Laboratoire de Physique Théorique

de la Matière Condensée

- Léo REGNIER : Complete visitation statistics of one-dimensional random walks
Doctorant en 2ème année, sous la supervision d'Olivier Bénichou

Random walks are often used to describe exploration processes of a spatial domain, such as dynamics on the web or relaxation in disordered media. one of the most fundamental observable to describe this process is the number of distinct sites visited up to time t, N(t). This quantity has been extensively studied in the physical and mathematical litterature: its average, variance, single time distribution P(N(t)), or even its covariance Cov(N(t_1),N(t_2)) have been characterized. However, little was known about the multiple time distribution which is crucial to fully describe the exploration process.
In my talk, I will present the results given in [1], in which we determine the complete statistical statistical behavior of the stochastic process (N(t))_t>0, namely the probability that n_1, n_2, n_3... distinct sites are visited at times t_1, t_2, t_3... From this multiple-time distribution, we show that the visitation statistics of 1d random walks are temporally correlated and we quantify the non-Markovian nature of the process. We exploit these ideas to derive unexpected results for the two-time trapping problem and also to determine the visitation statistics of two important stochastic processes, the run-and-tumble particle and the biased random walk.


- Adriano ANGELONE : Disorder-free quantum glasses in quasicrystalline lattices

Quasicrystals, ordered but not periodic structures, have been originally discovered in the context of solid state physics, and have since inspired considerable research efforts thanks to their peculiar geometric properties. Their experimental realization in photonic systems and cold atom setups has generated further interest, resulting in many studies investigating the properties of interacting quantum particles moving in quasicrystalline lattice structures. One of the most interesting results of these works has been the discovery of Bose Glass (BG) states, globally localized but displaying local patches of delocalized particles. In all of these studies, however, the quasicrystalline substrates were accompanied by disorder, to closely model cold atom experimental platforms; this leaves open the question as to if and how the typical phase diagram of these systems changes in the disorder-free case.
In this talk, I will discuss recent results by my co-workers and I on a system of particles interacting via finite-range interactions on a disorder-free quasicrystalline lattice. Using numerically exact Path Integral Monte Carlo simulations, we obtain the first approximation-free phase diagram of a model in this setting, confirming the existence of a BG phase in the absence of disorder (at odds with previous mean-field predictions). Our results are of great interest given the perspective of laboratory engineering of disorder-free quasicrystalline systems via photonic experiments.