## Integrable Probability and Related Fields from a Safe Distance

Seminar happens on Thursdays 2:30pm ET. During Spring 2021 this seminar will partially merge with the course 18.677 Topics in Stochastic Processes at MIT.

**Seminar description:**Talks should ideally highlight a key idea or technique which is interesting and comprehensible to those on other parts of the field, and we especially welcome talks which are largely or entirely expository.

Subscription to a mailing list.

If you don't want to get information emails and just want to get the Zoom link, please write an email to one of the organizers.

**Schedule:**

04/22/2021. Speaker: Jonas Arista (University of Chile)

Title:

*Exit distributions associated with loop-erased walks and random matrices.*Abstract: Non-intersecting processes in one dimension have long been an integral part of random matrix theory, at least since the pioneering work of F. Dyson in the 1960s. For example, if one considers n independent one-dimensional Brownian particles, started at the origin and conditioned not to intersect up to a fixed time T, then the locations of the particles at time T have the same distribution as the eigenvalues of a random real symmetric n×n matrix with independent centred Gaussian entries, with variance T on the diagonal and T/2 above the diagonal, this is known as the Gaussian Orthogonal Ensemble (or GOE). For planar (two-dimensional) state space processes, it is not clear how to generalise the above connections since the paths under consideration are allowed to have self-intersections (or loops). In this talk, we address this problem and consider systems of random walks in planar graphs constrained to a certain type of non-intersection between their loop-erased parts (this is closely related to connectivity probabilities of branches of the uniform spanning tree). We show that in a suitable scaling limit in terms of independent planar Brownian motions, certain exist distributions have also connections with random matrices, mainly Cauchy-type and circular ensembles. This is joint work with Neil O’Connell.

**Postponed.**Speaker: Mikhail Tikhonov (University of Virginia)

Title:

*Permuting the GUE corners eigenvalues*Abstract: The perturbed GUE corners ensemble is the joint distribution of eigenvalues of all principal submatrices of a matrix G+diag(a), where G is the random matrix from the Gaussian Unitary Ensemble (GUE), and diag(a) is a fixed diagonal matrix. We introduce Markov transitions based on exponential jumps of eigenvalues, and show that their successive application is equivalent in distribution to a deterministic shift of the matrix. This result also leads to a new distributional symmetry for a family of reflected Brownian motions with drifts coming from an arithmetic progression. Based on a joint work with L. Petrov.

**Previous talks:**

04/15/2021. Speaker: Mustazee Rahman (Durham University)

Title:

*A random growth model and its time evolution*(slides)Abstract: Planar random growth models are irreversible statistical mechanical systems with a notion of time evolution. It is of some interest to understand this evolution by studying joint distribution of points along the time-like direction. One of these models, polynuclear growth or last passage percolation, has allowed exact statistical calculations due to a close relation to determinantal processes. I will discuss recent works with Kurt Johansson where we calculate its time-like distribution using ideas surrounding determinantal processes. One can take a scaling limit of this distribution, which is then expected to be universal for the time distribution of many random growth models.

04/08/2021. Speaker: Mackenzie Simper (Stanford University)

Title:

*Induced Probability Distibutions on Double Cosets*(slides)Abstract: Suppose $H$ and $K$ are subgroups of a finite group $G$ and consider the $H-K$ double cosets. The uniform distribution on $G$ induces a probability distribution on this space of double cosets. I will discuss several examples where things are nice: the double cosets are indexed by classical combinatorial objects, and the induced distributions are well-known measures. When $G = S_n$ and $H$and $K$ are parabolic subgroups, the double cosets are contingency tables with fixed row and column sums. The induced distribution is the Fisher-Yates distribution, commonly used in statistical tests of independence. The random transpositions Markov chain on $S_n$ induces a natural Markov chain on contingency tables, for which we can study the eigenvalues and eigenfunctions. Joint work with Persi Diaconis.

03/25/2021. Speaker: Harriet Walsh (ENS de Lyon)

Abstract: Schur measures on integer partitions define a class of determinantal point processes, or free fermion models, by way of symmetric functions. They generalise the Plancherel measure, and generically have edge fluctuations with a 1/3 exponent characteristic of the KPZ class. We introduce “multicritical” Hermitian Schur measures with 1/(2n+1) critical exponents. Their asymptotic edge distributions are higher-order analogues of the Tracy-Widom GUE distribution first observed by Le Doussal, Majumdar and Schehr for the edge momenta of trapped fermions. We find explicit examples of these measures, and compute limit shapes. By relating Schur measures to unitary matrix models, we study the phase transitions corresponding to these new edge statistics, and explain a connection with certain models of string theory. Based on joint work with Dan Betea and Jérémie Bouttier.

02/25/2021. Speaker: Jimmy He (Stanford University)

Abstract: The Mallows measure on the symmetric group gives a way to generate random permutations which are more likely to be sorted than not. There has been a lot of recent work to try and understand limiting properties of Mallows permutations. I'll discuss some work on the joint distribution of descents, a statistic counting the number of "drops" in a permutation, and descents in its inverse, generalizing work of Chatterjee and Diaconis, and Vatutin. The proof uses Stein's method with a size-bias coupling as well as a regenerative representation of Mallows permutations due to Gnedin and Olshanski.

02/10/2021.

**(Wednesday)**Speaker: Yaroslav Kubivskyi (University of Geneva)Abstract: In this second talk we will discuss the general strategy of establishing the interface convergence to SLEs processes in statistical mechanics models. We will concentrate our attention on sufficient conditions, which guarantee the precompactness of the probability measures associated with the interfaces. One of these equivalent conditions is the following - the probability of an unforced crossing of any annulus A(z,r,R) = {y in C : r< |y-z|<R} by interface is less than 1/2. If the time will permit we will verify the conditions in some models, namely FK percolation.

**11/27/2020. Friday, 3pm**

Speaker: Nikolai Bobenko (University of Geneva)

Title:

*Schramm Loewner Evolution.*Abstract: In preparation for next week's talk I will deliver an overview over the central concepts and definitions in the theory of Schramm-Loewner evolution and its relation to conformal invariance.

11/19/2020.

**10:00 am.**Speaker: Jiyuan Zhang (University of Melbourne)

Title:

*Sums and products of invariant ensembles.*Abstract: An invariant ensemble is a random matrix that is unchanged under an adjoint action of a group of invariance. In this talk we will focus on sums/products of such ensembles, where general formulae for those eigenvalue probability density functions are provided, connecting eigenvalue PDFs to additive/multiplicative weights. There, a matrix addition/multiplication corresponds to convolution of their weights. Such connections can be shown for the additive spaces of the Hermitian, Hermitian antisymmetric, Hermitian anti-self-dual, and complex rectangular matrices as well as for the two multiplicative matrix spaces of the positive definite Hermitian matrices and of the unitary matrices. This is a joint work with Mario Kieburg (arXiv:2007.15259 [math-ph]).

11/12/2020.

**2:30pm.****Joint seminar with Columbia University.**Speaker: Kurt Johansson (KTH)

Title:

*Multivariate normal approximation for traces of random unitary matrices.*Abstract: Consider an n x n random unitary matrix U taken with respect to normalized Haar measure. It is a well known consequence of the strong Szego limit theorem that the traces of powers of U converge to independent complex normal random variables as n grows. I will discuss a recent result where we obtain a super-exponential rate of convergence in total variation distance between the traces of the first m powers of an n × n random unitary matrices and a 2m-dimensional Gaussian random vector. This generalizes previous results in the scalar case, which answered a conjecture by Diaconis, to the multivariate setting. We are especially interested in the regime where m grows with n. The problem on how the rate of convergence changes as m grows with n was raised recently by Sarnak. The result we obtain gives the dependence on the dimensions m and n in the estimate with explicit constants for m almost up to the square root of n. This is joint work with Gaultier Lambert.

11/05/2020. Speaker: Xiao Shen (University of Wisconsin)

Title:

*Coalescence estimates for the corner growth model with exponential weights.*Abstract: We establish estimates for the coalescence time of semi-infinite directed geodesics in the planar corner growth model with i.i.d. exponential weights. There are four estimates: upper and lower bounds on the probabilities of both fast and slow coalescence on the correct spatial scale with exponent 3/2. Our proofs utilize a geodesic duality introduced by Pimentel and properties of the increment-stationary last-passage percolation process. For fast coalescence our bounds are new and they have matching optimal exponential order of magnitude. For slow coalescence, we reproduce bounds proved earlier with integrable probability inputs, except that our upper bound misses the optimal order by a logarithmic factor. Joint work with Timo Seppäläinen.

10/29/2020. Speaker: Joonas Turunen (ENS de Lyon)

Title:

*Ising model on random triangulations of the half-plane: critical behavior and phase transition.*Abstract: We consider random planar triangulations of the disk coupled with an Ising model at a fixed temperature, emphasizing their local limits as the perimeter of the disk tends to infinity. In this model, the critical behavior of the partition functions is shown to differ from the one of the pure gravity university class at the critical temperature, and a phase transition is identified rigorously. We study how the interface geometry changes in this phase transition as well as find some scaling limits for the length of the infinite interface at the critical temperature which have interpretations in the continuum Liouville Quantum Gravity. The two key techniques in use are singularity analysis of rational parametrizations together with analytic combinatorics, as well as an exploration process of the Ising interface. Based on joint research with Linxiao Chen.

10/22/2020. Speaker: Yier Lin (Columbia University)

Title:

*Short time large deviations of the KPZ equation.*Abstract:

*We establish the Freidlin--Wentzell Large Deviation Principle (LDP) for the Stochastic Heat Equation with multiplicative noise in one spatial dimension. That is, we introduce a small parameter $\sqrt{\epsilon}$ to the noise, and establish an LDP for the trajectory of the solution. Such a Freidlin--Wentzell LDP gives the short-time, one-point LDP for the KPZ equation in terms of a variational problem. Analyzing this variational problem under the narrow wedge initial data, we prove a quadratic law for the near center tail and a 5/2 law for the deep lower tail. These power laws confirm existing physics predictions Kolokolov and Korshunov (2007), Kolokolov and Korshunov (2009), Meerson, Katzav, and Vilenkin (2016), Le Doussal, Majumdar, Rosso, and Schehr (2016), and Kamenev, Meerson, and Sasorov (2016). Joint work with Li-Cheng Tsai.*10/15/2020. Speaker: Alexander Moll (University of Massachusetts Boston)

Title:

*Multi-phase z-measures and their origins.*Abstract: We introduce n-phase z-measures on partitions and show that these measures admit piecewise-linear limit shapes with exactly 2n+1 interlacing local extrema. Â In the case n=1, we show that the z-measures of Borodin-Olshanski admit limit shapes that are rectangles. Â For most of the talk, we discuss the origin of these measures in the supercritical problem of perturbing the n-phase solutions of the Benjamin-Ono equation on the circle by the gradient of the log-correlated Gaussian field.

10/08/2020. Speaker: Matan Harel (Northeastern University)

Title:

*Exponential Decay of Correlations in the two dimensional Random Field Ising Model.*Abstract: In the mid 1970's, Y. Imry and S.K. Ma predicted that the addition of an arbitrarily weak random external field to the two-dimensional Ising model would "round out" the phase transition, so that there would be a unique Gibbs measure at all temperatures. This was rigorously proven by M. Aizenmann and J. Wehr in the late 1980's, though the proof did not offer any quantitative control of decay rates for this unique Gibbs measure. We use an extension of the Ising model to vertex and edge spins to produce exact representations of the order parameter of the Random Field Ising Model (RFIM). We use that percolation model to quantify the Aizenman-Wehr proof, and show that correlations decay exponentially fast in the RFIM. This is joint work with Ron Peled and Michael Aizenman.

10/01/2020. Speaker: Alisa Knizel (University of Chicago)

Title:

*Matrix Product Ansatz and ASEP.*Abstract: I will talk about a classical approach to study stationary distribution in a class of interacting particle systems called Matrix Product Ansatz. I will use open ASEP as the main example.

09/24/2020. Speaker: Promit Ghosal (MIT)

Title:

*Probabilistic Liouville conformal blocks and its properties*.Abstract: Conformal blocks are essential ingredients of conformal field theory. Liouville conformal blocks are especially important because of its connection with supersymmetric (SUSY) gauge theory. This talk will focus on a probabilistic definition of Liouville conformal block on the torus and its properties. In particular, I will discuss the relation between probabilistic conformal block and Nekrasov's partition function. If time permits, I will touch on an ongoing work on the modular transformation relations of the conformal blocks, a property closely related to the S-duality in SUSY gauge theory. This talk will be entirely based on joint works with Guillaume Remy, Xin Sun and Yi Sun.

09/17/2020. Speaker: Amol Aggarwal (Harvard University)

Title:

*Arctic Boundaries in Ice Models*.Abstract: Certain two-dimensional models in statistical mechanics are widely known or believed to exhibit arctic boundaries, which are sharp transitions from ordered (frozen) to disordered (temperate) phases. In this talk we will explain a general heuristic devised by Colomo-Sportiello in 2016, known as the geometric tangent method, for locating these arctic boundaries in such models. We will also outline a (more recent) mathematical justification of this tangent method for the domain-wall six-vertex model at ice point, which is based on a probabilistic analysis of non-crossing directed path ensembles.

09/10/2020. Speaker: Xuan Wu (University of Chicago)

Title:

*The Airy Sheet and the directed landscape*.*Part II.*Abstract: This is an expository talk where we will present recent progress by Dauvergne, Ortmann, and Virag on the KPZ Fixed Point and the directed landscape. These are universal objects which appear as long-time fluctuations in various probabilistic models such as TASEP, LPP, ballistic deposition, and the Eden model. The authors observed that this directed landscape is a sort of brownian motion in the (max, +) convolution algebra, which reduced the difficulty to constructing the time 1 distribution of this object (the Airy Sheet). To construct this, they used a coupling with the Airy line ensemble using a Busemann function. This in turn involved a continuum version of the RSK correspondence called the Melon Identity, as well as precise estimates for Dyson Brownian motion and the Airy Line ensemble which use only the determinantal structure and the Gibbs resampling property of these objects.

09/03/2020. Speaker: Shalin Parekh (Columbia University)

Title:

*The Airy Sheet and the directed landscape*.*Part I.*Abstract: This is an expository talk where we will present recent progress by Dauvergne, Ortmann, and Virag on the KPZ Fixed Point and the directed landscape. These are universal objects which appear as long-time fluctuations in various probabilistic models such as TASEP, LPP, ballistic deposition, and the Eden model. The authors observed that this directed landscape is a sort of brownian motion in the (max, +) convolution algebra, which reduced the difficulty to constructing the time 1 distribution of this object (the Airy Sheet). To construct this, they used a coupling with the Airy line ensemble using a Busemann function. This in turn involved a continuum version of the RSK correspondence called the Melon Identity, as well as precise estimates for Dyson Brownian motion and the Airy Line ensemble which use only the determinantal structure and the Gibbs resampling property of these objects.

08/27/2020. Speaker: Marcin Lis (University of Vienna)

Title:

*On topological correlations in planar Ashkin-Teller models*.Abstract: We generalize the switching lemma of Griffiths, Hurst, and Sherman to the random current representation of Ashkin–Teller models. We then use it together with properties of two-dimensional topology to derive linear relations for multi-point boundary spin correlations and bulk order-disorder correlations in planar models. We also show that the same linear relations are satisfied by products of Pfaffians. As a result a clear picture arises in the noninteracting case of two independent Ising models, where multi-point correlation functions are given by Pfaffians and determinants of their respective two-point functions. This gives a unified treatment of both the classical Pfaffian identities and recent total positivity inequalities of boundary correlations in the planar Ising model. We also derive the Simon and Gaussian inequality for general Ashkin–Teller models with negative four-body coupling constants.

08/20/2020. Speaker: Paul Melotti (University of Fribourg)

Title:

*The eight-vertex model via dimers*.Abstract: The eight-vertex model is an useful description that generalizes several spin systems such as the Ashkin-Teller model, as well as the more common six-vertex model, and others. In a special "free-fermion" regime, it is known since the work of Fan, Lin, Wu in the late 60s that the model can be mapped to non-bipartite dimers. However, no general theory is known for dimers in the non-bipartite case, contrary to the extensive rigorous description of Gibbs measures by Kenyon, Okounkov, Sheffield for bipartite dimers. In this talk I will show how to transform these non-bipartite dimers into bipartite ones, on generic planar graphs. I will mention a few consequences: computation of long-range correlations, criticality and critical exponents, and their "exact" application to Z-invariant regimes on isoradial graphs.

08/13/2020.

**6:00pm**Speaker: Matteo Mucciconi (Tokyo Institute of Technology)Title:

*Stochastic vertex models and interlacing arrays*.Abstract: For models "at positive temperature" in the KPZ class solvability usually comes in two flavors: Macdonald processes and quantum integrability. The first one leverages algebraic and combinatorial properties of Macdonald polynomials to study physical observables of the system. The second uses the Bethe Ansatz and the Yang-Baxter equation. In recent collaborations with A. Bufetov and L. Petrov we bridge this gap studying particle processes weighted by "spin" variants of Macdonald polynomials. Consequences of these constructions are two: 1) many models like the higher spin vertex model or the q-Hahn TASEP are recovered as marginals of two dimensional growth processes 2) unexpected properties of the spin q-Whittaker polynomials are proven or conjectured.

08/06/2020. Speaker: Cesar Cuenca (Caltech)

Title:

*Kerov's mixing construction and the representation ring of finite groups*Abstract: Kerov's mixing construction for the symmetric groups produces a list of specializations of the ring of symmetric functions that are positive on Schur functions. I will explain this construction for the finite groups GL(n, q), where q is a prime power. Then I will show that it can be degenerated and extended to other finite groups of Lie type. Most of this is classical stuff, and only the last part is joint work with Grigori Iosifovich Olshanski.

07/16/2020. Speaker: Alexey Bufetov (University of Bonn)

Title:

*ASEP through symmetric functions*.Abstract: In pioneering works Tracy and Widom proved several asymptotic results about ASEP with step initial condition, including t

^{1/3}fluctuations of the height function. I will discuss the proof of Borodin and Olshanski of some of these results, which are based on symmetric functions machinery. I will also discuss how their results can be applied to ASEP with finitely many particles.07/09/2020. Speaker: Mark Rychnovsky (Columbia University)

Title:

*The beta random walk in random environment.*07/02/2020. Speaker: Leonid Petrov (University of Virginia)

Title:

*Yang-Baxter equation and symmetric functions.*Abstract: I will discuss symmetric functions arising in the context of the sl(2) higher spin six vertex model. I will discuss some known results, and also a new refined Cauchy identity with a determinantal right-hand side.

6/18/2020. Speaker: Sayan Das (Columbia University)

Title:

*Probabilistic Aspects of the KPZ equation.*06/11/2020. OOPS Integrable Probability Workshop (no seminar)

06/04/2020. Speaker: Mikhail Khristoforov (University of Helsinki)

Title:

*Smirnov's proof of Cardy's formula and around.*05/28/2020. Speaker: Sergei Korotkikh (MIT)

Title:

*Stochastic 6-vertex model in integrable probability.*05/21/2020. Speaker: Roger Van Peski (MIT)

Title:

*Symmetric functions and random matrices.*05/14/2020. Speaker: Andrew Ahn (MIT)

Title:

*GFF in q*^{vol}plane partitions.05/07/2020. Speaker: Marianna Russkikh (MIT)

Title:

*Discrete complex analysis in lattice models. Part II.*05/01/2020. Speaker: Marianna Russkikh (MIT)

Title:

*Discrete complex analysis in lattice models. Part I.*