Freezing and decorated Poisson point processes

TL;DR

This paper characterizes the structure of extremal processes in branching models as decorated Poisson point processes, linking a key Laplace functional property to the freezing phenomenon in log-correlated fields.

Contribution

It proves that a specific Laplace functional property characterizes SDPPP structures and shows that the associated function must be a convolution of Gumbel distribution, illuminating the freezing phenomenon.

Findings

Laplace functional property characterizes SDPPP structure.

The function g must be a convolution of Gumbel distribution.

Provides a tool for proving SDPPP in various models.

Abstract

The limiting extremal processes of the branching Brownian motion (BBM), the two-speed BBM, and the branching random walk are known to be randomly shifted decorated Poisson point processes (SDPPP). In the proofs of those results, the Laplace functional of the limiting extremal process is shown to satisfy $L[\theta_y f]=g(y-\tau_f)$ for any nonzero, nonnegative, compactly supported, continuous function $f$, where $\theta_y$ is the shift operator, $\tau_f$ is a real number that depends on $f$, and $g$ is a real function that is independent of $f$. We show that, under some assumptions, this property characterizes the structure of SDPPP. Moreover, when it holds, we show that $g$ has to be a convolution of the Gumbel distribution with some measure. The above property of the Laplace functional is closely related to a `freezing phenomenon' that is expected by physicists to occur in a wide class of log-correlated fields, and which has played an important role in the analysis of various models. Our results shed light on this intriguing phenomenon and provide a natural tool for proving an SDPPP structure in these and other models.