Abstract
We study the dependence of the spectral gap for the generator of the Ginzburg–Landau dynamics for all mathcal O(n)-models with mean-field interaction and magnetic field, below and at the critical temperature on the number N of particles. For our analysis of the Gibbs measure, we use a one-step renormalization approach and semiclassical methods to study the eigenvalue-spacing of an auxiliary Schrödinger operator.
Highlights
Acting on spin configurations σ : {1, .., N } → spin σ : (Sn)−1 where MF is the mean-field Laplacian and h ∈ Rn an external magnetic field
For our study of spectral gaps, we consider the Ginzburg–Landau dynamics associated with the Gibbs measure dρ ∝ e−β H(σ ) with Hamilton function (1.1)
Inspired by the method in [BB19], we invoke the same one-step renormalization group procedure to reduce the high-dimensional problem to the study of a low-dimensional renormalized measure and a fluctuation measure
Summary
Acting on spin configurations σ : {1, .., N } → Sn−1 where MF is the mean-field Laplacian and h ∈ Rn an external magnetic field. The renormalized single spin potential Vn associated with the mean-field O(n)-model for φ ∈ Rn is defined as We start by showing that the inverse spectral gap in the Ising model in the case of subcritical magnetic fields, i.e. h < hc(β), converges at most exponentially fast to zero as the number of spins, N , increases.
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