Large Deviations in the Spherical Model: The Rate Functions

TL;DR

This paper analyzes large deviations of magnetization in the spherical model of a ferromagnet at low temperatures, deriving explicit rate functions for various domain geometries and revealing how these probabilities scale with system size and domain shape.

Contribution

It provides explicit rate functions for large deviations in the spherical model across different domain geometries at low temperature, highlighting the influence of domain shape and size.

Findings

Large deviation probabilities decay exponentially with system size for layers and blocks.

For rods, probabilities decay exponentially with a correction involving log n.

Rate functions depend explicitly on domain geometry and size.

Abstract

We study the spherical model of a ferromagnet in $d$-dimensional cubes $\Omega_n$ of volume $|\Omega_n|=n^d$ and investigate large deviations of the magnetization of various domains $D_k\subset \Omega_n$. We focus our attention on the low-temperature regime, $T<T_c$, and consider domains $D_k$ of three types: $(d-1)$-dimensional layers of width $k$, $(d-2)$-dimensional rods, and Kadanoff blocks. In the case of layers the large-deviation probabilities decay exponentially with $n^{d-2}$, and we obtain an explicit expression for the corresponding rate function. When the layer width $k\ll n$, the large-deviation probabilities are virtually independent of $k$. In the case of rods the probabilities of large deviations exhibit similar exponential decay, but this time it is distorted by $\log n$ corrections. In the case of Kadanoff blocks of size $k$ the large-deviation probabilities decay exponentially with $k^{d-2}$.