Universality in random structures: Interfaces, Matrices, Sandpiles
Mini-courses

    Week 1: 14 January -18 January

  1. Speaker: Charles Bordenave (Skype talk)

    Title: High trace methods in random matrix theory

    Abstract: 

    In 1955, Eugene Wigner has established the semi-circular law by computing expected traces of random matrices. In 1981, Füredi and Komlos have refined the computation of Wigner and studied the spectral radius of random matrices. Since then, there have been numerous successful extensions of their approach notably in connexion with the non-backtracking matrices. In this mini-course, we will introduce the high trace method of Furedi-Komlos and present some its latest developments: the use of non-backtracking matrices, the tangle-free random graphs and the comparison argument of Bandeira and Van Handel.

    Lecture Slides 1    Lecture Slides 2

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    Week 2: 21 January -25 January

  2. Speaker: Tomohiro Sasamoto

    Title: Integrable stochastic interacting systems

    Stochastic interacting particle systems show many intriguing phenomena due to the interaction among particles and have wide applications in various fields of science, but in general it is quite difficult to study their properties in detail. Over the last few decades, however, it has been gradually recognized that certain stochastic interacting particle systems can be “solved exactly”, meaning that they admit explicit calculations of various probabilities and expectation values, and behind this tractability lies the integrability of these systems. In particular there have been remarkable progress in the understanding of growth and transport models in the Kardar-Parisi-Zhang (KPZ) universality class, which have turned out to have deep connections with random matrix theory, representation theory, special functions and so on.

    Abstract:

    In these lectures, we will explain these developments, mainly focusing on the models in the KPZ class. In the first lecture, we introduce one of the most fundamental models on the subject, the asymmetric simple exclusion process (ASEP) and explain the connection between its totally asymmetric version (TASEP) and random matrix theory[1]. In the second lecture we introduce various models and discuss their integrability [2]. In the third lecture we introduce the notion of stochastic duality and explain its relation to the replica approach of using moments [3]. Then we elucidate how an exact formula for ASEP can be obtained by combining the duality and integrability of the model. In the fourth lecture we explain our approach introduced in [4], which does not rely on the moment calculation. We emphasize that the method can be applied to many models in parallel, including the stationary situation. In the last lecture we discuss our recent extension of the techniques to study a two spec ies exclusion process [5]. The implications to the nonlinear fluctuating hydrodynamics, which was proposed recently by H. van Beijren and H. Spohn, is also explained. Lastly we discuss some outstanding problems.

    References
    [1-1] K. Johansson, Shape fluctuations and random matrices, Commun. Math. Phys. (2009) 437-476. [arXiv:math/9903134]
    [1-2] T. Sasamoto, Fluctuations of the one-dimensional asymmetric exclusion process using random matrix 
    techniques, J. Stat. Mech. (2007) P07007. [arXiv:0705.2942]
    [2] A. Borodin, L. Petrov, Lectures on Integrable probability: Stochastic vertex models and symmetric functions.
    [arXiv: 1605.01349] 
    [3-1] G.M. Schutz, Duality Relations for Asymmetric Exclusion Processes, J. Stat. Phys. 86 (1997) 1265-1287. 
    [3-2] A. Borodin, I. Corwin, T. Sasamoto, From duality to determinants for q-TASEP and ASEP, Ann. Prob. 42 (2014) 2341– 2382. [arXiv:1207.5035]
    [3-3]G. Carinci, C. Giardina, F. Redig, T. Sasamoto, A generalized asymmetric exclusion process with Uq(sl2) stochastic duality, Prob. Th. Rel. Fields. 166, 887(2016). [arXiv: 1407.3367]
    [4]. T. Imamura, T. Sasamoto, Fluctuations for stationary q- TASEP, to appear in Prob. Th. Rel. Fields. 
        [arXiv:1701.05991]
    [5]. Z. Chen, J. de Gier, I. Hiki, T. Sasamoto, Exact confirmation of 1D nonlinear fluctuating hydrodynamics for a two-species exclusion process, Phys. Rev. Lett. 120, 240601 (2018). [arXiv:1803.06829] 

    3. Lecture 1 Notes Lecture 2 Notes Lecture 3 Notes Lecture 4 Notes Lecture 5 Notes 1 Lecture 5 Notes 2

     
     
     

    Week 3: 28 January - 1 February

  3. Speaker: Daniel Remenik

    Title: The KPZ fixed point

    Abstract: 

    The Kardar-Parisi-Zhang (KPZ) universality class is a broad class of models coming from mathematical physics which includes random interface growth, directed random polymers, interacting particle systems, and random stirred fluids. These models share a very special and rich asymptotic fluctuation behavior, which is loosely characterized by fluctuations which grow like t^{1/3} as time t evolves, decorrelate at a spatial scale of t^{2/3}, and have certain very special limiting distibutions; this fluctuation behavior is model independent but depends on the initial data, and in some important cases it is connected with distributions coming from random matrix theory.

    A somewhat vague conjecture in the field was that there should be a universal, scaling invariant limit for all models in the KPZ class, containing all the fluctuation behavior seen in the class. In these lectures I will describe joint work with K. Matetski and J. Quastel [5] where we were able to construct and give a complete description of this limiting process, known as the KPZ fixed point. This limiting universal process is a Markov process, taking values in real valued functions which look locally like Brownian motion.

    The construction follows from an novel exact solution for one of the most basic models in the KPZ class, the totally asymmetric exclusion process (TASEP), for arbitrary initial condition. This formula is given as the Fredholm determinant of a kernel involving the transition probabilities of a random walk forced to hit a curve defined by the initial data, and in the KPZ 1:2:3 scaling limit the formula leads in a transparent way to a Fredholm determinant formula given in terms of analogous kernels based on Brownian motion.

    Two sets of lecture notes [6,7] serve as a good complement to the mini-course.

    References:

    1. A. Borodin, P. L. Ferrari, M. Prähofer, and T. Sasamoto. Fluctuation properties of the TASEP with periodic initial configuration. J. Stat. Phys. 129.5-6 (2007), pp. 10551080.
    2. A. Borodin, I. Corwin, and D. Remenik. Multiplicative functionals on ensembles of non-intersecting paths. Ann. Inst. H. Poincaré Probab. Statist. 51.1 (2015), pp. 28–58.
    3. I. Corwin, J. Quastel, and D. Remenik. Continuum statistics of the Airy 2 process. Comm. Math. Phys. 317.2 (2013), pp. 347–362.
    4. J. Quastel, D. Remenik. How flat is flat in random interface growth? To appear in Trans. AMS. arXiv:1606.09228. 
    5. K. Matetski, J. Quastel and D. Remenik. The KPZ fixed point. arXiv:1701.00018.
    6. K. Matetski, J. Quastel. From TASEP to the KPZ fixed point. arXiv:1710.02635.
    7. D. Remenik. Course notes on the KPZ fixed point.
    8. T. Sasamot o. Spatial correlations of the 1D KPZ surface on a flat substrate. J. Phys. A 38.33 (2005), p. L549.

     

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    Week 4: 04 February - 08 February

  4. Speaker: Lionel Levine

    Title: Abelian networks and sandpile models

    Abstract: 

    In this mini-course, we explore particle systems with an “abelian property”: the order of certain interactions does not matter. Deepak Dhar [D] observed that many dynamical questions (what does this particle system do?) have a computational side (what can this network of automata compute?). The computational counterpart of the abelian property is a “least action principle”: the particles conspire to solve a certain optimization problem, by reaching stability in the most efficient possible way.

    We are interested in the phase transition between activity and fixation, and in universal properties of the “threshold state” that separates the two phases. The dynamical question “will this system fixate?” corresponds to the computational question “will this program halt?”. Alan Turing proved in 1936 that the latter question is undecidable in general. Hence, we should expect these systems to be hard, and there will be no completely satisfactory general theory; but questions in the neighborhood of an undecidable question are where the most fruitful mathematics lies!

    References: 
    [BL] Bond, Levine, https://arxiv.org/abs/1309.3445
    [C] Cairns, https://arxiv.org/abs/1508.00161
    [D] Dhar, https://doi.org/10.1016/j.physa.2006.04.004
    [H+] Holroyd et al., https://arxiv.org/abs/0801.3306
    [J] Jarai, https://arxiv.org/abs/1401.0354
    [L] Levine, https://arxiv.org/abs/1402.3283
    [LP] Levine, Peres, https://arxiv.org/abs/1611.00411
    [RSZ] Rolla,Sidoravicius,Zindy https://arxiv.org/abs/1707.06081