus. We also prove convergence of the threshold function to a constant in certain cases. In particular, we prove that $p_c=1/2$ in the case of middle dimension $i=d/2$ for both models. This gives finite-volume high-dimensional analogues of Kesten's theorems that $p_c=1/2$ for bond percolation on a square lattice and site percolation on a triangular lattice.

math.PR↗

arXiv2023

Topological Phases in the Plaquette Random-Cluster Model and Potts Lattice Gauge Theory

The $i$-dimensional plaquette random-cluster model on a finite cubical complex is the random complex of $i$-plaquettes with each configuration having probability proportional to $$p^{\text{# of plaquettes}}(1-p)^{\text{# of complementary plaquettes}}q^{\mathbf{ b}_{i-1}},$$ where $q\geq 1$ is a real parameter and $\mathbf{b}_{i-1}$ denotes the rank of the $(i-1)$-homology group with coefficients in a specified coefficient field. When $q$ is prime and the coefficient field is $\mathbb{F}_q$, this model is coupled with the $(i-1)$-dimensional $q$-state Potts lattice gauge theory. We prove that the probability that an $(i-1)$-cycle in $\mathbb{Z}^d$ is null-homologous in the plaquette random-cluster model equals the expectation of the corresponding generalized Wilson loop variable. This provides the first rigorous justification for a claim of Aizenman, Chayes, Chayes, Frölich, and Russo that there is an exact relationship between Wilson loop variables and the event that a loop is bounded by a surface in an interacting system of plaquettes. We also prove that the $i$-dimensional plaquette random-cluster model on the $2i$-dimensional torus exhibits a sharp phase transition at the self-dual point $p_{\mathrm{sd}} \mathrel{\vcenter{:}}= \frac{\sqrt{q}}{1+\sqrt{q}}$ in the sense of homological percolation. This implies a qualitative change in the generalized Swendsen--Wang dynamics from local to non-local behavior.

math.PR↗

arXiv2020

Topology and local geometry of the Eden model

The Eden cell growth model is a simple discrete stochastic process which produces a "blob" in $\mathbb{R}^d$: start with one cube in the regular grid, and at each time step add a neighboring cube uniformly at random. This process has been used as a model for the growth of aggregations, tumors, and bacterial colonies and the healing of wounds, among other natural processes. Here, we study the topology and local geometry of the resulting structure, establishing asymptotic bounds for Betti numbers. Our main result is that the Betti numbers grow at a rate between the conjectured rate of growth of the site perimeter and the actual rate of growth of the site perimeter. We also present the results of computational experiments on finer aspects of the geometry and topology, such as persistent homology and the distribution of shapes of holes.

math.PR↗

arXiv2020

Fractal Dimension and the Persistent Homology of Random Geometric Complexes

We prove that the fractal dimension of a metric space equipped with an Ahlfors regular measure can be recovered from the persistent homology of random samples. Our main result is that if $x_1,\ldots, x_n$ are i.i.d. samples from a $d$-Ahlfors regular measure on a metric space, and $E^0_α\left(x_1,\ldots,x_n\right)$ denotes the $α$-weight of the minimum spanning tree on $x_1,\ldots,x_n:$ \[E_α^0\left(x_1,\ldots,x_n\right)=\sum_{e\in T\left(x_1,\ldots,x_n\right)} |e|^α\,,\] then there exist constants $0<C_1\leq C_2$ so that \[C_1\leq n^{-\frac{d-α}{d}} E^0_α\left(x_1,\ldots,x_n\right)\leq C_2\,\] with high probability as $n\rightarrow \infty.$ In particular, \[\log\big(E^0_α(x_1,\ldots,x_n)\big)/\log(n)\longrightarrow (d-α)/d\,.\] This is a generalization of a result of Steele (1988) from the non-singular case to the fractal setting. Our result is best possible, in the sense that there exist Ahlfors regular measures for which the limit $\lim_{n\rightarrow\infty} n^{-\frac{d-α}{d}} E^0_α\left(x_1,\ldots,x_n\right)$ does not exist with high probability. We also prove analogous results for weighted sums defined in terms of higher dimensional persistent homology.

math.PR↗

arXiv2019

Statistical Topology of Bond Networks with Applications to Silica

Whereas knowledge of a crystalline material's unit cell is fundamental to understanding the material's properties and behavior, there are not obvious analogues to unit cells for disordered materials despite the frequent existence of considerable medium-range order. This article views a material's structure as a collection of local atomic environments that are sampled from some underlying probability distribution of such environments, with the advantage of offering a unified description of both ordered and disordered materials. Crystalline materials can then be regarded as special cases where the underlying probability distribution is highly concentrated around the traditional unit cell. Four descriptors of local atomic environments suitable for disordered bond networks are proposed and applied to molecular dynamics simulations of silica glasses. Each of them reliably distinguishes the structure of glasses produced at different cooling rates, with the $H_1$ barcode and coordination profile providing the best separation.

cs.CG↗

arXiv2019

Fractal Dimension Estimation with Persistent Homology: A Comparative Study

We propose that the recently defined persistent homology dimensions are a practical tool for fractal dimension estimation of point samples. We implement an algorithm to estimate the persistent homology dimension, and compare its performance to classical methods to compute the correlation and box-counting dimensions in examples of self-similar fractals, chaotic attractors, and an empirical dataset. The performance of the $0$-dimensional persistent homology dimension is comparable to that of the correlation dimension, and better than box-counting.

math.DS↗