Critical Properties and Glass Transitions in Randomly Coupled Fields

Abstract

We study an assembly of $N$ scalar fields coupled via a quenched random interaction, equivalent to a spin glass whose spins are promoted to $d$-dimensional fields. When $N$ is large, the random coupling matrix produces a family of Gaussian universality classes whose critical exponents are determined by the behavior of the eigenvalue density near the spectral edge. For a Wigner matrix, the correlation length diverges at criticality but the susceptibility remains finite, and one-loop replica analysis supports an upper critical dimension of unity. In an exactly solvable spherical variant, the heat capacity jumps at the transition in $d>1$, and correlations remain pinned to their critical form throughout the glass phase, giving generic scale invariance despite the absence of a Goldstone mode.