Related papers: Some perspectives on (non)local phase transitions …
In the paper we present simple examples of linear random fields defined on $\ZZ^2$ and $\ZZ^3$ which exhibit the scaling transition phenomenon. These examples lead to more general definition of the scaling transition and allow to understand…
This article explains a program to study complete and properly embedded minimal surfaces in $\mathbb{R}^3$ developed jointly with W.H. Meeks and A. Ros in the last three decades. It follows closely the structure of my invited ICM talk with…
Continuous phase transitions in equilibrium statistical mechanics were successfully described 50 years ago with the development of the renormalization group framework. This framework was initially developed in the context of phase…
This article surveys some of the work done in recent years on random simplicial complexes. We mostly consider higher-dimensional analogs of the well known phase transition in $G(n, p)$ theory that occurs at $p = \frac 1n$ . Our main…
I review some open problems on the ever-growing field of non-equilibrium phase transitions, paying special attention to the formulation of such problems in terms of Langevin equations or, equivalently, field-theoretical descriptions, and…
The present paper has two goals. First to present a natural example of a new class of random fields which are the variable neighborhood random fields. The example we consider is a partially observed nearest neighbor binary Markov random…
The Landau theory of phase transitions has been re-examined under the framework of a modified mean field theory in ferroelectrics. By doing so, one can see that there are two atomic movements involved in the ferroelectric phase transition;…
It is pointed out that despite of the non-linearity of the underlying equations, there do exist rather general methods that allow to generate new minimal surfaces from known ones.
We investigate nonequilibrium phase transitions in the presence of disorder that locally breaks the symmetry between two equivalent macroscopic states. In low-dimensional equilibrium systems, such "random-field" disorder is known to have…
A mean-field theory is developed for the scale-invariant length distributions observed during the coarsening of one-dimensional faceted surfaces. This theory closely follows the Lifshitz-Slyozov-Wagner theory of Ostwald ripening in…
We present some long-range interaction models for phase coexistence which have recently appeared in the literature, recalling also their relation to classical interface and capillarity problems. In this note, the main focus will be on the…
In this work we study the level sets of entire solutions of the Allen-Cahn equation and we prove minimality of the zero level set with respect to a certain perimeter functional with density. This provides a direct relationship between phase…
Classical Landau theory considers structural phase transitions and crystallization as a condensation of several critical density waves whose wave vectors are symmetrically equivalent. Analyzing the simplest nonequilibrium Landau potentials…
We consider a non-local phase transition equation set in a periodic medium and we construct solutions whose interface stays in a slab of prescribed direction and universal width. The solutions constructed also enjoy a local minimality…
One of the main problems of the theory of dynamical systems is the determination of the existence of periodic orbits of a self-map and more generally, the structure of the set of periods. Define the minimum period of a class os self-maps of…
In this brief review we introduce the methods of quantum field theory out of equilibrium and study the non-equilibrium aspects of phase transitions. Specifically we critically study the picture of the ``slow-roll'' phase transition in the…
We propose a phenomenological equation to describe kinetic roughening of a growing surface in presence of long range interactions. The roughness of the evolving surface depends on the long range feature, and several distinct scenarios of…
In the setting of fractional minimal surfaces, we prove that if two nonlocal minimal sets are one included in the other and share a common boundary point, then they must necessarily coincide. This strict maximum principle is not obvious,…
The nonlinear stability of Minkowski spacetime has been one of the central achievements in the mathematical theory of general relativity and, more broadly, in the analysis of nonlinear geometric wave equations. Since the seminal work of…
We study nematic equilibria on rectangular domains, in a reduced two-dimensional Landau-de Gennes framework. These reduced equilibria carry over to the three-dimensional framework at a special temperature. There is one essential model…