Jekyll2021-10-10T13:46:21-04:00https://lpetrov.cc/posts/feed.xmlLeonid PetrovLeonid Petrov. Integrable ProbabilityLeonid PetrovFree Fermion Six Vertex Model: Symmetric Functions and Random Domino Tilings2021-09-13T00:00:00-04:002021-09-13T00:00:00-04:00https://lpetrov.cc/2021/09/sl11<p>Our work deals with symmetric rational functions and probabilistic models based on the fully inhomogeneous six vertex (ice type) model satisfying the free fermion condition. Two families of symmetric rational functions $F_\lambda,G_\lambda$ are defined as certain partition functions of the six vertex model, with variables corresponding to row rapidities, and the labeling signatures $\lambda=(\lambda_1\ge \ldots\ge \lambda_N)\in \mathbb{Z}^N$ encoding boundary conditions. These symmetric functions generalize Schur symmetric polynomials, as well as some of their variations, such as factorial and supersymmetric Schur polynomials. Cauchy type summation identities for $F_\lambda,G_\lambda$ and their skew counterparts follow from the Yang–Baxter equation. Using algebraic Bethe Ansatz, we obtain a double alternant type formula for $F_\lambda$ and a Sergeev–Pragacz type formula for $G_\lambda$.</p>
<p>In the spirit of the theory of Schur processes, we define probability measures on sequences of signatures with probability weights proportional to products of our symmetric functions. We show that these measures can be viewed as determinantal point processes, and we express their correlation kernels in a double contour integral form. We present two proofs: The first is a direct computation of Eynard–Mehta type, and the second uses non-standard, inhomogeneous versions of fermionic operators in a Fock space coming from the algebraic Bethe Ansatz for the six vertex model.</p>
<p>We also interpret our determinantal processes as random domino tilings of a half-strip with inhomogeneous domino weights. In the bulk, we show that the lattice asymptotic behavior of such domino tilings is described by a new determinantal point process on $\mathbb{Z}^{2}$, which can be viewed as an doubly-inhomogeneous generalization of the extended discrete sine process.</p>Leonid PetrovOur work deals with symmetric rational functions and probabilistic models based on the fully inhomogeneous six vertex (ice type) model satisfying the free fermion condition. Two families of symmetric rational functions $F_\lambda,G_\lambda$ are defined as certain partition functions of the six vertex model, with variables corresponding to row rapidities, and the labeling signatures $\lambda=(\lambda_1\ge \ldots\ge \lambda_N)\in \mathbb{Z}^N$ encoding boundary conditions. These symmetric functions generalize Schur symmetric polynomials, as well as some of their variations, such as factorial and supersymmetric Schur polynomials. Cauchy type summation identities for $F_\lambda,G_\lambda$ and their skew counterparts follow from the Yang–Baxter equation. Using algebraic Bethe Ansatz, we obtain a double alternant type formula for $F_\lambda$ and a Sergeev–Pragacz type formula for $G_\lambda$. In the spirit of the theory of Schur processes, we define probability measures on sequences of signatures with probability weights proportional to products of our symmetric functions. We show that these measures can be viewed as determinantal point processes, and we express their correlation kernels in a double contour integral form. We present two proofs: The first is a direct computation of Eynard–Mehta type, and the second uses non-standard, inhomogeneous versions of fermionic operators in a Fock space coming from the algebraic Bethe Ansatz for the six vertex model. We also interpret our determinantal processes as random domino tilings of a half-strip with inhomogeneous domino weights. In the bulk, we show that the lattice asymptotic behavior of such domino tilings is described by a new determinantal point process on $\mathbb{Z}^{2}$, which can be viewed as an doubly-inhomogeneous generalization of the extended discrete sine process.MSRI program Fall 20212021-07-19T00:00:00-04:002021-07-19T00:00:00-04:00https://lpetrov.cc/2021/07/MSRI<div>In Fall 2021, I am in residence at the MSRI Program <a href="https://www.msri.org/programs/328">Universality and Integrability in Random Matrix Theory and Interacting Particle Systems</a>, Berkeley, CA, <b>August 16 - December 17, 2021.</b></div>Leonid PetrovIn Fall 2021, I am in residence at the MSRI Program Universality and Integrability in Random Matrix Theory and Interacting Particle Systems, Berkeley, CA, August 16 - December 17, 2021.MATRIX program 20212021-04-26T00:00:00-04:002021-04-26T00:00:00-04:00https://lpetrov.cc/2021/04/MATRIX21<div>Program "<a href="https://sites.google.com/view/intcombfintemp2021/home">Integrability and Combinatorics at Finite Temperature</a>", <a href="https://www.matrix-inst.org.au/events/integrability-and-combinatorics-at-finite-temperature/">MATRIX Institute</a>, Australia, <b>June 7-18, 2021</b> (online).</div>Leonid PetrovProgram "Integrability and Combinatorics at Finite Temperature", MATRIX Institute, Australia, June 7-18, 2021 (online).Particle Systems course2021-01-29T00:00:00-05:002021-01-29T00:00:00-05:00https://lpetrov.cc/2021/01/part-sys-announce<div>MATH 7370 <b>Probability II. Particle Systems</b> (Spring 2021) • <a href="https://github.com/lenis2000/HTML_ParticleSysS21">Course page</a></div>Leonid PetrovMATH 7370 Probability II. Particle Systems (Spring 2021) • Course pageMATH 7370 Probability Theory II: Particle Systems2020-12-21T00:00:00-05:002020-12-21T00:00:00-05:00https://lpetrov.cc/2020/12/particle-sysLeonid Petrov2022 travel2020-10-11T00:00:00-04:002020-10-11T00:00:00-04:00https://lpetrov.cc/2020/10/travel-2022<!-- ##### January -->
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<!-- ##### December -->Leonid PetrovRadnom polymers and symmetric functions2020-10-06T00:00:00-04:002020-10-06T00:00:00-04:00https://lpetrov.cc/2020/10/beta-polymers<p>Surveys integrable random polymers (based on gamma / inverse gamma or beta distributed weights), and explains their connection to symmetric functions. Based on a <a href="https://lpetrov.cc/2020/03/sqW/">joint work with Matteo Mucciconi</a>.</p>
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<p><a href="https://storage.lpetrov.cc/research_files/talks/beta-polymers.pdf" target="_blank">PDF (12.2 MB)</a></p>Leonid PetrovSurveys integrable random polymers (based on gamma / inverse gamma or beta distributed weights), and explains their connection to symmetric functions. Based on a joint work with Matteo Mucciconi.MATH 3100 • Introduction to Probability (2 sections)2020-08-20T00:00:00-04:002020-08-20T00:00:00-04:00https://lpetrov.cc/2020/08/probLeonid PetrovRefined Cauchy identity for spin Hall-Littlewood symmetric rational functions2020-07-20T00:00:00-04:002020-07-20T00:00:00-04:00https://lpetrov.cc/2020/07/refined-sHL<p>Fully inhomogeneous spin Hall-Littlewood symmetric rational functions <span class="kdmath">$\mathsf{F}_\lambda$</span> arise in the context of $\mathfrak{sl}(2)$ higher spin six vertex models, and are multiparameter deformations of the classical Hall-Littlewood symmetric polynomials. We obtain a refined Cauchy identity expressing a weighted sum of the product of two $\mathsf{F}_\lambda$’s as a determinant. The determinant is of Izergin-Korepin type: it is the partition function of the six vertex model with suitably decorated domain wall boundary conditions. The proof of equality of two partition functions is based on the Yang-Baxter equation.</p>
<p>We rewrite our Izergin-Korepin type determinant in a different form which includes one of the sets of variables in a completely symmetric way. This determinantal identity might be of independent interest, and also allows to directly link the spin Hall-Littlewood rational functions with (the Hall-Littlewood particular case of) the interpolation Macdonald polynomials. In a different direction, a Schur expansion of our Izergin-Korepin type determinant yields a deformation of Schur symmetric polynomials.</p>
<p>In the spin-$\frac12$ specialization, our refined Cauchy identity leads to a summation identity for eigenfunctions of the ASEP (Asymmetric Simple Exclusion Process), a celebrated stochastic interacting particle system in the Kardar-Parisi-Zhang universality class. This produces explicit integral formulas for certain multitime probabilities in ASEP.</p>Leonid PetrovFully inhomogeneous spin Hall-Littlewood symmetric rational functions $\mathsf{F}_\lambda$ arise in the context of $\mathfrak{sl}(2)$ higher spin six vertex models, and are multiparameter deformations of the classical Hall-Littlewood symmetric polynomials. We obtain a refined Cauchy identity expressing a weighted sum of the product of two $\mathsf{F}_\lambda$’s as a determinant. The determinant is of Izergin-Korepin type: it is the partition function of the six vertex model with suitably decorated domain wall boundary conditions. The proof of equality of two partition functions is based on the Yang-Baxter equation. We rewrite our Izergin-Korepin type determinant in a different form which includes one of the sets of variables in a completely symmetric way. This determinantal identity might be of independent interest, and also allows to directly link the spin Hall-Littlewood rational functions with (the Hall-Littlewood particular case of) the interpolation Macdonald polynomials. In a different direction, a Schur expansion of our Izergin-Korepin type determinant yields a deformation of Schur symmetric polynomials. In the spin-$\frac12$ specialization, our refined Cauchy identity leads to a summation identity for eigenfunctions of the ASEP (Asymmetric Simple Exclusion Process), a celebrated stochastic interacting particle system in the Kardar-Parisi-Zhang universality class. This produces explicit integral formulas for certain multitime probabilities in ASEP.Online conference on Statistical Mechanics, Integrable Systems and Probability2020-04-14T00:00:00-04:002020-04-14T00:00:00-04:00https://lpetrov.cc/2020/04/SMISP<div>"<a href="http://mtikhonov.com/smisp/">Online conference on Statistical Mechanics, Integrable Systems and Probability</a>", <b>April 27 - May 1, 2020</b>.</div>Leonid Petrov"Online conference on Statistical Mechanics, Integrable Systems and Probability", April 27 - May 1, 2020.