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Six-vertex model with rare corners and random restricted permutations

Vadim Gorin, Richard Kenyon

TL;DR

This work analyzes the $c\to0$ limit of the six-vertex model with domain-wall-like boundaries and its equivalence to restricted Mallows permutations, formulating a variational principle for the resulting limit shapes. The limit shapes are governed by a permuton energy and the Euler-Lagrange equations, which in general produce piecewise-algebraic, often discontinuous solutions controlled by a hyperbolic Liouville PDE. For convex restriction data, the EL system reduces to a tractable algebraic set that admits explicit closed-form descriptions and a constructive algorithm, including a Sinkhorn-like method at $r=0$ and a numerical continuation for small $|r|$. The paper provides extensive explicit examples, including convex and non-convex restriction matrices and a non-rectangular domain, illustrating the versatility of the approach and its potential for explicit limit-shape computations in a non-free-fermion regime.

Abstract

We study limit shapes in two equivalent models: the six-vertex model in the $c\to0$ limit and the random Mallows permutation with restricted permutation matrix. We give the Euler-Lagrange equation for the limit shape and show how to solve it for a class of rectilinear polygonal domains. Its solutions are given by piecewise-algebraic functions with lines of discontinuities.

Six-vertex model with rare corners and random restricted permutations

TL;DR

This work analyzes the limit of the six-vertex model with domain-wall-like boundaries and its equivalence to restricted Mallows permutations, formulating a variational principle for the resulting limit shapes. The limit shapes are governed by a permuton energy and the Euler-Lagrange equations, which in general produce piecewise-algebraic, often discontinuous solutions controlled by a hyperbolic Liouville PDE. For convex restriction data, the EL system reduces to a tractable algebraic set that admits explicit closed-form descriptions and a constructive algorithm, including a Sinkhorn-like method at and a numerical continuation for small . The paper provides extensive explicit examples, including convex and non-convex restriction matrices and a non-rectangular domain, illustrating the versatility of the approach and its potential for explicit limit-shape computations in a non-free-fermion regime.

Abstract

We study limit shapes in two equivalent models: the six-vertex model in the limit and the random Mallows permutation with restricted permutation matrix. We give the Euler-Lagrange equation for the limit shape and show how to solve it for a class of rectilinear polygonal domains. Its solutions are given by piecewise-algebraic functions with lines of discontinuities.
Paper Structure (19 sections, 18 theorems, 112 equations, 10 figures)

This paper contains 19 sections, 18 theorems, 112 equations, 10 figures.

Table of Contents

  1. Introduction
  2. Problem setup
  3. Six-vertex model with rare corners
  4. Restricted random permutations
  5. Between the six-vertex and permutation points of view
  6. Asymptotic theorems
  7. Asymptotic regime
  8. Variational principle
  9. Maximizers via Euler-Lagrange equations
  10. Smooth solutions to Euler-Lagrange equations
  11. Consequences of Definition \ref{['Definition_smooth_solution']}
  12. Algebraic equations for convex $I$
  13. The case $r=0$
  14. Numeric algorithms for small $r$
  15. Domains with unique solution to algebraic equations
  16. ...and 4 more sections

Key Result

Theorem 2.11

Take the $(X,Y,I)$ data of Definitions Def_restrictions_6v, Def_restrictions_perm with convex array $I$ and assume that there exists at least one permutation restricted by $\Omega^{X,Y,I}$. Consider the correspondence between configurations $\sigma^{6v}$ of the six-vertex model in domain $\Omega^{X,

Figures (10)

  • Figure 1: The six types of vertices and their weights.
  • Figure 2: $N\times N$ square with Domain Wall Boundary Conditions for $N=5$ and two possible configurations of paths with $c$--type vertices emphasized.
  • Figure 3: Domain $\Omega^{X,Y,I}$ with $N=6$, $k=\ell=3$, $X=(0,2,4,6)$, $Y=(0,2,4,6)$, $I=110111011$, and one possible configuration of paths with $c$--type vertices emphasized and the values of the height function in light gray.
  • Figure 4: Domain $\Omega^{X,Y,I}$ as in Figure \ref{['Figure_domain_ex']}, but with $I=110011011$.
  • Figure 5: Possible permutations for two choices of $\Omega^{X,Y,I}$ with $N=6$, $k=\ell=3$, $X=(0,2,4,6)$, $Y=(0,2,4,6)$. Points $(\sigma(m),m)$ are prohibited in red shaded regions. $I=111101111$ on the left and $I=110111011$ on the right with $\sigma=(152653)$ and $\sigma=(546132)$, respectively.
  • ...and 5 more figures

Theorems & Definitions (64)

  • Definition 2.1
  • Remark 2.2
  • Example 2.3
  • Definition 2.4
  • Definition 2.5
  • Remark 2.7
  • Definition 2.8
  • Remark 2.9
  • Remark 2.10
  • Theorem 2.11
  • ...and 54 more