arXiv subjects

Catherine Wolfram

Publications and source records attributed to Catherine Wolfram.

4 recordsLinked to original sources

arXiv2025

The multinomial dimer model

The dimer model is a classical statistical mechanics model which is exactly solvable in two dimensions, but about which little is known in higher dimensions. In analogy with large $N$ limits in lattice gauge theory, we study a large $N$ limit of the dimer model in any dimension $d$. The dependence on $N$ comes from the multinomial tiling model introduced by Kenyon and Pohoata, which gives a general framework for adding a dependence on $N$ to a tiling model. We study the behavior of this model on periodic bipartite graphs in ${\mathbb R}^d$, in the scaling limit as the multiplicity $N$ and then the size of the graph go to infinity. In this iterated limit, in any dimension $d$, we prove a variational principle and show that random configurations concentrate on a limit shape which is the unique solution to an associated system of Euler-Lagrange equations. The rate function of the variational principle is the integral of a surface tension function, which we can compute explicitly for lattices in any dimension $d$ as the Legendre dual of the free energy for the model on the torus. We give a unified methodology for computing the surface tension and Euler-Lagrange equations in any dimension $d$. A new structure called the critical gauge also emerges in the large $N$ limit. We show that the critical gauge functions converges in the scaling limit to a limiting gauge function which is the unique solution to a dual Euler-Lagrange equation. This limiting gauge function determines the limit shape and vice versa. We further use our techniques to compute explicit limit shapes in some two and three dimensional examples, such as the Aztec diamond and "Aztec cuboid". This is one of the first stat mech models in dimensions $d\ge3$ where limit shapes can be computed explicitly.

math.PR↗

arXiv2025

Epstein curves and holography of the Schwarzian action

We apply Epstein's construction of hypersurfaces in the hyperbolic disk $\mathbb D$ to prove identities between the Schwarzian action on $\operatorname{PSL}_2(\mathbb R)\backslash \mathrm{Diff}^3 (\mathbb S^1)$, the length of the corresponding Epstein curve in $\mathbb D$, and the area enclosed by the Epstein curve. These results are inspired by the holographic duality between Jackiw--Teitelboim gravity and Schwarzian field theory. We also show that the horocycle truncation used in the construction of the Epstein curve defines a renormalized length of hyperbolic geodesics in $\mathbb D$, which coincides with the logarithm of the bi-local observable of Schwarzian field theory. The construction of the Epstein curve also extends to the coadjoint orbits $\operatorname{PSL}_2^{(n)}(\mathbb R)\backslash \mathrm{Diff}^3 (\mathbb S^1)$, and we obtain the same identities for the analog of the Schwarzian action on these coadjoint orbits. Furthermore, we show that the Schwarzian action is the derivative of the Loewner energy of the welded Jordan curve. This energy is the action functional of Schramm--Loewner evolutions and holographically expressed as a renormalized volume in hyperbolic $3$-space. As a by-product of these relations, we obtain two immediate proofs of the non-negativity of the Schwarzian action using the isoperimetric inequality and the monotonicity of the Loewner energy.

math-ph↗

arXiv2023

Circle homeomorphisms with square summable diamond shears

We introduce and the study the space of homeomorphisms of the circle (up to Möbius transformations) which are in $\ell^2$ with respect to modular coordinates called diamond shears along the edges of the Farey tessellation. Diamond shears are related combinatorially to shear coordinates, and are also closely related to the $\log Λ$-lengths of decorated Teichmüller space introduced by Penner. We obtain sharp results comparing this new class to the Weil-Petersson class and Hölder classes of circle homeomorphisms. We also express the Weil-Petersson metric tensor and symplectic form in terms of infinitesimal shears and diamond shears.

math.CV↗

arXiv2023

Large deviations for the 3D dimer model

In 2000, Cohn, Kenyon and Propp studied uniformly random perfect matchings of large induced subgraphs of $\mathbb Z^2$ (a.k.a. dimer configurations or domino tilings) and developed a large deviation theory for the associated height functions. We establish similar results for large induced subgraphs of $\mathbb Z^3$. To formulate these results, recall that a perfect matching on a bipartite graph induces a flow that sends one unit of current from each even vertex to its odd partner. One can then subtract a "reference flow'' to obtain a divergence-free flow. We show that the flow induced by a uniformly random dimer configuration converges in law (when boundary conditions on a bounded $R \subset \mathbb R^3$ are controlled and the mesh size tends to zero) to the deterministic divergence-free flow $g$ on $R$ that maximizes $$\int_{R} \text{ent}(g(x)) \,dx$$ given the boundary data, where $\text{ent}(s)$ is the maximal specific entropy obtained by an ergodic Gibbs measure with mean current $s$. The function $\text{ent}$ is not known explicitly, but we prove that it is continuous and {\em strictly concave} on the octahedron $\mathcal O$ of possible mean currents (except on the edges of $\mathcal O$) which implies (under reasonable boundary conditions) that the maximizer is uniquely determined. We further establish two versions of a large deviation principle, using the integral above to quantify how exponentially unlikely the discrete random flows are to approximate other deterministic flows. The planar dimer model is mathematically rich and well-studied, but many of the most powerful tools do not seem readily adaptable to higher dimensions. Our analysis begins with a smaller set of tools, which include Hall's matching theorem, the ergodic theorem, non-intersecting-lattice-path formulations, and double-dimer cycle swaps.

math.PR↗