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Spin systems with hyperbolic symmetry: a survey

Roland Bauerschmidt, Tyler Helmuth

Abstract

Spin systems with hyperbolic symmetry originated as simplified models for the Anderson metal--insulator transition, and were subsequently found to exactly describe probabilistic models of linearly reinforced walks and random forests. In this survey we introduce these models, discuss their origins and main features, some existing tools available for their study, recent probabilistic results, and relations to other well-studied probabilistic models. Along the way we discuss some of the (many) open questions that remain.

Spin systems with hyperbolic symmetry: a survey

Abstract

Spin systems with hyperbolic symmetry originated as simplified models for the Anderson metal--insulator transition, and were subsequently found to exactly describe probabilistic models of linearly reinforced walks and random forests. In this survey we introduce these models, discuss their origins and main features, some existing tools available for their study, recent probabilistic results, and relations to other well-studied probabilistic models. Along the way we discuss some of the (many) open questions that remain.

Paper Structure

This paper contains 29 sections, 13 theorems, 46 equations.

Table of Contents

  1. Introduction
  2. Magic Formula for Edge-Reinforced Random Walk
  3. Hyperbolic spin systems
  4. The $\mathbb{H}^{2|0}$ model
  5. The $\mathbb{H}^{0|2}$ model
  6. The $\mathbb{H}^{2|2}$ model
  7. Beyond: $\mathbb{H}^{n|2m}$
  8. Symmetries
  9. Spontaneous symmetry breaking
  10. Horospherical coordinates
  11. Physical background: Anderson transition
  12. Dictionary
  13. Linearly reinforced walks and $\mathbb{H}^{2|2}$
  14. Hyperbolic symmetry and the VRJP
  15. Isomorphism theorems
  16. ...and 14 more sections

Key Result

theorem 1.1

Let $G=(\Lambda,E)$ be finite. Edge-reinforced random walk with $X_{0}=0$ and initial weights $\alpha= (\alpha_{ij})$ is a random walk in random environment: The environment $\mu_\alpha$ has density proportional to with respect to Lebesgue measure on the unit simplex in $[0,\infty)^{E}$, and where $\alpha_{i}=\sum_{j}\alpha_{ij}$.

Theorems & Definitions (14)

  • theorem 1.1: Magic formula for ERRW
  • theorem 2.1: SUSY localisation for $\mathbb{H}^{2|2}$
  • theorem 4.1: Magic formula for VJRP MR3420510
  • theorem 4.2: SSB for $\mathbb{H}^{2|2}$ MR2728731
  • theorem 4.3: Localisation for $\mathbb{H}^{2|2}$ MR2736958
  • theorem 4.4
  • proof : Sketch of proof
  • theorem 4.5
  • theorem 4.6
  • theorem 4.7
  • ...and 4 more