Project development

The team consists of four research units, based at the universities of Rome Tre, Sapienza, Milan-Bicocca and Padua. All the people in the units (see section 5 below for a list) share a strong common background mainly in the theory of large deviations, mathematical statistical physics, Markov chains, random walks and interacting particle systems. At the same time there are important exclusive expertise in each unit and specific synergies among the units. More specifically we recall that: (i) the mathematical theory of kinetically constrained models has been pioneered and successfully developed by the two units in Rome,

1. (ii) several breakthroughs on the large scale behaviour of disordered systems, random surfaces and random triangulations have been made separately by the units in Rome and in Milan,
2. (iii) metastability and Markov chains mixing times have seen important advances by the (joint and separate) efforts by the units in Rome and in Padua,
3. (iv) the unit of the PI has contributed to several important recent progresses concerning the research line B, while line D is mostly developed within the units in Padua and Rome-La Sapienza.

We also emphasize that the four units are already quite well integrated among them, with a good record of scientific collaboration on specific research papers and on the organization of some very successful workshops and summer schools (cf. the series of probability workshops in "La Pietra, Florence", '04, '06, '08, '11).

In conclusion, the whole team has the perfect blend of diversity and integration to successfully carry out the proposed project. Part of the funding will be used to hire two postdocs in order to give a further boost to the project. These postdocs, based at the units of Rome Tre and Milan, will have the opportunity to interact with all the people in the team, providing a further element of synergy.

The role of each unit w.r.t. the expected targets

We will now describe more specifically the role of each unit with regards to the expected targets. For simplicity, the latter will be referred to according to the listing given in the previous section.

A. The main efforts towards the targets belonging to research line A will be done jointly by the units in Rome (the PI, A. Faggionato and N. Cancrini) for what concerns A.1, A.2 and A.3 and by the unit in Padua regarding A.4. Their efforts will also be supported by E. Scoppola, a leading expert in metastability for Markov processes. We forecast intense collaboration on A.1, A.2 and A.3 with P. Chleboun (Warwick), C. Toninelli (CNRS, Paris and PI of an ERC grant on Glassy Systems), O. Blondel (Lyon), R. Morris (IMPA), a leading figure in bootstrap percolation, and A.Smith (Ottawa). For the same targets we also intend to keep a strict contact with some group of physicists working in the field (most likely R. Jack and J. Garrahan in UK) in order to test our results and identify interesting possible new directions within the above research lines. For what concerns A.4 we forecast collaborations with S. Dommers (Bochum), C. Giardinà (Modena), A. Gaudilliére (CNRS Marseille) and S. Grosskinsky (Warwick)

B. The leading scientist for the core of this part of the project will be P. Caputo (Rome Tre) but we expect some pivotal contributions by F. Zucca and D. Bertacchi (Milan) for interesting random walks on Erdös-Renyi graphs (e.g the Frog and Rumor model) and by M. Campanino (Padua) . We forecast strong collaborations with two leading figures in the field, C. Bordenave (CNRS Toulouse) and D. Chafai (Paris), and with J. Salez (Paris), D. Petritis (Rennes), F.Machado et al. (San Paulo).

C. The leading scientist for the first two targets C.1 and C.2 will be F. Caravenna. We also expect some significant exchange of expertise with the units in Rome (the PI, P. Caputo, L. Bertini and M. Mariani), because of some substantial overlap of key results on random polymers and surface models. For this line of research we forecast collaborations with N. Zygouras (Warwick) and R. Sun (Singapore). For what concerns C.3, C4 and C.5 most of the activity will be done by the units in Rome, with L. Bertini more focused on the dynamical part and P. Caputo together with the PI on the discrete Gaussian model. For what concerns C.3 we forecast collaboration with F.L.Toninelli (Lyon), a leading figure of the field and a close collaborator of the units in Rome and Milan, and with A. Stauffer (Bath) and A. Sinclair (Berkeley) for C.5. Finally we expect also some significant advance on the topic of "anomalous diffusion of random walks in a "Lévy" random environment" by the unit in Padua.

D. The leading group for the first part D.1 will be the unit in La Sapienza (L. Bertini and M. Mariani) while the responsibility for the second part D.2 will be instead taken by the unit in Padua. We forecast collaborations with A. Boritchev and C.E.Brehier (Lyon) for D.1 and with A. Asselah (Marseille), E. Locherbach (Cergy), M. Tolotti (Venezia), M. Formentin, F. Collet (Bologna), L. Campi (London) and F. Delarue (Nice), a leading international figure for stochastic games, for what concerns D.2. Several Ph.D students in Padua will also be somewhat involved in the second part.

Modalities of integration and collaboration

We plan to create a web site dedicated to the project and collecting all the related news, events (seminars, workshops, visitors), preprints and video recorded lectures. In order to enhance and strengthen the "interaction network" among the research units, we plan different levels of integration and collaboration.

1. A first basic level of integration will occur with little efforts because of geographical proximity. The two units in Rome already share a long history of scientific collaborations, joint seminars and international visitors. Similarly, although to a somewhat lesser extent, for the units in Padua and Milan. As a first step to greatly ameliorate this situation, we shall video-record all the seminars/lectures that will occur within the project and to collect them in the project web site. That will require a rather low level equipment, part of which is already available in the four departments. Integration and collaboration will also be boosted by sending team members of different units to international workshops dedicated to the main topics of the project (e.g. Oberwolfach 2016 "Large Scale Stochastic Dynamics").

2. As a second level of integration we shall organise, at the very beginning of the project, a series of keynote lectures across the units on the detailed state of the art of the main targets of the project. These lectures will be delivered by the leading scientists for each topic and the majority of the team members will attend them. In the second half of each year we plan a two day workshop across the units every year, where each sub-group will illustrate intermediate results and discuss future developments.

3. A third level of integration consists in a series of invitations of leading figures related to the main targets of the project to deliver mini-courses on advanced topics. We will organise them in a way to maximise the attendance by the team members and by graduate students and postdocs. The geographical location will rotate among Rome, Padua and Milan. Among the names we have in mind we mention R. Cerf (population dynamics and percolation), C. Garban (noise sensitivity), A. Smith (mixing times for constrained models), F.L.Toninelli (dimers and random surfaces), D. Chafai (random matrices and graphs), I. Corwin (KPZ qquation) and F. Delarue (stochastic games).

4. Finally, towards the end of the third year, we plan to organise an international event with a format in between a workshop and a summer school (i.e. a mixture of seminars and mini-courses).