### Attention : désormais les séminaires ont lieu tous les lundis à 10h45 en salle 523 du LPTMC - Tour 12-13

**Nicolas Behr (IRIF Paris-Diderot et LPTMC)**

**Stochastic Mechanics of Graph Rewriting Systems for Physicists**

Consider a statistical system evolving on a state space of graphical structures, such as e.g. a social network system. Given a set of transitions on such a system, where each transition consists of a local transformation pattern applied at random to the system's state (e.g. adding a new edge, deleting an edge,...), one may define a continuous-time Markov chain in order to study the stochastic evolution of the system. Our novel approach to this problem involves an extension of Doi's description of chemical reaction systems in terms of boson creation and annihilation operators (which later evolved into the Doi-Peliti formalism) to a general stochastic mechanics framework based on the idea of so-called rule algebras. Assuming no prior familiarity with the underlying concept of graph rewriting and related mathematics, I will give an introduction to the formalism and present a number of application examples.

**SALLE INHABITUELLE: salle de séminaire de l'INSP tour 22-23 salle 317 (3ème étage).**

**Lara Benfatto (CNR et Université de Rome Sapienza)**

**Berezinskii-Kosterlitz-Thouless transition in 2D superconductors**

Lecture I: The Beresinskii-Kosterlitz-Thouless transition: the two-dimensional world and its peculiarities

More than 40 years after the seminal work by Berezinskii, Kosterlitz and Thouless the

BKT transition remains one of the most spectacular phenomena in condensed matter

systems, as it has been acknowledged by the 2016 Nobel Prize. Even though it was originally formulated within the context of the two dimensional XY model for classical spins, it represents the paradigm for the superfluid transition in two dimensions. As such, it has been the subject of an intense theoretical and experimental investigation in a variety of systems, ranging from thin films of superconductors to artificial heterostructures and cold atoms.

In the first lecture I will give an introduction to the basic mathematical ingredients needed to understand the occurrence of a BKT transition within the context of the classical 2D XY model. After discussing the difference between order and rigidity for a second-order phase transition, I will discuss the peculiar role of vortices in 2D and I will derive the mapping onto the Coulomb-gas model. Finally, I will sketch the main outcomes of the renormalization-group approach for the BKT phase transition.

Lecture II: Applications to superfluids. What we should expect to see in real systems?

The Beresinskii-Kosterlitz-Thouless transition is expected to describe the metal-to-superconductor thermal transition in quasi-2D systems. However, despite many efforts along the years its signatures remain rather elusive. In this second lectures I will give an overview of the numerous attempts we made along the years to identify the mechanisms which may hinder a clear-cut observation of BKT physics in 2D films of conventional and unconventional superconductors. In particular I will discuss the role of the vortex-core energy and of the spontaneous inhomogeneity of the superconducting background which naturally emerge in disordered thin films. These effects must be seriously taken into account while addressing the famous universal jump of the superfluid density, the non-linear IV characteristics near Tc, or the paraconductivity effects while approaching it from above. Finally, I will make a comparison with some recent results within the context of cold atoms, underlying differences and analogies between the two classes of systems.

**Sophie Marbach (LPS-ENS)**

**Out-of-equilibrium Transport of Particles within Fluctuating Nanotubes**

In Nature exceptional permeability and selectivity properties are reached, for example ion channels are able to distinguish with high throughput very similar ions like Sodium and Potassium. The paradigm change as compared to nanoscale technology is that these biological filters are out-of-equilibrium, submitted to either thermal or active fluctuations – for example of the pore constriction. Here we investigate how out-of-equilibrium fluctuations of a pore may affect the translocation dynamics, in particular dispersion coefficients. Our findings demonstrate a complex interplay between transport and surface wiggling and elucidate the impact of pore agitation in a broad range of artificial and biological porins, but also, at larger scales, in vascular motion in fungi, intestinal contractions and microfluidic surface waves. These results open up the possibility that transport across membranes can be actively tuned by external stimuli, with potential applications to nanoscale pumping, osmosis and dynamical ultrafiltration.

**Vardan Kaladzhyan (KTH Royal Institute of Technology, Stockholm)**

**Topology from Triviality**

We show that bringing into proximity two topologically trivial systems can give rise to a topological phase. More specifically, we study a 1D metallic nanowire proximitised by a 2D superconducting substrate with a mixed s-wave and p-wave pairing, and we demonstrate both analytically and numerically that the phase diagram of such a setup can be richer than reported before. Thus, apart from the two "expected" well-known phases (i.e. where the substrate and the wire are both simultaneously trivial or topological), we show that there exist two peculiar phases in which the nanowire can be in a topological regime while the substrate is trivial, and vice versa.

**Marylou Gabrié (LPS-ENS Paris)**

**Entropy and mutual information in models of deep neural networks**

The successes and the multitude of applications of deep learning methods have spurred efforts towards quantitative modeling of the performance of deep neural networks. In particular, an information-theoretic approach has been receiving increasing interest. Nevertheless, it is in practice computationally intractable to compute entropies and mutual informations in industry-sized neural networks. In this talk, we will consider instead a class of models of deep neural networks, for which an expression for these information-theoretic quantities can be derived from the replica method. We will examine how mutual informations between hidden and input variables can be reported along the training of such neural networks on synthetic datasets. Finally we will discuss the numerical results of a few training experiments.