Large Deviation Analysis for Canonical Gibbs Measures

TL;DR

The paper develops a large-deviation framework for canonical Gibbs point processes in the continuum, focusing on binomial bases and growing periodic windows. By conditioning a Poisson process on a fixed point count and introducing a novel move-based coupling to mimic conditioning, the authors derive a full LDP for bounded local observables (and bounds for unbounded cases) across three Gibbs models: local bounded interactions, nonnegative increasing unbounded interactions, and hard-core interactions. The main results include explicit rate functions of the form I(P)+P^o(V) and a finite limit for the partition function Z_n, with extensions to varying boundary conditions and concrete examples such as the binomial Strauss process. These results bridge the gap between grand-canonical and canonical analyses, providing tools for understanding free-energy-like quantities and tail behaviors in canonical continuum Gibbs systems.

Abstract

In this paper, we present a large-deviation theory developed for functionals of canonical Gibbs processes, i.e., Gibbs processes with respect to the binomial point process. We study the regime of a fixed intensity in a sequence of increasing windows. Our method relies on the traditional large-deviation result for local bounded functionals of Poisson point processes noting that the binomial point process is obtained from the Poisson point process by conditioning on the point number. Our main methodological contribution is the development of coupling constructions allowing us to handle delicate and unlikely pathological events. The presented results cover three types of Gibbs models - a model given by a bounded local interaction, a model given by a non-negative possibly unbounded increasing local interaction and the hard-core interaction model. The derived large deviation principle is formulated for the distributions of individual empirical fields driven by canonical Gibbs processes, with its special case being a large deviation principle for local bounded observables of the canonical Gibbs processes. We also consider unbounded non-negative increasing local observables, but the price for treating this more general case is that we only get large-deviation bounds for the tails of such observables. Our primary setting is the one with periodic boundary condition, however, we also discuss generalizations for different choices of the boundary condition.

Figures (1)

• Figure 1: Graphic representation of the partition of the window $W_n$ from the proof of Lemma \ref{['lemma:rarebdensepoints']} in dimension $d = 2$. Left: The partition corresponding to the choice $\left\lfloor\frac{w_n}{L}\right\rfloor = 2$, i.e. $c_n = 3^2 \left\lfloor\frac{w_n}{L}\right\rfloor^2 =36$. Different shades of gray correspond to the partition of the indices $\{1,\dots,36\}$ into subsets $A_1,\dots, A_9$ of size $K_n =\left\lfloor\frac{w_n}{L}\right\rfloor^2= 4$. Right: A point configuration $\omega$ (round points) in the partitioned window $W_n$ into cubes $Q_{a_n}(z_{1,n}),\dots, Q_{a_n}(z_{36,n})$. The crosses are the centers of the cubes $z_{i,n}$. Highlighted cubes correspond to the cube $Q_{3a_n}^{(n)}(z_{1,n})$. For this configuration we get that $M_{1,n}(\omega) = 0$ and $M'_{1,n}(\omega) = 11$.

Theorems & Definitions (84)

• Definition 2.1
• Definition 2.2
• Definition 2.3
• Definition 2.4
• Lemma 2.5
• proof
• Lemma 2.6
• proof
• Theorem 2.7: The LDP in the bounded case
• Remark 2.8