Statistical physics of disordered systems.

Summary This thesis presents several aspects of the stochastic growth of interfaces, through its most studied model, the Kardar-Parisi-Zhang (KPZ) equation. Although very simple in expression, this equation conceals a great phenomenological richness and has been the object of intensive research for decades. This has led to the emergence of a new class of universals, containing some of the most common growth models, such as the Eden model or the Polynuclear Growth Model. The KPZ equation is also related to optimization problems in the presence of disorder (the Directed Polymer), or to the turbulence of fluids (Burger equation), reinforcing its interest. However, the limits of this class of universalities are still poorly understood. The purpose of this thesis is, after presenting the most recent progress in the field, to test the limits of this class of universality. The thesis is articulated in four parts:i) First, we present theoretical tools that allow us to finely characterize the evolution of the interface. These tools show a great flexibility, which we illustrate by considering the case of a confined geometry (an interface growing along a wall).ii) We then focus on the influence of disorder, and more particularly the importance of extreme events in the growth mechanics. Large fluctuations in disorder deform the interface and lead to a noticeable modification of the scaling exponents. We pay particular attention to the consequences of such disorder on optimization strategies in disordered media.iii) The presence of correlations in disorder is of immediate experimental interest. Although they do not modify the universality class, they greatly influence the average growth rate of the interface. This part is dedicated to the study of this average speed, often neglected because it is difficult to define, and to the existence of a growth optimum closely linked to the competition between exploration and exploitation.iv) Finally, we consider an experimental example of stochastic growth (which does not belong to the KPZ class) and develop a phenomenological formalism to model the propagation of a chemical interface in a disordered porous medium. Throughout the manuscript, the consequences of the observed phenomena in various fields, such as optimization strategies, population dynamics, turbulence or finance, are detailed.