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Exponential decay of correlations at high temperature in $H^{2|2n}$ nonlinear sigma models

Margherita Disertori, Javier Durán Fernández, Luca Fresta

Abstract

We consider a family of nonlinear sigma models on $\mathbb{Z}^{d}$ whose target space is the hyperbolic super manifold $H^{2|2n}$, $n >1$, introduced by Crawford as an extension of Zirnbauer's $H^{2|2}$ model for disordered systems. We prove exponential decay of the two-point correlation function in the high-temperature regime $β\leq C n^{-1}$, with $C>0$ a universal constant, for any $n>1$ and any dimension $d\geq 1$, with mass $\log β^{-1}$. We also consider models with long-range interaction and prove fast decay in the same high-temperature regime. The proof is based on the reduction to a marginal fermionic theory and combines a high-temperature cluster expansion, exact combinatorics and bounds derived via Grassmann norms.

Exponential decay of correlations at high temperature in $H^{2|2n}$ nonlinear sigma models

Abstract

We consider a family of nonlinear sigma models on whose target space is the hyperbolic super manifold , , introduced by Crawford as an extension of Zirnbauer's model for disordered systems. We prove exponential decay of the two-point correlation function in the high-temperature regime , with a universal constant, for any and any dimension , with mass . We also consider models with long-range interaction and prove fast decay in the same high-temperature regime. The proof is based on the reduction to a marginal fermionic theory and combines a high-temperature cluster expansion, exact combinatorics and bounds derived via Grassmann norms.

Paper Structure

This paper contains 13 sections, 17 theorems, 134 equations, 1 figure.

Table of Contents

  1. Introduction
  2. The model and its probabilistic marginals
  3. Main result
  4. Proof of the main theorem
  5. High-temperature expansion
  6. Grassmann norms
  7. Proof of the main result from the activity estimates
  8. Norm estimates
  9. Single-point model
  10. Proof of the optimal activity estimates
  11. Proof of the high-temperature bound
  12. $H^{0|2}$ model and the arboreal gas
  13. Acknowledgements

Key Result

Theorem 1.1

Let $\langle \,\cdot\,\rangle^{\Lambda}_{\beta,\varepsilon,n}$ denote the Gibbs measure of the $H^{2|2n}$ nonlinear sigma model on $\Lambda \subset \mathbb{Z}^{d}$ finite box at inverse temperature $\beta >0$ and external field $\varepsilon >0$. For all $d \in \mathbb{N}$ there exists a constant $C_

Figures (1)

  • Figure :

Theorems & Definitions (35)

  • Theorem 1.1
  • Remark 1.2
  • Remark 1.3: Extensions to other observables
  • Lemma 2.1
  • proof
  • Lemma 2.2: High-temperature expansion
  • Remark 2.3
  • proof
  • Proposition 2.4
  • Lemma 2.5: $\ell^{1}$-type norm Fresta2021
  • ...and 25 more