2025/05/30 by Du, Hang, Zhou, Yanxin
#60K35 #82B20 #82B27 #FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph) #Probability (math.PR)
paper · doi:10.48550/arxiv.2505.24283
For any integers d,q≥ 3, we consider the q-state ferromagnetic Potts model with an external field on a sequence of expander graphs that converges to the d-regular tree \mathttTd in the Benjamini-Schramm sense. We show that along the critical line, any subsequential local weak limit of the Potts measures is a mixture of the free and wired Potts Gibbs measures on \mathttTd. Furthermore, we show the possibility of an arbitrary extent of strong phase coexistence: for any α∈ [0,1], there exists a sequence of locally \mathttTd-like expander graphs \Gn\, such that the Potts measures on \Gn\ locally weakly converges to the (α,1-α)-mixture of the free and wired Potts Gibbs measures. Our result extends results of \citeHJP23 which restrict to the zero-field case and also require q to be sufficiently large relative to d, and results of \citeBDS23 which restrict to the even d case. We also confirm the phase coexistence prediction of \citeBDS23, asserting that the Potts local weak limit is a genuine mixture of the free and wired states in a generic setting. We further characterize the subsequential local weak limits of random cluster measures on such graph sequences, for any cluster parameter q>2 (not necessarily integer).