Workshop on Random walks, folding transitions, and related topics

Firenze, Villa Finaly
September 11-13, 2017


Par Steve Hersey — http://flickr.com/photos/sherseydc/2954982676/, CC BY 2.0, https://commons.wikimedia.org/w/index.php?curid=5099526

Folding occurs in a variety of contexts in which a random walk is submitted to an attractive path interaction or conditioned to have an atypical small volume. In other situations, the walk unfolds due to self-repellence. Finally, in some instance the walk may exhibit a phase transition between these two behaviours. One of the key challenges is then to characterize the geometry of such folded/unfolded paths. These models are often motivated by problems arising from biology and physics, such as protein folding or phase transitions of polymers. The aim of this workshop is to gather probabilists around this topic.

Mini course by Artem Sapozhnikov (Leipzig):
Random walk on a torus and random interlacements

Abstract: We discuss geometric properties of the simple random walk on a large discrete torus on time scales proportional to the volume of the torus. A considerable progress has been recently made by exploring connections between the random walk and the random interlacements. The geometry of the range (vertices visited by the walk) is by now well understood. The vacant set (unvisited vertices) undergoes a percolation phase transition, which makes its analysis more challenging, and a number of interesting open problems still stand. Incidentally, some are linked to open questions about the geometry of a simple random walk on the infinite integer lattice.

Participants:

Organizers:

Schedule (tentative):

(1h = 45' talk + 5' questions + 10' break)

Monday 11th of September:
9h00 - 9h15 : Welcome
9h15 - 10h30: Mini-course by Artem Sapozhnikov
10h45 - 11h45: Perla Sousi
11h45 - 12h45: Dirk Erhard
13:00 - 14h00: LUNCH
15h30 - 16h30: Paolo Dai Pra
16h30 - 17h30: Francesca Nardi
20h00 : WORKSHOP DINNER

Tuesday, 12th of September:
9h15 - 10h30: Mini-course by Artem Sapozhnikov
10h45 - 11h45: Yueyun Hu
11h45 - 12h45: Amine Asselah
13:00 - 14h00: LUNCH
15h30 - 16h30: Quentin Berger
16h30 - 17h30: Julien Poisat
20h00 : DINNER

Wednesday, 13th of September:
9h15 - 10h30: Mini-course

Abstracts

Speaker: Amine Asselah
Title: On an inequality and a strategy to probe localisation.
Abstract: We present the strategy we developed with B.Schapira (arxiv 2016) to probe the most likely scenario a walk adopts to produce a smaller than typical volume of its range. We present a key inequality for the occupation time for the random walk which is of independent interest.

Speaker: Paolo Dai Pra
Title : Synchronisation and functional central limit theorems for interacting reinforced random walks
Abstract. We obtain Central Limit Theorems in Functional form for a class of time-inhomogeneous interacting random walks. Due to a reinforcement mechanism and interaction, the walks are strongly correlated and converge almost surely to the same, possibly random, limit. We study random walks interacting through a mean-field rule and compare the rate they converge to their limit with the rate of synchronization, i.e. the rate at which their mutual distances converge to zero. We show that, under certain conditions, synchronization is faster than convergence. Even if our focus is on theoretical results, we propose as main motivations two contexts in which such results could directly apply: urn models and opinion dynamics in a random network evolving via preferential attachment.

Speaker: Francesca Nardi
Title : Competing Metastable States for general rare transition dynamics and applications to Blume-Capel model and Probabilistic Cellular Automata.
Abstract : The study of systems with multiple (not necessarily degenerate) metastable states presents subtle difficulties from the mathematical point of view related to the variational problem that has to be solved in these cases. First, we prove sufficient conditions to identify multiple metastable states. Since this analysis typically involves non-trivial technical issues, we give different conditions that can be chosen appropriately depending on the specific model under study. We show how these results can be used to attack the problem of multiple metastable states. Second, we consider the problem of non degenerate in energy metastable states forming a series in the framework of reversible finite state space Markov chains. We assume that, starting from the state at higher energy, the system visits with high probability the second one before reaching the stable state. In this framework we give a sharp estimate of the transition time from the metastable state at higher energy to the stable state and, on the proper exponential time scale, we prove an addition rule. As application of the theory, we study the Blume-Capel model in the zero chemical potential case and to a probabilistic cellular automata that happen to have two multiple not degenerate in energy metastable states. We estimate in probability, in law and in expectation the time for the transition from the metastable states to the stable state. Moreover, we identify the set of critical configurations that represent the minimal gate for the transition.

Speaker: Perla Sousi
Title: Capacity of random walk and Wiener sausage in 4 dimensions
Abstract: In four dimensions we prove a non-conventional CLT for the capacity of the range of simple random walk and a strong law of large numbers for the capacity of the Wiener sausage. This is joint work with Amine Asselah and Bruno Schapira.

Speaker: Julien Poisat
Title : Charged polymers
Abstract: Charged polymer models aim to describe the folding/unfolding transition undergone by polyelectrolytes, which are polymer chains that self-interact via electric charges carried by their monomers. I will review recent results about the phase transition in the annealed model (integrated charges) and discuss open problems and conjectures. Attention will be brought to dimensions d \ge 2. Based on joint works with Quentin Berger, Francesco Caravenna, Frank den Hollander and Nicolas Pétrélis.

Speaker : Dirk Erhard
Title : Survival probability of a simple random walk among a system of weakly killing exclusion particles.
Abstract : In this talk we consider the following model. Given are a simple random walk and independently of it a simple symmetric exclusion process. Whenever the random walker is on top of an exclusion particle it gets killed at rate \epsilon. We consider the regime in which time gets appropiately rescaled with \epsilon and prove asymptotics for the survival probability of the walk. This is joint work in progress with Martin Hairer.

Speaker : Yueyun Hu
Title : On a hierarchical renormalization model on trees
Abstract : This talk is based on a joint work with Zhan Shi (Paris 6). We study a simple max-type recursive model introduced by Derrida and Retaux (2014) in the context of a physical problem, and find a wide range for the exponent in the free energy in the nearly supercritical regime.

Speaker : Quentin Berger
Title : A bivariate renewal model for DNA : denaturation and condensation phase transitions
Abstract : I will introduce the so-called generalized Poland Scheraga model, used to describe loops that appear in the binding of two DNA strands. This relatively recent model is based on bivariate renewal processes. I will characterise the denaturation transition (or depinning transition), and stress the existence of a so-called condensation transition (i.e. the appearance of a macroscopic loop) which is absent from the original Poland Scheraga model.

Practical details:

Link to Villa Finaly

Soutiens : LIA – LYSM “Laboratory Ypatia of Mathematical Sciences", PEPS (Projets Exploratoires Premier Soutien)