Jekyll2020-07-21T21:13:34-04:00https://lpetrov.cc/posts/feed.xmlLeonid PetrovLeonid Petrov. Integrable ProbabilityLeonid 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.Spin deformation of q-Whittaker polynomials and Whittaker functions2020-04-13T00:00:00-04:002020-04-13T00:00:00-04:00https://lpetrov.cc/2020/04/sqW<p>Spin $q$-Whittaker symmetric polynomials labeled by partitions $\lambda$ were recently introduced by Borodin and Wheeler (<a href="https://arxiv.org/abs/1701.06292">arXiv:1701.06292</a>) in the context of integrable $\mathfrak{sl}_2$ vertex models. They are a one-parameter deformation of the $t=0$ Macdonald polynomials. We present a new, more convenient modification of spin $q$-Whittaker polynomials and find two Macdonald type $q$-difference operators acting diagonally in these polynomials with eigenvalues, respectively, $q^{-\lambda_1}$ and $q^{\lambda_N}$ (where $\lambda$ is the polynomial’s label). We study probability measures on interlacing arrays based on spin $q$-Whittaker polynomials, and match their observables with known stochastic particle systems such as the $q$-Hahn TASEP.</p>
<p>In a scaling limit as $q\nearrow 1$, spin $q$-Whittaker polynomials turn into a new one-parameter deformation of the $\mathfrak{gl}_n$ Whittaker functions. The rescaled Pieri type rule gives rise to a one-parameter deformation of the quantum Toda Hamiltonian. The deformed Hamiltonian acts diagonally on our new spin Whittaker functions. On the stochastic side, as $q\nearrow 1$ we discover a multilevel extension of the beta polymer model of Barraquand and Corwin (<a href="https://arxiv.org/abs/1503.04117">arXiv:1503.04117</a>), and relate it to spin Whittaker functions.</p>
<p>Based on a <a href="https://lpetrov.cc/2020/03/sqW/">joint work with Matteo Mucciconi</a>
and earlier works with <a href="https://lpetrov.cc/2017/12/YBfield/">Alexey Bufetov</a> and <a href="https://lpetrov.cc/2019/05/BMP_YB/">both of them</a>.</p>
<!-- these references correspond to the CV PDF, papers basically in order of arXiv -->
<!--more-->
<p><a href="https://storage.lpetrov.cc/research_files/talks/sqw.pdf" target="_blank">PDF (33.6 MB)</a></p>Leonid PetrovSpin $q$-Whittaker symmetric polynomials labeled by partitions $\lambda$ were recently introduced by Borodin and Wheeler (arXiv:1701.06292) in the context of integrable $\mathfrak{sl}_2$ vertex models. They are a one-parameter deformation of the $t=0$ Macdonald polynomials. We present a new, more convenient modification of spin $q$-Whittaker polynomials and find two Macdonald type $q$-difference operators acting diagonally in these polynomials with eigenvalues, respectively, $q^{-\lambda_1}$ and $q^{\lambda_N}$ (where $\lambda$ is the polynomial’s label). We study probability measures on interlacing arrays based on spin $q$-Whittaker polynomials, and match their observables with known stochastic particle systems such as the $q$-Hahn TASEP. In a scaling limit as $q\nearrow 1$, spin $q$-Whittaker polynomials turn into a new one-parameter deformation of the $\mathfrak{gl}_n$ Whittaker functions. The rescaled Pieri type rule gives rise to a one-parameter deformation of the quantum Toda Hamiltonian. The deformed Hamiltonian acts diagonally on our new spin Whittaker functions. On the stochastic side, as $q\nearrow 1$ we discover a multilevel extension of the beta polymer model of Barraquand and Corwin (arXiv:1503.04117), and relate it to spin Whittaker functions. Based on a joint work with Matteo Mucciconi and earlier works with Alexey Bufetov and both of them.Spin q-Whittaker polynomials and deformed quantum Toda2020-03-31T00:00:00-04:002020-03-31T00:00:00-04:00https://lpetrov.cc/2020/03/sqW<p>Spin $q$-Whittaker symmetric polynomials labeled by partitions $\lambda$ were recently introduced by Borodin and Wheeler (<a href="https://arxiv.org/abs/1701.06292">arXiv:1701.06292</a>) in the context of integrable $\mathfrak{sl}_2$ vertex models. They are a one-parameter deformation of the $t=0$ Macdonald polynomials. We present a new, more convenient modification of spin $q$-Whittaker polynomials and find two Macdonald type $q$-difference operators acting diagonally in these polynomials with eigenvalues, respectively, $q^{-\lambda_1}$ and $q^{\lambda_N}$ (where $\lambda$ is the polynomial’s label). We study probability measures on interlacing arrays based on spin $q$-Whittaker polynomials, and match their observables with known stochastic particle systems such as the $q$-Hahn TASEP.</p>
<p>In a scaling limit as $q\nearrow 1$, spin $q$-Whittaker polynomials turn into a new one-parameter deformation of the $\mathfrak{gl}_n$ Whittaker functions. The rescaled Pieri type rule gives rise to a one-parameter deformation of the quantum Toda Hamiltonian. The deformed Hamiltonian acts diagonally on our new spin Whittaker functions. On the stochastic side, as $q\nearrow 1$ we discover a multilevel extension of the beta polymer model of Barraquand and Corwin (<a href="https://arxiv.org/abs/1503.04117">arXiv:1503.04117</a>), and relate it to spin Whittaker functions.</p>Leonid PetrovSpin $q$-Whittaker symmetric polynomials labeled by partitions $\lambda$ were recently introduced by Borodin and Wheeler (arXiv:1701.06292) in the context of integrable $\mathfrak{sl}_2$ vertex models. They are a one-parameter deformation of the $t=0$ Macdonald polynomials. We present a new, more convenient modification of spin $q$-Whittaker polynomials and find two Macdonald type $q$-difference operators acting diagonally in these polynomials with eigenvalues, respectively, $q^{-\lambda_1}$ and $q^{\lambda_N}$ (where $\lambda$ is the polynomial’s label). We study probability measures on interlacing arrays based on spin $q$-Whittaker polynomials, and match their observables with known stochastic particle systems such as the $q$-Hahn TASEP. In a scaling limit as $q\nearrow 1$, spin $q$-Whittaker polynomials turn into a new one-parameter deformation of the $\mathfrak{gl}_n$ Whittaker functions. The rescaled Pieri type rule gives rise to a one-parameter deformation of the quantum Toda Hamiltonian. The deformed Hamiltonian acts diagonally on our new spin Whittaker functions. On the stochastic side, as $q\nearrow 1$ we discover a multilevel extension of the beta polymer model of Barraquand and Corwin (arXiv:1503.04117), and relate it to spin Whittaker functions.MATRIX program 20202020-01-26T00:00:00-05:002020-01-26T00:00:00-05:00https://lpetrov.cc/2020/01/MATRIX<div>(<b>program postponed due to COVID-19</b>) Program "<a href="https://www.matrix-inst.org.au/events/integrability-and-combinatorics-at-finite-temperature/">Integrability and Combinatorics at Finite Temperature</a>",
MATRIX Institute, University of Melbourne, <b>June 1-19, 2020</b>.</div>Leonid Petrov(program postponed due to COVID-19) Program "Integrability and Combinatorics at Finite Temperature", MATRIX Institute, University of Melbourne, June 1-19, 2020.2021 travel2020-01-17T00:00:00-05:002020-01-17T00:00:00-05:00https://lpetrov.cc/2020/01/travel-2021<!-- ##### January -->
<!--more-->
<h5 id="february">February</h5>
<p>22-26 • San Jose, CA • AIM SQuaRE</p>
<!-- ##### March -->
<!-- ##### April -->
<h5 id="may">May</h5>
<p>(<strong>postponed due to COVID-19</strong>)
3-21
•
Florence, Italy
•
Program “Randomness, Integrability and Universality” at Galileo Galilei Institute</p>
<!-- ##### June -->
<h5 id="july">July</h5>
<p>19-23
•
Seoul, South Korea
•
<a href="http://wc2020.org/index.php">10th World Congress in Probability and Statistics</a>
•
(<em>rescheduled from 2020</em>)</p>
<!-- ##### August -->
<!-- ##### September -->
<!-- ##### October -->
<!-- ##### November -->
<!-- ##### December -->Leonid PetrovMATH 3340 • Complex Variables with Applications2020-01-04T00:00:00-05:002020-01-04T00:00:00-05:00https://lpetrov.cc/2020/01/complexLeonid PetrovMATH 7310 • Real Analysis and Linear Spaces I2020-01-04T00:00:00-05:002020-01-04T00:00:00-05:00https://lpetrov.cc/2020/01/grad-realLeonid PetrovParameter symmetry in perturbed GUE corners process and reflected drifted Brownian motions2019-12-17T00:00:00-05:002019-12-17T00:00:00-05:00https://lpetrov.cc/2019/12/pert_GUE_symm<p>The perturbed GUE corners ensemble is the joint distribution of eigenvalues of all principal submatrices of a matrix $G+\mathrm{diag}(\mathbf{a})$, where $G$ is the random matrix from the Gaussian Unitary Ensemble (GUE), and $\mathrm{diag}(\mathbf{a})$ 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.</p>
<p>The construction we present may be viewed as a random matrix analogue of the <a href="https://lpetrov.cc/2019/07/backwards_TASEP/">recent results</a> of the first author and Axel Saenz.</p>Leonid PetrovThe perturbed GUE corners ensemble is the joint distribution of eigenvalues of all principal submatrices of a matrix $G+\mathrm{diag}(\mathbf{a})$, where $G$ is the random matrix from the Gaussian Unitary Ensemble (GUE), and $\mathrm{diag}(\mathbf{a})$ 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. The construction we present may be viewed as a random matrix analogue of the recent results of the first author and Axel Saenz.Parameter permutation symmetry in particle systems and random polymers2019-12-12T00:00:00-05:002019-12-12T00:00:00-05:00https://lpetrov.cc/2019/12/symm_IPS<p>Many integrable stochastic particle systems in one space dimension (such as TASEP - Totally Asymmetric Simple Exclusion Process - and its various deformations, with a notable exception of ASEP) remain integrable when we equip each particle $x_i$ with its own jump rate parameter $\nu_i$. It is a consequence of integrability that the distribution of each particle $x_n(t)$ in a system started from the step initial configuration depends on the parameters $\nu_j$, $j\le n$, in a symmetric way. A transposition $\nu_n \leftrightarrow \nu_{n+1}$ of the parameters thus affects only the distribution of $x_n(t)$. For $q$-Hahn TASEP and its degenerations (namely, $q$-TASEP and beta polymer) we realize the transposition $\nu_n \leftrightarrow \nu_{n+1}$ as an explicit Markov swap operator acting on the single particle $x_n(t)$. For beta polymer, the swap operator can be interpreted as a simple modification of the lattice on which the polymer is considered. Our main tools are Markov duality and contour integral formulas for joint moments.</p>
<!--more-->
<p>In particular, our constructions lead to a continuous time Markov process $\mathsf{Q}^{(\mathsf{t})}$ preserving the time $\mathsf{t}$ distribution of the $q$-TASEP (with step initial configuration, where $\mathsf{t}\in \mathbb{R}_{>0}$ is fixed). The dual system is a certain transient modification of the stochastic $q$-Boson system. We identify asymptotic survival probabilities of this transient process with $q$-moments of the $q$-TASEP, and use this to show convergence of the process $\mathsf{Q}^{(\mathsf{t})}$ with arbitrary initial data to its stationary distribution.</p>
<p>Setting $q=0$, we recover the results about the usual TASEP established recently in <a href="https://lpetrov.cc/2019/07/backwards_TASEP/">this paper</a> by a different approach based on Gibbs ensembles of interlacing particles in two dimensions.</p>Leonid PetrovMany integrable stochastic particle systems in one space dimension (such as TASEP - Totally Asymmetric Simple Exclusion Process - and its various deformations, with a notable exception of ASEP) remain integrable when we equip each particle $x_i$ with its own jump rate parameter $\nu_i$. It is a consequence of integrability that the distribution of each particle $x_n(t)$ in a system started from the step initial configuration depends on the parameters $\nu_j$, $j\le n$, in a symmetric way. A transposition $\nu_n \leftrightarrow \nu_{n+1}$ of the parameters thus affects only the distribution of $x_n(t)$. For $q$-Hahn TASEP and its degenerations (namely, $q$-TASEP and beta polymer) we realize the transposition $\nu_n \leftrightarrow \nu_{n+1}$ as an explicit Markov swap operator acting on the single particle $x_n(t)$. For beta polymer, the swap operator can be interpreted as a simple modification of the lattice on which the polymer is considered. Our main tools are Markov duality and contour integral formulas for joint moments.