Generalized Random Energy Model at Complex Temperatures

TL;DR

This paper analyzes the complex-temperature behavior of the Generalized Random Energy Model (GREM), revealing phase structures, zero distributions, and confirming physics predictions through rigorous mathematical results.

Contribution

It provides a rigorous analysis of GREM at complex temperatures, describing phase phases, fluctuations, and zeros, extending prior physics-based predictions.

Findings

Identified 2-3 types of phases in GREM at complex temperatures.

Described the distribution of zeros of the partition function in the complex plane.

Confirmed the correspondence between zeros and phase transitions.

Abstract

Motivated by the Lee--Yang approach to phase transitions, we study the partition function of the Generalized Random Energy Model (GREM) at complex inverse temperature $\beta$. We compute the limiting log-partition function and describe the fluctuations of the partition function. For the GREM with $d$ levels, in total, there are $\frac 12 (d+1)(d+2)$ phases, each of which can symbolically be encoded as $G^{d_1}F^{d_2}E^{d_3}$ with $d_1,d_2,d_3\in\mathbb{N}_0$ such that $d_1+d_2+d_3=d$. In phase $G^{d_1}F^{d_2}E^{d_3}$, the first $d_1$ levels (counting from the root of the GREM tree) are in the glassy phase (G), the next $d_2$ levels are dominated by fluctuations (F), and the last $d_3$ levels are dominated by the expectation (E). Only the phases of the form $G^{d_1}E^{d_3}$ intersect the real $\beta$ axis. We describe the limiting distribution of the zeros of the partition function in the complex $\beta$ plane (= Fisher zeros). It turns out that the complex zeros densely touch the positive real axis at $d$ points at which the GREM is known to undergo phase transitions. Our results confirm rigorously and considerably extend the replica-method predictions from the physics literature.