Asymptotic behavior of the critical temperature at large Rényi index in the 2D Ising model

Determine analytically the α→∞ limit of the critical temperature T_c(α) for the two-dimensional nearest-neighbor ferromagnetic Ising model (with coupling J=1) when equilibrium is defined by the α-Rényi ensemble. Establish whether T_c(α) approaches 1 in this limit and, if so, derive the precise asymptotic value and the rate at which T_c(α) converges.

Background

The authors perform Monte Carlo simulations of the two-dimensional Ising model in the α-Rényi ensemble and estimate the critical temperature T_c(α) via finite-size scaling collapses. They observe that as α increases, T_c(α) decreases and appears to approach an asymptote near 1 (in units with J=1). However, they do not provide an analytical derivation for this behavior.

Clarifying the large-α asymptotics of T_c(α) would solidify the understanding of how the α-Rényi ensemble modifies thermal criticality relative to the Gibbs case (α→1), and would provide a theoretical benchmark for numerical observations reported in the paper.