Dimers, Markov chains and Critical Phenomena

Final Report Summary - DMCP (Dimers, Markov chains and Critical Phenomena)

The scientific project focuses on the probabilistic study of dynamical and equilibrium properties of discrete random interfaces, spin models and dimer models. One of the main goals was to derive, starting from a microscopic stochastic dynamics, a macroscopic deterministic evolution (often, an anisotropic mean curvature flow) in the scaling limit. Another important issue are equilibrium fluctuations of random discrete interfaces and dimer models. This research has tight links with mathematical physics, discrete geometry and rigorous quantum field theory. Mathematical tools to be employed include Markov Chains, conformal invariance, random walks on graphs, rigorous Renormaliazation Group techniques... More specific problems that were to be attacked include:

GOAL 1) Dynamics and equilibrium fluctuations of random dimer coverings (perfect matchings) of bipartite graphs: these are a central object in combinatorics and discrete geometry and can be seen as discrete interfaces. How does their geometry evolve under stochastic dynamics? What is the dynamical consequence of the conformal invariance properties of their equilibrium fluctuations? How quickly can one sample a random perfect matching (mixing time)?

GOAL 2) Universality for two-dimensional statistical mechanics models. The project was to study the critical properties of weakly interacting dimers on the square lattice (the non-interacting case being exactly solvable). As a long-term goal, the project was to study the spin-spin correlation functions of two-dimensional, non-integrable critical Ising models.

During his visit, F. Toninelli collaborated both with local researchers in Roma Tre (notably P. Caputo, A. Giuliani and myself) and with researchers from other institutions (notably V. Mastropietro (Milan), A. Borodin (MIT), I. Corwin (Columbia Univ., New York), P. Ferrari (Bonn), N. Torri (Lyon), B. Laslier (Cambridge UK), F. Caravenna (Milan)). The collaborations of F. Toninelli were greatly helped by his visits to Lyon, New York and Boston. Also, internationally acknowledged experts in this research area were invited to Rome in the framework of the DMCP project: Y. Spinka (Tel Aviv), T. Richthammer (Hildesheim), B. Laslier (Cambridge, UK) among others. These visitors collaborated with F. Toninelli as well as with local scientists including myself and gave lectures in Roma Tre. Short relations on the invitations and on Toninelli's scientific visits outside Rome are attached to this report.

SCIENTIFIC RESULTS

These interactions, occurring during the visit of F. Toninelli to Rome, gave rise to 8 scientific manuscripts: 3 published in journals, 3 submitted to journals and published in the Preprint Archive http://arxiv.org and two in preparation.

The works [2] and [3] (numbering corresponds to the list given at the end of this section), as well as the work in progress [7], represent an important contribution in the understanding of (2+1)-dimensional discrete growth models (GOAL 1 above). In [3], F. Toninelli introduces and studies a model of 2-dimensional stochastic interface growth and identifies its stationary measures. This is an important step in understanding the universality class of the so-called anisotropic KPZ equation in two dimensions. In [2,7], with his coauthors, he extends this study to a more general class of models.

The works [1] and [4] deal with random polymer models.While random polymer models are not directly the main focus of this research project, they are indirectly related to the study of random interface models, (GOAL 1). Indeed, level lines of random interfaces can be seen as interacting polymers and understanding the effect of such interactions on the geometrical properties of the interface was the main motivation of [4].

The works [5,6,8] by F. Toninelli, A. Giuliani and V. Mastropietro contain breakthrough results on the universality of height fluctuations for non-integrable models of dimers on two-dimensional lattices, and their converenge to a massless Gaussian Free Field (GOAL 2). Work [6] contains the full proof of the results on the square lattice, while [5] is a shorter, less technical, version that was aimed at disseminating these results in the theoretical physics community. Finally, in the ongoing work [8] the same authors extend their results to more general lattices, pursuing their study of universality of height fluctuations, and they study the effect of non-periodic boundary conditions.
As for work [6], a first version was submitted to the journal Ann. Institut H Poincare before the beginning of the Marie Curie project. The journal requested deep revisions; the revision work kept F. Toninelli, together with his coworkers A. Giuliani and V. Mastropietro, busy for several weeks. In the end the article was substantially strengthened (and its size increased by over 20 pages) and it was accepted for publication by the same journal.

The Roma Tre PhD School in Mathematics and Physics took advantage of F. Toninelli's presence to increase the scientific offer to its PhD students with a 12-hour course on "Dimer models: thermodynamics, fluctuations and Glauber dynamics". These topics are of great interest in both probability theory and mathematical phyisics. The course was attended mostly by PhD students, but also by some undergraduate students and faculty staff. The program of the course is attached to this report.

The main impact of this visit is certainly scientific. It gave rise to scientific results which will
remain to indicate the commitment of the EC in the progress of pure science. This was very
important for the diffusion of scientific ideas. F. Toninelli, during his stay, gave several seminars at
different Universities in Europe and US and during International Conferences or Workshops in the field. A
long range impact will be due to his PhD course which transferred information previously not
available in Rome scientific community.

PUBLICATIONS

[1] F. Caravenna, F. L. Toninelli, N. Torri, Universality for the pinning
model in the weak coupling regime, http://arxiv.org/abs/1505.04927 submitted to Annals of Probability

[2] I. Corwin, F. L. Toninelli, Stationary measure of the driven two-dimensional
q-Whittaker particle system on the torus, http://arxiv.org/abs/1509.01605 submitted to Annals of Applied Probability

[3] F. L. Toninelli, A (2+1)-dimensional growth process with explicit stationary measures, arXiv:1503.05339 submitted to Annals of Probability

[4] P. Caputo, F. Martinelli, F. Toninelli, Multi-level pinning problems for random walks and self-avoiding lattice paths, Elect. J. Probab. 20 (2015),1{29

[5] A. Giuliani, V. Mastropietro, F. Toninelli, Height fluctuations in non-integrable classical dimers, Europhys. Lett.
109 (2015), 60004

[6] A. Giuliani, V. Mastropietro, F. L. Toninelli, Height fluctuations and interacting dimers, arXiv:1406.7710 to appear on Ann. Inst. Henri Poincare' (Prob. Stat)

[7] A. Borodin, I. Corwin, P. L. Ferrari, F. Toninelli, in preparation

[8] A. Giuliani, V. Mastropietro, F. Toninelli, in preparation

Final Report Summary - DMCP (Dimers, Markov chains and Critical Phenomena)

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