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Algorithms for the ferromagnetic Potts model on expanders

Charlie Carlson, Ewan Davies, Nicolas Fraiman, Alexandra Kolla, Aditya Potukuchi, Corrine Yap

TL;DR

Algorithms for approximating the partition function of the ferromagnetic Potts model on d-regular expanding graphs, using extremal graph theory and applications of Karger's algorithm to counting cuts that may be of independent interest are given.

Abstract

We give algorithms for approximating the partition function of the ferromagnetic $q$-color Potts model on graphs of maximum degree $d$. Our primary contribution is a fully polynomial-time approximation scheme for $d$-regular graphs with an expansion condition at low temperatures (that is, bounded away from the order-disorder threshold). The expansion condition is much weaker than in previous works; for example, the expansion exhibited by the hypercube suffices. The main improvements come from a significantly sharper analysis of standard polymer models; we use extremal graph theory and applications of Karger's algorithm to count cuts that may be of independent interest. It is \#BIS-hard to approximate the partition function at low temperatures on bounded-degree graphs, so our algorithm can be seen as evidence that hard instances of \#BIS are rare. We also obtain efficient algorithms in the Gibbs uniqueness region for bounded-degree graphs. While our high temperature proof follows more standard polymer model analysis, our result holds in the largest known range of parameters $d$ and $q$.

Algorithms for the ferromagnetic Potts model on expanders

TL;DR

Algorithms for approximating the partition function of the ferromagnetic Potts model on d-regular expanding graphs, using extremal graph theory and applications of Karger's algorithm to counting cuts that may be of independent interest are given.

Abstract

We give algorithms for approximating the partition function of the ferromagnetic -color Potts model on graphs of maximum degree . Our primary contribution is a fully polynomial-time approximation scheme for -regular graphs with an expansion condition at low temperatures (that is, bounded away from the order-disorder threshold). The expansion condition is much weaker than in previous works; for example, the expansion exhibited by the hypercube suffices. The main improvements come from a significantly sharper analysis of standard polymer models; we use extremal graph theory and applications of Karger's algorithm to count cuts that may be of independent interest. It is \#BIS-hard to approximate the partition function at low temperatures on bounded-degree graphs, so our algorithm can be seen as evidence that hard instances of \#BIS are rare. We also obtain efficient algorithms in the Gibbs uniqueness region for bounded-degree graphs. While our high temperature proof follows more standard polymer model analysis, our result holds in the largest known range of parameters and .
Paper Structure (27 sections, 23 theorems, 99 equations, 2 figures)

This paper contains 27 sections, 23 theorems, 99 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Our results
  3. An overview of techniques
  4. Related work
  5. Organization
  6. Preliminaries
  7. Abstract polymer models
  8. Algorithms from convergent cluster expansions
  9. Low-Temperature Algorithms
  10. Sketch of low-temperature argument
  11. Polymer model
  12. Proof of Lemma \ref{['lem:polymerpf']}
  13. The triangle-free case
  14. Proof of Lemma \ref{['lem:polymerapprox']}
  15. Enumerative results
  16. ...and 12 more sections

Key Result

Theorem 1

For every $\epsilon > 0$, there exists $d_0(\epsilon)$ such that for $d \ge d_0(\epsilon)$, $q \ge d^c$ where $c$ is an absolute constant, and positive $\beta \notin ((1-\epsilon)\beta_o, (1+\epsilon)\beta_o)$, there exist for $G$ in the class of $d$-regular $2$-expander graphs, and for $G$ in the class of triangle-free $d$-regular $1$-expander graphs.

Figures (2)

  • Figure 1: The three different cases for an edge $e$ to be in $N_J(u_i) \cap N_J(w_i)$
  • Figure 2: An example of part of the path $P'$ in red arrows.

Theorems & Definitions (41)

  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Theorem 4
  • Theorem 5
  • Theorem 6: Kotecký--Preiss KP86
  • Theorem 7: JKP20 and HPR19b
  • Lemma 8
  • Lemma 9
  • Lemma 10
  • ...and 31 more