Jekyll2020-01-18T09:42:27-05:00https://lpetrov.cc/posts/feed.xmlLeonid PetrovLeonid Petrov. Integrable ProbabilityLeonid Petrov2021 travel2020-01-17T00:00:00-05:002020-01-17T00:00:00-05:00https://lpetrov.cc/2020/01/travel-2021<!-- ##### January -->
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<h5 id="may">May</h5>
<p>3-21
•
Florence, Italy
•
Program “Randomness, Integrability and Universality” at Galileo Galilei Institute</p>
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<!-- ##### 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>
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<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.PushTASEP in inhomogeneous space2019-10-20T00:00:00-04:002019-10-20T00:00:00-04:00https://lpetrov.cc/2019/10/pushTASEP<p>We consider the PushTASEP (pushing totally asymmetric simple exclusion process, also sometimes called long-range TASEP) with the step initial configuration evolving in an inhomogeneous space. That is, the rate of each particle’s jump depends on the location of this particle. We match the distribution of the height function of this PushTASEP with Schur processes. Using this matching and determinantal structure of Schur processes, we obtain limit shape and fluctuation results which are typical for stochastic particle systems in the Kardar-Parisi-Zhang universality class. PushTASEP is a close relative of the usual TASEP. In inhomogeneous space the former is integrable, while the integrability of the latter is not known.</p>Leonid PetrovWe consider the PushTASEP (pushing totally asymmetric simple exclusion process, also sometimes called long-range TASEP) with the step initial configuration evolving in an inhomogeneous space. That is, the rate of each particle’s jump depends on the location of this particle. We match the distribution of the height function of this PushTASEP with Schur processes. Using this matching and determinantal structure of Schur processes, we obtain limit shape and fluctuation results which are typical for stochastic particle systems in the Kardar-Parisi-Zhang universality class. PushTASEP is a close relative of the usual TASEP. In inhomogeneous space the former is integrable, while the integrability of the latter is not known.Mapping TASEP back in time2019-09-09T00:00:00-04:002019-09-09T00:00:00-04:00https://lpetrov.cc/2019/09/TASEP<p>We obtain a new relation between the distributions $\mu_t$ at different times $t ≥ 0$ of the continuous-time TASEP (Totally Asymmetric Simple Exclusion Process) started from the step initial configuration. Namely, we present a continuous-time Markov process with local interactions and particle-dependent rates which maps the TASEP distributions μt backwards in time. Under the backwards process, particles jump to the left, and the dynamics can be viewed as a version of the discrete-space Hammersley process. Combined with the forward TASEP evolution, this leads to a stationary Markov dynamics preserving $\mu_t$ which in turn brings new identities for expectations with respect to $\mu_t$. Based on a <a href="https://lpetrov.cc/2019/07/backwards_TASEP/">joint work with Axel Saenz</a>.</p>
<p>Note that the original keynote presentation contained videos which are not included in the PDF download.</p>
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<p><a href="https://storage.lpetrov.cc/research_files/talks/TASEP_back_Osaka.pdf" target="_blank">PDF (13.5 MB)</a></p>Leonid PetrovWe obtain a new relation between the distributions $\mu_t$ at different times $t ≥ 0$ of the continuous-time TASEP (Totally Asymmetric Simple Exclusion Process) started from the step initial configuration. Namely, we present a continuous-time Markov process with local interactions and particle-dependent rates which maps the TASEP distributions μt backwards in time. Under the backwards process, particles jump to the left, and the dynamics can be viewed as a version of the discrete-space Hammersley process. Combined with the forward TASEP evolution, this leads to a stationary Markov dynamics preserving $\mu_t$ which in turn brings new identities for expectations with respect to $\mu_t$. Based on a joint work with Axel Saenz. Note that the original keynote presentation contained videos which are not included in the PDF download.Special Session on Integrable Probability 20202019-08-06T00:00:00-04:002019-08-06T00:00:00-04:00https://lpetrov.cc/2019/08/AMS-UVA<div><a href="http://www.ams.org/meetings/sectional/2273_program_ss27.html">Special Session on Integrable Probability at the 2020 AMS Spring Southeastern Sectional Meeting at University of Virginia, March 13-15, 2020</a></div>Leonid PetrovSpecial Session on Integrable Probability at the 2020 AMS Spring Southeastern Sectional Meeting at University of Virginia, March 13-15, 2020Course notes on random matrices2019-07-29T00:00:00-04:002019-07-29T00:00:00-04:00https://lpetrov.cc/2019/07/rmt-announce<div>MATH 8380 Random Matrices (Fall 2019) • <a href="https://rmt-fall2019.s3.amazonaws.com/rmt-fall2019.pdf">PDF course notes</a> • <a href="https://lpetrov.cc/rmt19/">Course page</a></div>Leonid PetrovMATH 8380 Random Matrices (Fall 2019) • PDF course notes • Course pageMATH 8380 • Random Matrices2019-07-28T00:00:00-04:002019-07-28T00:00:00-04:00https://lpetrov.cc/rmt<h3 id="course-notes--found-typo-or-mistake-let-me-know"><a href="https://rmt-fall2019.s3.amazonaws.com/rmt-fall2019.pdf">Course notes</a> • Found typo or mistake? Let me know!</h3>
<div><object data="https://rmt-fall2019.s3.amazonaws.com/up.txt" style="height:30px"></object></div>
<ul>
<li>Chapter 1. Some history</li>
<li>Chapter 2. Gaussian Unitary Ensemble and Dyson Brownian Motion</li>
<li>Interlude. Markov chains and stochastic differential equations</li>
<li>Chapter 3. Two ways to derive the GUE eigenvalue distribution</li>
<li>Chapter 4. Wigner Semicircle Law</li>
<li>Chapter 5. Orbital measures and free operations</li>
<li>Chapter 6. Representation-theoretic discrete analogue of the GUE</li>
<li>Chapter 7. Determinantal point processes</li>
<li>Chapter 8. Asymptotics via contour integrals</li>
<li>Chapter 9. Harish-Chandra-Itsykson-Zuber integral and random matrix distributions</li>
<li>Chapter 10. Universality</li>
<li>Chapter 11. Markov maps on corners processes and generalizations</li>
</ul>
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<hr />
<h3 id="syllabus">Syllabus</h3>
<p><strong>Instructor.</strong> Leonid Petrov. Contact information is at <a href="https://lpetrov.cc"><code class="highlighter-rouge">https://lpetrov.cc</code></a></p>
<p>The class meets on Tuesdays and Thursdays at 9:30-10:45 in Kerchof 128.</p>
<p>Office hours Tuesdays and Thursdays 11:30-1 (or just drop in at any time). Office is Kerchof 209</p>
<p><strong>Description.</strong> Study of random matrices is an exciting topic with first major advances in the mid-20th century in connection with statistical (quantum) physics. Since then it found numerous connections to algebra, geometry, combinatorics, as well as to the core of the probability theory. The applications are also numerous: e.g., statistics, number theory, engineering, neuroscience; with more of them discovered every month. The course will discuss fundamental problems and results of Random Matrix Theory, and their connections to tools of algebra and combinatorics.</p>
<p><strong>Course homepage.</strong> The course homepage is at <a href="https://lpetrov.cc/rmt19/"><code class="highlighter-rouge">https://lpetrov.cc/rmt19/</code></a>. It contains
the syllabus, link to course notes, and other relevant information.</p>
<p><strong>Structure.</strong> The course discusses:</p>
<ol>
<li>Limit shape results for random matrices (such as Wigner’s Semicircle Law). Connections to Free Probability.</li>
<li>Concrete ensembles of random matrices (GUE, circular, and Beta ensembles). Bulk and edge asymptotics via exact computations. Connection to determinantal point processes.</li>
<li>Unitary invariant Hermitian matrices. Interlacing arrays of reals
and their boundary.</li>
<li>Dynamics on matrices and spectra. Dyson’s Brownian Motion.</li>
<li>Universality of random matrix asymptotics.</li>
<li>(optional) Discrete analogues of random matrix models: random permutations, random tilings, interacting particle systems.</li>
<li>(optional) Applications to machine learning, neural networks.</li>
</ol>
<p><strong>References.</strong> There are several textbooks which I will consult while teaching the course. It is not required to buy any of them to successfully participate in the course.</p>
<ol>
<li>Mehta, M.L. <em>“Random Matrices”</em>.</li>
<li>Anderson, G.W., Guionnet, A. and Zeitouni, O. <em>“An Introduction to Random Matrices”</em>.</li>
<li>Pastur, L. and Shcherbina, M. <em>“Eigenvalue Distribution of Large Random Matrices”</em>.</li>
<li>Tao, T. <em>“Topics in random matrix theory”</em>.</li>
</ol>
<p>Course notes will be posted on this website, and updated regularly.
Direct download link is <a href="https://rmt-fall2019.s3.amazonaws.com/rmt-fall2019.pdf"><code class="highlighter-rouge">https://rmt-fall2019.s3.amazonaws.com/rmt-fall2019.pdf</code></a></p>
<p><strong>Grading.</strong>
The course grade is based on homework and class engagement
(your participation in in-class discussions; asking questions in class
and at office hours;
volunteering to type up homework solutions;
possibly volunteering to give short expository talks detailing
an aspect in the course; etc).
There is no midterm or final exam.</p>
<p>The homework will be assigned in the course notes (look for
green background). The deadline for each problem is 2.5 or 3 weeks,
which means:</p>
<ul>
<li>if a problem relates to a lecture on Tuesday in week $n$, then it is
due on Thursday on week $n+3$ (but no later than the official
final exam
date for the course);</li>
<li>if a problem relates to a lecture on Thursday in week $n$, then it is still
due on Thursday on week $n+3$ (but no later than the official
final exam
date for the course);</li>
</ul>
<p>Level of homework problems ranges from easy to very difficult.
It is understood that you won’t turn in all problems all the time,
but
putting an adequate effort into solving homework
problems and
communicating your solutions clearly is
of paramount importance for your learning.</p>
<p>Homework can be submitted either by email (scan or typeset, and send; this is the
preferred method); or turned in in class (in which case please still
scan the homework to keep a copy).</p>
<hr />
<p><sub><strong>Required official statement.</strong> All students with special needs requiring accommodations should present the appropriate paperwork from the Student Disability Access Center (SDAC). It is the student’s responsibility to present this paperwork in a timely fashion and follow up with the instructor about the accommodations being offered. Accommodations for test-taking (e.g., extended time) should be arranged at least 5 business days before an exam.</sub></p>Leonid PetrovCourse notes • Found typo or mistake? Let me know! Chapter 1. Some history Chapter 2. Gaussian Unitary Ensemble and Dyson Brownian Motion Interlude. Markov chains and stochastic differential equations Chapter 3. Two ways to derive the GUE eigenvalue distribution Chapter 4. Wigner Semicircle Law Chapter 5. Orbital measures and free operations Chapter 6. Representation-theoretic discrete analogue of the GUE Chapter 7. Determinantal point processes Chapter 8. Asymptotics via contour integrals Chapter 9. Harish-Chandra-Itsykson-Zuber integral and random matrix distributions Chapter 10. Universality Chapter 11. Markov maps on corners processes and generalizations