Exact solutions for the statistics of extrema of some random 1D landscapes, Application to the equilibrium and the dynamics of the toy model

TL;DR

This paper extends the real-space renormalization group method to analyze the extrema statistics of various 1D landscapes, applying it to a toy model to understand equilibrium properties and non-equilibrium dynamics.

Contribution

It provides exact solutions for the extrema statistics of 1D landscapes using RSRG, including a detailed analysis of a toy model with quadratic and Brownian potentials.

Findings

Explicit measure of the renormalized landscape via Airy functions

Statistics of the absolute minimum and nearly degenerate minima at low temperature

Distribution of equilibration times and long-time dynamics properties

Abstract

The real-space renormalization group (RSRG) method introduced previously for the Brownian landscape is generalized to obtain the joint probability distribution of the subset of the important extrema at large scales of other one-dimensional landscapes. For a large class of models we give exact solutions obtained either by the use of constrained path-integrals in the continuum limit, or by solving the RSRG equations via an Ansatz which leads to the Liouville equation. We apply in particular our results to the toy model energy landscape, which consists in a quadratic potential plus a Brownian potential. The measure of the renormalized landscape is obtained explicitly in terms of Airy functions, and allows to study in details the Boltzmann equilibrium of a particle at low temperature as well as its non-equilibrium dynamics. For the equilibrium, we give results for the statistics of the absolute minimum which dominates at zero temperature, and for the configurations with nearly degenerate minima which govern the thermal fluctuations at very low-temperature. For the dynamics, we compute the distribution over samples of the equilibration time, or equivalently the distribution of the largest barrier in the system. We also study the properties of the rare configurations presenting an anomalously large equilibration time which govern the long-time dynamics. We compute the disorder averaged diffusion front, which interpolates between the Kesten distribution of the Sinai model at short rescaled time and the reaching of equilibrium at long rescaled time. Finally, the method allows to describe the full coarsening (i.e. many domain walls) of the 1D RFIM in a field gradient as well as its equilibrium.