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A few observations around Gaussian domination and continuous symmetry breaking for spin O(N) model

Xiao Han

TL;DR

This work analyzes Gaussian domination for the spin $O(N)$ model on general finite graphs, introducing a modified partition function $Z^*_{G,N,\beta}(h)$ and proving that Gaussian domination yields a concrete lower bound on spin correlations in terms of the Green's function: $\mathbb{E}_{\mu_{G,N,\beta}} \sigma_x \cdot \sigma_y \ge 1 - \frac{N u_{xy}(x)}{2\beta d(x)}$. It establishes a phase picture with high- and low-temperature regimes where domination holds on arbitrary finite graphs, and proves that at low temperature the rescaled spin components converge to a vector-valued Gaussian free field, with moment convergence and rate estimates. The paper also provides several counterexamples showing that Gaussian domination is not robust to small graph changes, can fail on simple structures like binary trees, and cannot be deduced from a merely bounded Green's function. These results illuminate the boundaries of elementary methods for Gaussian domination and deepen understanding of when Gaussian domination can imply long-range order in $O(N)$ spin systems beyond highly symmetric settings.

Abstract

We investigate the notion of Gaussian domination for the spin $O(N)$ model on general finite graphs. We begin by proving a general inequality for spin correlations under the assumption of Gaussian domination, which directly implies long-range order at low temperatures for graphs with bounded Green's function. Usually, Gaussian domination is proved via reflection positivity, but this requires strict symmetries and is very rigid. In this article we also probe the boundaries of elementary methods for proving Gaussian domination. Although we did not find a way to get uniform bounds, we do offer new views for Gaussian domination at low and high temperatures for finite graphs, and a few counterexamples illustrating the interplay between correlation estimates and Gaussian domination and how local changes in the graph structure can affect the presence of Gaussian domination.

A few observations around Gaussian domination and continuous symmetry breaking for spin O(N) model

TL;DR

This work analyzes Gaussian domination for the spin model on general finite graphs, introducing a modified partition function and proving that Gaussian domination yields a concrete lower bound on spin correlations in terms of the Green's function: . It establishes a phase picture with high- and low-temperature regimes where domination holds on arbitrary finite graphs, and proves that at low temperature the rescaled spin components converge to a vector-valued Gaussian free field, with moment convergence and rate estimates. The paper also provides several counterexamples showing that Gaussian domination is not robust to small graph changes, can fail on simple structures like binary trees, and cannot be deduced from a merely bounded Green's function. These results illuminate the boundaries of elementary methods for Gaussian domination and deepen understanding of when Gaussian domination can imply long-range order in spin systems beyond highly symmetric settings.

Abstract

We investigate the notion of Gaussian domination for the spin model on general finite graphs. We begin by proving a general inequality for spin correlations under the assumption of Gaussian domination, which directly implies long-range order at low temperatures for graphs with bounded Green's function. Usually, Gaussian domination is proved via reflection positivity, but this requires strict symmetries and is very rigid. In this article we also probe the boundaries of elementary methods for proving Gaussian domination. Although we did not find a way to get uniform bounds, we do offer new views for Gaussian domination at low and high temperatures for finite graphs, and a few counterexamples illustrating the interplay between correlation estimates and Gaussian domination and how local changes in the graph structure can affect the presence of Gaussian domination.

Paper Structure

This paper contains 9 sections, 18 theorems, 84 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Main results
  3. From Gaussian domination to long-range correlations
  4. Phase transition for Gaussian domination
  5. Gaussian domination at high temperature
  6. Gaussian domination at low temperature
  7. A Counterexample of Gaussian domination
  8. A Counterexample in binary trees
  9. From bounded Green's function to symmetric breaking

Key Result

Theorem 2.1

Given $N\geq1$, a connected graph $G=(V, E)$, and $\beta \in [0, +\infty)$, if there is a Gaussian domination in the spin $O(N)$ model on $G$ at inverse temperature $\beta$, we have that $\mathbb{E}_{\mu_{G,N,\beta}} \sigma_x \cdot \sigma_y \geq 1-\frac{Nu_{xy}(x)}{2\beta d(x)}$ for any $x,y \in V$.

Figures (3)

  • Figure 1: Counterexample
  • Figure 2: Binary Tree
  • Figure 3: Parallel Paths

Theorems & Definitions (33)

  • Definition 1: Spin $O(N)$ model
  • Definition 2: Modified partition function
  • Definition 3: Gaussian domination
  • Theorem 2.1
  • Theorem 2.2
  • Theorem 2.3
  • Theorem 2.4
  • Theorem 2.5
  • Theorem 2.6
  • Theorem 2.7
  • ...and 23 more