Soutenance de thèse Maximilien Bernard

Lieu de la soutenance : LPENS, 24 Rue Lhomond, 75005, Paris, salle Conf IV

This thesis studies how disorder and fluctuations shape large-scale growth phenomena in three related settings: fluctuating interfaces, random spring chains, and random multiplicative growth. The first part concerns interface growth. In standard kinetic roughening, an initially flat interface becomes rough and develops correlations over increasing length scales. This behavior is characterized by scaling exponents, which describe how height fluctuations grow in space and time. These exponents can be measured either through global observables, probing the whole interface, or through local observables restricted to a finite window. In many experimental and numerical systems, however, these two descriptions do not agree, a phenomenon known as anomalous scaling. To clarify its origin, we study two models in which this mismatch has different causes. The first is a heterogeneous elastic line, where we show that the apparent anomaly is a purely statistical effect. The second is the stochastic porous medium equation, a strongly nonlinear model in which the anomaly is genuine: the interfaces display distinct local and global laws that we determine. The second part concerns Anderson localization, which appears here through the spectral properties of the heterogeneous elastic lines. In a homogeneous chain, vibrational eigenmodes are extended over the whole system. In the presence of quenched disorder, they may instead become spatially localized, with amplitudes concentrated near a localization center and decaying away from it. We study chains with both random masses and random spring constants, with particular emphasis on the strong-disorder regime, where standard weak-disorder expansions break down. We develop a new combinatorial method to probe this regime. The last part concerns random multiplicative growth with redistribution. In such models, changes in wealth, population, or mass, for instance, are proportional to the amount already present, so that small differences are amplified over time. This naturally generates broad distributions and provides a simple mechanism for the emergence of inequality and concentration, with applications to population dynamics, wealth distribution, city growth, and ecology. We in particular study models in which each site has its own quenched growth rate, representing a persistent advantage or disadvantage, and is also subject to transient fluctuations. Redistribution competes with these mechanisms by homogenizing the system. This competition leads to rich phase diagrams, with localized and delocalized phases, as well as a new intermediate phase, which we call partially localized, and which arises from the interplay between quenched heterogeneity and temporal noise.

Jury: Yan Fyodorov, Pierre Le Doussal (directeur de thèse), Cécile Monthus, Romualdo Pastor Satorras, Aleksandra Petkovic, Alberto Rosso (co-directeur de thèse), Stefano Zapperi