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The $H^{2|2}$ monotonicity theorem revisited

Yichao Huang, Xiaolin Zeng

Abstract

We use supersymmetric localization and integration by parts to derive variational and convex correlation inequalities in statistical physics. As a primary application, we give an alternative proof of the monotonicity theorem for the $H^{2|2}$ supersymmetric hyperbolic sigma model. This recovers a result of Poudevigne without relying on probabilistic couplings.

The $H^{2|2}$ monotonicity theorem revisited

Abstract

We use supersymmetric localization and integration by parts to derive variational and convex correlation inequalities in statistical physics. As a primary application, we give an alternative proof of the monotonicity theorem for the supersymmetric hyperbolic sigma model. This recovers a result of Poudevigne without relying on probabilistic couplings.

Paper Structure

This paper contains 14 sections, 7 theorems, 67 equations.

Table of Contents

  1. Introduction
  2. Overview of our method
  3. A short proof of the H2|2 monotonicity theorem
  4. Sign convention
  5. Slepian-type inequality with supersymmetry
  6. Supersymmetric proof of the H2|2 monotonicity theorem
  7. Definition and statement
  8. Example: rooted one-point graph
  9. General H2|2 monotonicity theorem
  10. A useful lemma
  11. Boundary variations
  12. Bulk variations
  13. Replacing Poudevigne's coupling argument by supersymmetric integration by parts
  14. Perspective function of a jointly convex function

Key Result

Theorem 1

Fix an integer $N\geq 1$, consider the vertex set $V=\{1,\dots,N,\delta\}$ with a root vertex $\delta$, unoriented edges $(ij)=(ji)$ for $i,j\in V$, and symmetric positive edge weights $W_{ij}=W_{ji}>0$ for $i,j\in V$. Define for $t=(t_1,\dots,t_N)$ and $t_{\delta}=0$, where the sum is over the spanning trees $\mathcal{T}$ of the complete graph over $\{1,\dots,N,\delta\}$. Consider the effective

Theorems & Definitions (16)

  • Theorem 1: Monotonicity theorem for the $H^{2|2}$ model
  • Remark 2
  • Proposition 3: The simplest Gaussian convex inequality
  • proof
  • Theorem 4: Monotonicity theorem for $H^{2|2}$ MR4721024
  • Remark 5
  • Theorem 6: Generalized form of the $H^{2|2}$ monotonicity theorem
  • Lemma 7: Switching lemma
  • proof
  • proof : Proof of Theorem \ref{['th:better']} in the boundary case
  • ...and 6 more