ICTS

Consider an $n \times n$ matrix with i.i.d.~Bernoulli($p$) entries. It is well known that for $p= \Omega(1)$, i.e.~$p$ is bounded below by some positive constant, the matrix is invertible with high probability. If $p \ll \frac{\log n}{n}$ then the matrix contains zero rows and columns with high probability and hence it is singular with high probability. In this talk, we will discuss the sharp transition of the invertibility of this matrix at $p =\frac{\log n}{n}$. This phenomenon extends to the adjacency matrices of directed and undirected Erd\H{o}s-R\'{e}nyi graphs, and random bipartite graph. This is joint work with Mark Rudelson.

For fixed positive integers m, we consider the product of m independent n by n random matrices with iid entries as in the limit as n tends to infinity. Under suitable assumptions on the entries of each matrix, it is known that the limiting empirical distribution of the eigenvalues is described by the m-th power of the circular law. Moreover, this same limiting distribution continues to hold if each iid random matrix is additively perturbed by a bounded rank deterministic error. However, the bounded rank perturbations may create one or more outlier eigenvalues. We describe the asymptotic location of the outlier eigenvalues, which extends a result of Terence Tao for the case of a single iid matrix. Our methods also allow us to consider several other types of perturbations, including multiplicative perturbations.

In this talk, I will present the framework of the so-called nonlinear large deviations introduced by Chatterjee and Dembo. In a seminal paper, they provided a sufficient criterion in order that the large deviations of a function on the discrete hypercube to be due by only changing the mean of the background measure. This sufficient condition was formulated in terms of the complexity of the gradient of the function of interest. I will present general nonlinear large deviation estimates similar to Chatterjee-Dembo's original bounds except that we do not require any second order smoothness. The approach relies on convex analysis arguments and is valid for a broad class of distributions. Then, I will detail an application of this nonlinear large deviations bounds to the problem of estimating the upper tail of cycles counts in sparse Erdös-Renyi graphs down to the connectivity parameter $n^{-1/2}$.

Products of random matrices have always been a topic of interest in  Mathematics, Physics and Statistics for various reasons. In this talk, we shall discuss, along with their relevance, the  exact eigenvalue distributions of certain product random matrix models and also the  asymptotic behaviour  of the eigenvalues of products of random matrices with the  matrix sizes fixed  and the number of matrices in the product increasing.

What is the typical inner product between eigenvectors of non-normal matrices from the invariant ensembles with density proportional to $e^{-\mbox{ \rm Tr} V(XX^*)} dX$? In the Ginibre case (i.e. $V(x)=x$), when the eigenvectors are chosen to correspond to specific eigenvalues,
a CLT can be proved (after proper re-scaling). In the general case, the scale is known, but no limit law is known. I will describe the known (to me)  results and their proofs. I will also  describe a particular large deviations problem that seems essential for th e general case. 
(Joint work with Florent Benaych-Georges)

In random matrix theory, after defining a family of random matrix one of the first question one asks is about the existence and regularity of limiting empirical spectral distrib ution (LSD). Here, I will talk about the absolute continuity and bound on the density of LSD for random Hankel and Toeplitz matrices.

We establish an exact mapping between the positions of N non interacting fermions in a 2d rotating harmonic trap in its ground-state and the eigenvalues of the NxN complex Ginibre ensemble of Random Matrix Theory (RMT). Using RMT techniques, we make precise predictions for the statistics of the positions of the fermions, both in the bulk as well as at the edge of the trapped Fermi gas. In addition, we compute exactly, for any finite N, the R\'enyi entanglement entropy and the number variance of a disk of radius r in the ground-state. We show that while these two quantities are proportional to each other in the (extended) bulk, this is no longer the case very close to the trap center nor at the edge. Near the edge, and for large N, we provide exact expressions for the scaling functions associate d with these two observables.

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.

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.

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.
 

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.

The dimer model is a model from statistical mechanics corresponding to random perfect matchings on graphs. Circle patterns are a class of embeddings of planar graphs such that every face admits a circumcircle. In this talk I describe a correspondence between dimer models on planar bipartite graphs and circle pattern embeddings of these graphs. As special cases of this correspondence we recover the Tutte embeddings (a.k.a harmonic embeddings) for resistor networks and the s-embeddings for Ising models. This correspondence is also the key for studying Miquel dynamics, a discrete integrable system on circle patterns. This is joint work with Richard Kenyon (Brown University), Wai Yeung Lam (Brown University) and Marianna Russkikh (University of Geneva).

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.
 

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.

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.

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.

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.

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.

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

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

The frog model is a branching random walk on a graph in which particles branch only at unvisited sites. I will discuss a collection of results that are joint with Tobias Johnson and Matthew Junge. These papers all discuss the frog model on regular trees. On infinite trees we consider the question of transience and recurrence. On finite depth trees we examine the time it takes before every vertex on the graph is visited.

We will discuss set of models where the random walk (or Brownian motion) moves in a restricted domain D(t), which itself evolves in time. This evolution could be independent of random walk evolution, but still affecting its motion, or evolution of D itself could be affected by random walk.
Then I will focus on the phase transition (recurrence vs transience) for once reinforced random walks and some other self interacting processes.

Modern statistical mechanics offers a large class of driven-dissipative stochastic systems that naturally evolve to a critical state, of which Activated Random Walks are perhaps the best example. The main pursuit in this field is to show universality of critical parameters, describe the critical behavior, the scaling relations and critical exponents of such systems, and the connection between driven-dissipative dynamics and conservative dynamics in infinite space. The study of this model was an open challenge for a long time, then it had signifi cant partial progress a decade ago, and got stuck again. Through the last 5 years it has seen exciting progress thanks to contributions by Asselah, Basu, Cabezas, Ganguly, Hoffman, Schapira, Sidoravicius, Sousi, Stauffer, Taggi, Teixeira, Tournier, Zindy, and myself. These covered most of the questions regarding existence of an absorbing and an active phase for different ranges of parameters, and current efforts are drifting towards the description of critical states, scaling limits, etc. We will summarize the current state of art and discuss some of the many open problems.

We describe the growing patterns formed in the rotor-router model, starting from noisy initial conditions. By the detailed study of two cases, we show that: (a) the boundary of the pattern for a certain class of initial conditions displays KPZ fluctuations with a Tracy-Widom distribution, (b) by changing the amount of randomness, one can induce a transition in which the rotor-router path changes from recurrent to transient. We show that this transition falls in the 3D Anisotropic Directed Percolation universality class.