Correlation length lower bound for the random-field Potts model with the greedy lattice animal

Abstract

Motivated by recent developments over the past few years in the study of the correlation length of the random-field Ising model due to Ding and Wirth in a paper first available in 2020, we pursue one natural direction of research that the authors propose is of interest, namely in confirming that the same scaling for the correlation length for the random-field Ising model equals that of the random-field Potts model. To demonstrate that the $\frac{4}{3}$ emergence in the correlation length scaling for the random-field Potts model coincides with that correlation length scaling for the random-field Ising model, we refer to arguments due to Talagrand, in which an upper bound on the greedy lattice animal readily provides a corresponding lower bound on the correlation length. Contributions from the $\frac{4}{3}$ exponent appearing in the correlation length scaling for the random-field Ising, and random-field Potts models alike are reminiscent of $\frac{4}{3}$ exponents in upper bounds taken under the Liouville quantum gravity metric in high temperature.