Non-local random deposition models for earthquakes and energy propagation

TL;DR

This paper introduces and analyzes non-local random deposition models with heavy-tailed object sizes, focusing on their limiting behavior and fluctuations, with applications to earthquakes and energy propagation.

Contribution

It presents new non-local deposition models with heavy-tailed features and characterizes their asymptotic laws and fluctuations, extending understanding of seismic and energy transfer processes.

Findings

Identified the limit in law of the surface profiles for three models.

Determined the fluctuation limits of the surface profiles.

Analyzed models with applications to earthquakes and stellar energy emission.

Abstract

We investigate a new class of non-local random deposition models, initially introduced by physicists to study the field of mechanical constraints (stress) applied along a line or on a given area located in a seismic zone. The non-local features are twofold. First, the falling objects have random and heavy-tailed dimensions. Second, the locations where the objects are falling are at least for some of the models that we consider, depending on the shape of the surface before deposition. We consider $(h_N)_{N\in \N}$ a sequence of random $(d+1)$-dimensional surfaces defined on $[0,D]^d$ for $d\in \{1,2\}$. Thus, the process $h_{N}$ is obtained by adding to $h_{N-1}$ an object $$\s\in [0,D]^d \mapsto Z_{N}^{\alpha-1} \psi\Big(\frac{v_{Y_N}(\s)}{Z_N}\Big),$$ where $Z=(Z_i)_{i\in \N}$ is an i.i.d. sequence of Pareto random variables, where $\psi:[0,\infty)\mapsto \mathbb{R}^+$ determines the global shape of the object, where $v_{\y}(\x)$ is the distance between $\x$ and $\y$ on the torus and where $Y=(Y_i)_{i\in \N}$ are random variables in $[0,D]^d$ that provide the location of each falling object. In the present paper we focus on three variations of this model. First, the rand-model for which $Y$ is an i.i.d. sequence of uniform random variables. Then, the min-model, that introduces an important property of the physics of earthquakes but is also harder to tract since a strong correlation appears between the $N$-th falling object and the shape of the profile $h_{N-1}$. Finally we consider a variant of the rand-model: the stellar model, which allows us to study the intensity of the microwaves emitted by stellar clouds and measured at the Earth surface. For those three models, our results identify the limit in law of $(h_N)_{N\in \N}$ viewed as a sequence of continuous random functions rescaled properly. We also determine the limit in law of the fluctuations of $(h_N)_{N\in \N}$.