arXiv subjects

Paul Duncan

Publications and source records attributed to Paul Duncan.

11 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↗

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↗

arXiv2025

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↗

arXiv2025

Reality Check: A New Evaluation Ecosystem Is Necessary to Understand AI's Real World Effects

Conventional AI evaluation approaches concentrated within the AI stack exhibit systemic limitations for exploring, navigating and resolving the human and societal factors that play out in real world deployment such as in education, finance, healthcare, and employment sectors. AI capability evaluations can capture detail about first-order effects, such as whether immediate system outputs are accurate, or contain toxic, biased or stereotypical content, but AI's second-order effects, i.e. any long-term outcomes and consequences that may result from AI use in the real world, have become a significant area of interest as the technology becomes embedded in our daily lives. These secondary effects can include shifts in user behavior, societal, cultural and economic ramifications, workforce transformations, and long-term downstream impacts that may result from a broad and growing set of risks. This position paper argues that measuring the indirect and secondary effects of AI will require expansion beyond static, single-turn approaches conducted in silico to include testing paradigms that can capture what actually materializes when people use AI technology in context. Specifically, we describe the need for data and methods that can facilitate contextual awareness and enable downstream interpretation and decision making about AI's secondary effects, and recommend requirements for a new ecosystem.

cs.CY↗

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↗

arXiv2014

Understanding Machine-learned Density Functionals

Kernel ridge regression is used to approximate the kinetic energy of non-interacting fermions in a one-dimensional box as a functional of their density. The properties of different kernels and methods of cross-validation are explored, and highly accurate energies are achieved. Accurate {\em constrained optimal densities} are found via a modified Euler-Lagrange constrained minimization of the total energy. A projected gradient descent algorithm is derived using local principal component analysis. Additionally, a sparse grid representation of the density can be used without degrading the performance of the methods. The implications for machine-learned density functional approximations are discussed.

physics.chem-ph↗

arXiv2011

Pattern avoidance in ascent sequences

Ascent sequences are sequences of nonnegative integers with restrictions on the size of each letter, depending on the number of ascents preceding it in the sequence. Ascent sequences have recently been related to (2+2)-free posets and various other combinatorial structures. We study pattern avoidance in ascent sequences, giving several results for patterns of lengths up to 4, for Wilf equivalence and for growth rates. We establish bijective connections between pattern avoiding ascent sequences and various other combinatorial objects, in particular with set partitions. We also make a number of conjectures related to all of these aspects.

math.CO↗