Large Deviations in the Spherical Model: The Rate Functions

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 $k^{d-2}$.