The glassy phase of the complex branching Brownian motion energy model

Abstract

We identify the fluctuations of the partition function for a class of random energy models, where the energies are given by the positions of the particles of the complexvalued branching Brownian motion (BBM). Specifically, we provide the weak limit theorems for the partition function in the so-called "glassy phase" -the regime of parameters, where the behaviour of the partition function is governed by the extrema of BBM. We allow for arbitrary correlations between the real and imaginary parts of the energies. This extends the recent result of Madaule, Rhodes and Vargas [19], where the uncorrelated case was treated. In particular, our result covers the case of the real-valued BBM energy model at complex temperatures.The glassy phase of the complex temperature BBM Figure 1: Phase diagram of the REM (and conjecturally of the BBM energy model). The grey curves are the level lines of the limiting log-partition function, cf. (1.18). This paper mainly deals with phase B 2 .Koukiou [15]. The full phase diagram of this model at complex temperatures including the fluctuations and zeros of the partition function were identified by Kabluchko and one of us in [11]. In particular, the case of arbitrary correlations between the imaginary and real parts of the energies was considered in [11]. The same authors answered in [12] similar questions about the Generalized Random Energy model (GREM) -a model with hierarchical correlations -and obtained the full phase diagram. In the complex GREM, the phase diagram turned out to have a much richer structure than that of the complex REM. This sheds some light on the phase diagrams of the models beyond the complex REM universality class.(1.4) 1 This implies that p 0 = 0, so none of the particles ever dies. 2 The latter assumption is just a matter of normalization. Any expected number of children greater than 1 (= the supercritical regime) is allowed and the results of this paper remain valid with appropriate modifications of constants.3 Under the stated conditions, the convergence of the extremal process of BBM, on which we rely, is proven in [3]. For the case of branching random walk, using truncation techniques, Madaule [18] has shown the same under conditions that would in the Gaussian case imply finiteness of k p k k(ln k) 3 . This could probably be carried over to BBM. It is not clear whether the result holds under the Kesten-Stigum condition k p k k ln k < ∞. For a discussion on these issues, we refer to the lecture notes by Shi [23]. In the present paper, we are not concerned with improving the conditions on the offspring distribution.ECP 0 (2015), paper 0.

The phase diagram of the complex branching Brownian motion energy model

“…Derrida [15] and [21]. However, the fluctuations of the partition function of this model are already influenced by the strong correlations and differ from those of the REM in all phases of the model, as we show in this work (and in [18]).…”

“…Convergence in probability for β ∈ B 1 and B 3 in (1.9) follows from Theorems 1.2 and 1.4 (ii) by [21, Lemma 3.9 (1)]. Convergence for the glassy phase β ∈ B 2 was shown in [18]. For the boundaries between all three phases, the formula (1.9) follows from the continuity of the limiting log-partition function.…”

“…This paper deals with phases B 1 and B 3 and the boundaries. For a treatment of phase B 2 , see [18].…”

Secular Coefficients and the Holomorphic Multiplicative Chaos

“…1 We note that for θ ≥ 1, [LRV15, Conjectures 5.2,5.3] suggest that an additional logarithmic normalization is required for convergence, analogously to logarithmic factors needed for convergence of the critical GMC. Indeed, in work of [MRV16,HK15] (see also [HK18]), an analogous statement is proven for a complex random energy model built from branching processes. Note this in constrast to HMC θ , which requires no further normalization to converge for any θ > 0.…”

The structure of extreme level sets in branching Brownian motion

“…For instance, interpreting the heights as energy levels in a spin-glass system, (negative) extreme values capture the lowest energy states. The latter carry the corresponding Gibbs distribution at low temperature (glassy-phase) [12,24,30].…”