Non-local random deposition models for earthquakes and energy propagation

TL;DR

This work develops and analyzes a class of non-local, heavy-tailed random deposition models to capture energy and stress propagation in seismic and astrophysical contexts. It introduces three variants—rand, min, and stellar—to model different physical observables, formalizes their recursive deposition rules with Pareto-distributed widths and spatial centroids, and studies the limit behavior of the evolving interface h_N and its fluctuations under precise scalings. The main contributions are rigorous distributional limit theorems that reveal phase transitions governed by the exponents $\zeta=\frac{\alpha-d}{\beta-1}$ and $\kappa=\frac{\alpha-n-d}{\beta-1}$, yielding stable or Gaussian limits depending on regime, as well as detailed constructions of limiting laws $\mu_d$ and $\mu_{\text{stel}}$ for the fluctuation fields. These results connect complex non-local deposition dynamics to physically relevant phenomena (earthquake stress evolution, energy absorption, and stellar radiation) and provide a solid mathematical framework for convergence in function spaces, with open questions left for the min-model fluctuations.

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 $$\s\in [0,D]^d \mapsto Z_{N}^{α-1} ψ\Big(\frac{v_{Y_N}(\s)}{Z_N}\Big),$$ where $Z=(Z_i)_{i\in \N}$ is an i.i.d. sequence of Pareto random variables, where $ψ:[0,\infty)\mapsto \mathbb{R}^+$ 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 $(h_N)_{N\in \N}$.

Figures (5)

• Figure 2: Schematics of minus the variation of the energy of a wave that propagates in straight line from the top to the bottom parallel to the black arrow. The wave energy is displayed as a green line and consists in any electromagnetic radiation assumed to propagate in a straight line and absorbed by aerosols in the atmosphere. It is decreased when the ray interact with an aerosol. The absorption by aerosol is assumed to be proportional to their width and independent of the energy, as expected when their effect is small. The energy after the $(i)$-th encounter is displayed for $i=0$ to $3$. Note that the process is quite generic and this figure describes also the emission of any quantity that propagates in straight line and encounters objects that are additive sources of amplitude proportional to their width.
• Figure 3: In dimension $d=1$ and for $D=60$, sampling of the rand model after $4$ transformations (i.e. $h_4$) on the left and of the min model after $4$ transformations (i.e., $h_4^{\text{min}}$) on the right. Note that we have used the triangle function $\psi(x)=(1-x) \, {1}_{[0,1]}(x)$ and the same random variables $Z_1,Z_2,Z_3,Z_4$ for both figures.
• Figure 4: Phase diagram of the rand model in dimension $d=1$ for a function $\psi \in {\mathcal{H}}^1$. Note that in zones A and D, the profile growth is super-ballistic whereas it is simply ballistic in zones B and C. The fluctuations, in turn are non Gaussian in zones A and B whereas they are Gaussian in zones C and D.
• Figure 5:
• Figure 6:

Theorems & Definitions (43)

• Remark 2.2.1
• Remark 2.4.1
• Remark 2.4.2
• Definition 3.1.1
• Definition 3.1.2
• Theorem 3.2.1
• Remark 3.2.2
• Remark 3.2.3: Heuristics of the critical point
• Theorem 3.2.4
• Remark 3.2.5