Jekyll2019-05-17T09:27:53-04:00https://lpetrov.cc/posts/feed.xmlLeonid PetrovLeonid Petrov. Integrable ProbabilityLeonid PetrovYang-Baxter random fields and stochastic vertex models2019-05-15T00:00:00-04:002019-05-15T00:00:00-04:00https://lpetrov.cc/2019/05/BMP_YB<p>Bijectivization refines the Yang-Baxter equation into a pair of local Markov moves which randomly update the configuration of the vertex model. Employing this approach, we introduce new Yang-Baxter random fields of Young diagrams based on spin $q$-Whittaker and spin Hall-Littlewood symmetric functions. We match certain scalar Markovian marginals of these fields with (1) the stochastic six vertex model; (2) the stochastic higher spin six vertex model; and (3) a new vertex model with pushing which generalizes the $q$-Hahn PushTASEP introduced recently by Corwin-Matveev-Petrov (2018). Our matchings include models with two-sided stationary initial data, and we obtain Fredholm determinantal expressions for the $q$-Laplace transforms of the height functions of all these models. Moreover, we also discover difference operators acting diagonally on spin $q$-Whittaker or (stable) spin Hall-Littlewood symmetric functions.</p>Leonid PetrovBijectivization refines the Yang-Baxter equation into a pair of local Markov moves which randomly update the configuration of the vertex model. Employing this approach, we introduce new Yang-Baxter random fields of Young diagrams based on spin $q$-Whittaker and spin Hall-Littlewood symmetric functions. We match certain scalar Markovian marginals of these fields with (1) the stochastic six vertex model; (2) the stochastic higher spin six vertex model; and (3) a new vertex model with pushing which generalizes the $q$-Hahn PushTASEP introduced recently by Corwin-Matveev-Petrov (2018). Our matchings include models with two-sided stationary initial data, and we obtain Fredholm determinantal expressions for the $q$-Laplace transforms of the height functions of all these models. Moreover, we also discover difference operators acting diagonally on spin $q$-Whittaker or (stable) spin Hall-Littlewood symmetric functions.Simulations of the q reversal in q-vol lozenge tilings of the hexagon2019-04-30T00:00:00-04:002019-04-30T00:00:00-04:00https://lpetrov.cc/2019/04/q-vol-simulations<div><a href="https://lpetrov.cc/2019/04/q-vol-simulations/">Simulations of the $q$ parameter reversal in $q^{volume}$ lozenge tilings of the hexagon</a></div>
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<ul>
<li>
<h2 id="1---simulations-of-a-new-left-jumping-dynamics"><a href="https://lpetrov.cc/simulations/2019-04-30-qvol/">1</a> - Simulations of a new left-jumping dynamics</h2>
</li>
<li>
<h2 id="2---a-naive-mirroring-dynamics"><a href="https://lpetrov.cc/simulations/2019-05-02-qvol-mirroring/">2</a> - A naive mirroring dynamics</h2>
</li>
</ul>Leonid PetrovSimulations of the $q$ parameter reversal in $q^{volume}$ lozenge tilings of the hexagonFrom infinite random matrices over finite fields to square ice2019-03-12T08:00:00-04:002019-03-12T08:00:00-04:00https://lpetrov.cc/GLnq_6V<p>Asymptotic representation theory of symmetric groups is a rich and beautiful subject with deep connections with probability, mathematical physics, and algebraic combinatorics. A one-parameter deformation of this theory related to infinite random matrices over a finite field leads to a randomization of the classical Robinson-Schensted correspondence between words and Young tableaux. Exploring such randomizations we find unexpected applications to six vertex (square ice) type models and traffic systems on a 1-dimensional lattice.</p>
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<p><a href="https://storage.lpetrov.cc/research_files/talks/GLnqSquareIce.pdf" target="_blank">PDF (32 MB)</a></p>Leonid PetrovAsymptotic representation theory of symmetric groups is a rich and beautiful subject with deep connections with probability, mathematical physics, and algebraic combinatorics. A one-parameter deformation of this theory related to infinite random matrices over a finite field leads to a randomization of the classical Robinson-Schensted correspondence between words and Young tableaux. Exploring such randomizations we find unexpected applications to six vertex (square ice) type models and traffic systems on a 1-dimensional lattice.Reading seminar on integrable probability in Spring 20192019-01-16T00:00:00-05:002019-01-16T00:00:00-05:00https://lpetrov.cc/reading-seminar<div><a href="https://lpetrov.cc/reading-2019/">Reading seminar on integrable probability</a> in Spring 2019</div>
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<h2 class="mt-4 mb-3"> Fridays, 11:00am - 12:30pm, Kerchof 128</h2>
<p>We will discuss the classical problem of describing analogues of irreducible representations of the infinite-dimensional unitary group (Edrei-Voiculescu theorem), and its connections with</p>
<ul>
<li>representations of finite-dimensional unitary groups</li>
<li>symmetric functions</li>
<li>random lozenge tilings and asymptotic combinatorics</li>
<li>total positivity</li>
<li>classical analysis (e.g. B-splines)</li>
<li>von Neumann factor and spherical representations of the infinite-dimensional unitary group</li>
<li>random matrices</li>
</ul>
<hr />
<h3 id="schedule-of-meetings">Schedule of meetings</h3>
<p>The schedule is subject to change as we progress with the topics.</p>
<ol>
<li>
<p>January 18, 2019, special location - <strong>Ruffner Hall 127</strong>.</p>
<p><em>Leo Petrov</em> - Introduction “The many facets of the Edrei-Voiculescu theorem”</p>
</li>
<li>
<p>January 25, 2019, special location - <strong>Ruffner Hall 123</strong>.</p>
<p><em>Liron Speyer</em> - $U(N)$: weights, irreducible characters, Schur polynomials, branching
<br />
[standard textbooks like the H. Weyl’s], a summary is in [section 2.2 in arXiv:1310.8007]</p>
</li>
<li>
<p>February 1, 2019</p>
<p><em>George Seelinger</em> - Extreme characters of $U(\infty)$: definition and two elementary examples [<a href="https://ghseeli.github.io/grad-school-writings/presentations/two-elementary-examples-of-extreme-characters-of-u-infty.pdf">PDF notes</a>]
<br />
[for example, section 4.1.10 in <a href="https://d3m0khvr0ybm92.cloudfront.net/courses/7382F12/LectureNotes.pdf">these draft lecture notes</a>]</p>
</li>
<li>
<p>February 8, 2019</p>
<p><em>Axel Saenz</em> - Functional equation for irreducible characters and the ring theorem for extreme characters of $U(\infty)$
<br />
[G. Olshanski, On semigroups related to infinite-dimensional groups. In: Topics in Representation Theory (A. A. Kirillov, ed.). Advances in Soviet Math.] or [A. Borodin, G. Olsahnski “Representations of the Infinite Symmetric Group”, exercises 1.5 and 4.3]</p>
</li>
<li>
<p>February 15, 2019</p>
<p><em>Ethan Zell</em> - Projection formula for Gibbs measures on tilings (formula for the number of trapezoidal Gelfand-Tsetlin schemes)
<br />
[arXiv:1109.1412 and arXiv:1208.3443]</p>
</li>
<li>
<p>February 22, 2019</p>
<p><em>Leo Petrov</em> - Idea of proof of Edrei-Voiculescu theorem
<br />
[arXiv:1109.1412]</p>
</li>
<li>
<p>March 1, 2019</p>
<p>No meeting
<br /></p>
</li>
<li>
<p>March 8, 2019</p>
<p><em>Joseph Eisner</em> - Characters and von Neumman factor representations. GNS construction
<br /></p>
</li>
<li>
<p>March 22, 2019</p>
<p><em>Axel Saenz</em> - Pascal triangle as a branching graph <a href="https://storage.lpetrov.cc/courses/integrable_seminar/Axel-Pascal-notes.pdf">[PDF notes]</a>
<br />
[<a href="https://link.springer.com/article/10.1007%2FBF01480687">Kerov, “Combinatorial examples in the theory of AF-algebras”</a>],
[A. Borodin, G. Olsahnski “Representations of the Infinite Symmetric Group”],
<a href="https://d3m0khvr0ybm92.cloudfront.net/courses/7382F12/LectureNotes.pdf">these draft lecture notes</a></p>
</li>
<li>
<p>March 29, 2019</p>
<p><em>Arun Kannan</em> - Diffusions from dynamics on the Pascal triangle
<br />
[<a href="https://arxiv.org/pdf/0706.1034.pdf">arXiv:0706.1034</a>, section 2]</p>
</li>
<li>
<p>April 5, 2019</p>
<p>No meeting
<br /></p>
</li>
<li>
<p>April 12, 2019</p>
<p><em>George Seelinger</em> - A q-analogue of de Finetti’s theorem
<br />
[<a href="https://arxiv.org/pdf/0905.0367.pdf">arXiv:0905.0367</a>]</p>
</li>
<li>
<p>April 19, 2019</p>
<p><em>Liron Speyer</em> - Young graph and its boundary
<br />
[A. Borodin, G. Olsahnski “Representations of the Infinite Symmetric Group”]</p>
</li>
<li>
<p>April 26, 2019</p>
<p><em>Joseph Eisner</em> - Construction of extreme representations of the infinite symmetric group
<br />
Chapter 10, [A. Borodin, G. Olsahnski “Representations of the Infinite Symmetric Group”]</p>
</li>
</ol>Leonid PetrovReading seminar on integrable probability in Spring 2019Erratum to “Spectral theory for interacting particle systems solvable by coordinate Bethe ansatz”2018-12-19T00:00:00-05:002018-12-19T00:00:00-05:00https://lpetrov.cc/2018/12/BCPS-2-erratum<p>(<a href="https://storage.lpetrov.cc/research_files/Petrov-publ/erratum_1407.pdf">PDF</a>)</p>
<p>This is a correction to Theorems 7.3 and 8.12 in <a href="https://lpetrov.cc/2014/07/BCPS-2/">our paper</a>. These statements claimed to deduce the spatial Plancherel formula (spatial biorthogonality) of the ASEP and XXZ eigenfunctions from the corresponding statements for the eigenfunctions of the q-Hahn system. Such a reduction is wrong.</p>Leonid Petrov(PDF) This is a correction to Theorems 7.3 and 8.12 in our paper. These statements claimed to deduce the spatial Plancherel formula (spatial biorthogonality) of the ASEP and XXZ eigenfunctions from the corresponding statements for the eigenfunctions of the q-Hahn system. Such a reduction is wrong.MATH 7310 • Real Analysis and Linear Spaces I2018-12-18T00:00:00-05:002018-12-18T00:00:00-05:00https://lpetrov.cc/2018/12/grad-realLeonid PetrovGibbs measures, arctic curves, and random interfaces2018-11-30T07:00:00-05:002018-11-30T07:00:00-05:00https://lpetrov.cc/Gibbs<p>This talk outlines connections between 2-dimensional Gibbs measures with a height function
and particle systems in the Kardar-Parisi-Zhang universality class.</p>
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<p><a href="https://storage.lpetrov.cc/research_files/talks/Gibbs2018.pdf" target="_blank">PDF (35 MB)</a></p>Leonid PetrovThis talk outlines connections between 2-dimensional Gibbs measures with a height function and particle systems in the Kardar-Parisi-Zhang universality class.The q-Hahn PushTASEP2018-11-15T00:00:00-05:002018-11-15T00:00:00-05:00https://lpetrov.cc/2018/11/qHahnPush<p>We introduce the $q$-Hahn PushTASEP — an integrable stochastic interacting particle system which is a 3-parameter generalization of the PushTASEP, a well-known close relative of the TASEP (Totally Asymmetric Simple Exclusion Process). The transition probabilities in the $q$-Hahn PushTASEP are expressed through the $_4\phi_3$ basic hypergeometric function. Under suitable limits, the $q$-Hahn PushTASEP degenerates to all known integrable (1+1)-dimensional stochastic systems with a pushing mechanism. One can thus view our new system as a pushing counterpart of the $q$-Hahn TASEP introduced by <a href="https://arxiv.org/abs/1308.3250">Povolotsky</a>. We establish Markov duality relations and contour integral formulas for the $q$-Hahn PushTASEP. We also take a $q\to1$ limit of our process arriving at a new beta polymer-like model.</p>Leonid PetrovWe introduce the $q$-Hahn PushTASEP — an integrable stochastic interacting particle system which is a 3-parameter generalization of the PushTASEP, a well-known close relative of the TASEP (Totally Asymmetric Simple Exclusion Process). The transition probabilities in the $q$-Hahn PushTASEP are expressed through the $_4\phi_3$ basic hypergeometric function. Under suitable limits, the $q$-Hahn PushTASEP degenerates to all known integrable (1+1)-dimensional stochastic systems with a pushing mechanism. One can thus view our new system as a pushing counterpart of the $q$-Hahn TASEP introduced by Povolotsky. We establish Markov duality relations and contour integral formulas for the $q$-Hahn PushTASEP. We also take a $q\to1$ limit of our process arriving at a new beta polymer-like model.Virginia Integrable Probability Summer School2018-10-28T00:00:00-04:002018-10-28T00:00:00-04:00https://lpetrov.cc/2018/10/vipss<div><a href="http://vipss.int-prob.org/">Virginia Integrable Probability Summer School</a> will be held
at University of Virginia
from <b>May 27 to June 8, 2019</b></div>Leonid PetrovVirginia Integrable Probability Summer School will be held at University of Virginia from May 27 to June 8, 2019Generalizations of TASEP in discrete and continuous inhomogeneous space2018-08-29T00:00:00-04:002018-08-29T00:00:00-04:00https://lpetrov.cc/2018/08/Schur-vertex<p>We investigate a rich new class of exactly solvable particle systems generalizing the Totally Asymmetric Simple Exclusion Process (TASEP). Our particle systems can be thought of as new exactly solvable examples of tandem queues, directed first- or last passage percolation models, or Robinson-Schensted-Knuth type systems with random input. One of the novel features of the particle systems is the presence of spatial inhomogeneity which can lead to the formation of traffic jams.</p>
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<p>For systems with special step-like initial data, we find explicit limit shapes, describe hydrodynamic evolution, and obtain asymptotic fluctuation results which put the systems into the Kardar-Parisi-Zhang universality class. At a critical scaling around a traffic jam in the continuous space TASEP, we observe deformations of the Tracy-Widom distribution and the extended Airy kernel, revealing the finer structure of this novel type of phase transitions.</p>
<p>A homogeneous version of a discrete space system we consider is a one-parameter deformation of the geometric last passage percolation, and we obtain extensions of the limit shape parabola and the corresponding asymptotic fluctuation results.</p>
<p>The exact solvability and asymptotic behavior results are powered by a new nontrivial connection to Schur measures and processes.</p>Leonid PetrovWe investigate a rich new class of exactly solvable particle systems generalizing the Totally Asymmetric Simple Exclusion Process (TASEP). Our particle systems can be thought of as new exactly solvable examples of tandem queues, directed first- or last passage percolation models, or Robinson-Schensted-Knuth type systems with random input. One of the novel features of the particle systems is the presence of spatial inhomogeneity which can lead to the formation of traffic jams.