Jekyll2018-12-11T16:57:05-05:00https://lpetrov.cc/posts/feed.xmlLeonid PetrovLeonid Petrov. Integrable ProbabilityLeonid 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://d3m0khvr0ybm92.cloudfront.net/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 evolve in discrete or continuous space and 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 features of the particle systems we consider 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 their hydrodynamic evolution, and obtain asymptotic fluctuation results which put our generalized TASEPs 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 our discrete space system is a one-parameter deformation of the geometric last-passage percolation (Johansson, 2000), and we obtain an extension of the last-passage percolation limit shape parabola.</p>
<p>The exact solvability and asymptotic behavior of generalizations of TASEP we study 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 evolve in discrete or continuous space and 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 features of the particle systems we consider is the presence of spatial inhomogeneity which can lead to the formation of traffic jams.Workshop on Representation Theory, Combinatorics, and Geometry2018-07-29T00:00:00-04:002018-07-29T00:00:00-04:00https://lpetrov.cc/2018/07/okounkov-workshop<div><a href="http://math.virginia.edu/ims/workshop-fall-2018/">Workshop on Representation Theory, Combinatorics, and Geometry</a>
will be at University of Virginia on <b>October 19-21, 2018</b>. The workshop precedes
the <a href="http://math.virginia.edu/ims/lectures/andrei-okounkov/">Virginia Mathematics Lectures by Andrei Okounkov</a> (October 22-24)</div>Leonid PetrovWorkshop on Representation Theory, Combinatorics, and Geometry will be at University of Virginia on October 19-21, 2018. The workshop precedes the Virginia Mathematics Lectures by Andrei Okounkov (October 22-24)MATH 3100 • Introduction to Probability (2 sections)2018-07-25T08:00:00-04:002018-07-25T08:00:00-04:00https://lpetrov.cc/2018/07/probLeonid Petrov2019 travel2018-05-24T23:30:00-04:002018-05-24T23:30:00-04:00https://lpetrov.cc/2018/05/travel-2019<h5 id="january">January</h5>
<p>8-10 •
Moscow, Russia •
Conference
<a href="https://crei.skoltech.ru/cas/calendar/conf190108/">“New Frontiers in Representation Theory”</a>
dedicated to the 70th birthday of G.I.Olshanski,
at SkolTech Center for Advances Studies</p>
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<h5 id="march">March</h5>
<p>10-15 •
Banff, Alberta, Canada
•
<a href="https://www.birs.ca/events/2019/5-day-workshops/19w5220">BIRS Workshop “Asymptotic Algebraic Combinatorics”</a></p>
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<h5 id="may">May</h5>
<p>13-14 • Durham, NC • SouthEastern Probability Conference 2019</p>
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<h5 id="september">September</h5>
<p>9-13 or 17-20 • Osaka, Japan • New Trends in Integrable Systems 2019</p>
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<!-- ##### December -->Leonid PetrovJanuary 8-10 • Moscow, Russia • Conference “New Frontiers in Representation Theory” dedicated to the 70th birthday of G.I.Olshanski, at SkolTech Center for Advances Studies March 10-15 • Banff, Alberta, Canada • BIRS Workshop “Asymptotic Algebraic Combinatorics” May 13-14 • Durham, NC • SouthEastern Probability Conference 2019[quick link] My big BiBTeX file2018-04-15T09:30:00-04:002018-04-15T09:30:00-04:00https://lpetrov.cc/2018/04/bibtex<div><a href="https://lpetrov.cc/bib/bib.html">The big bibliography (BiBTeX) file I'm using</a></div>Leonid PetrovThe big bibliography (BiBTeX) file I'm usingHow to enable SSL for a homepage hosted on S32018-04-13T08:00:00-04:002018-04-13T08:00:00-04:00https://lpetrov.cc/2018/04/SSL-for-website<p>As Google Chrome
<a href="https://www.theverge.com/2018/2/8/16991254/chrome-not-secure-marked-http-encryption-ssl">will mark</a>
all HTTP websites unsafe later this year,
it is time to figure out how to enable SSL on my websites.
I currently have 3 websites under active management:</p>
<ul>
<li>this homepage <a href="https://lpetrov.cc"><code class="highlighter-rouge">lpetrov.cc</code></a></li>
<li>the <a href="http://frg.int-prob.org">FRG website</a></li>
<li>the <a href="http://math.virginia.edu">UVA Math department website</a></li>
</ul>
<p>All three of them are hosted through AWS, but the homepage is by far the
easiest as it only involves S3 and no EC2 instances.
So at first I decided to turn on SSL at the homepage,
which I succeeded with.</p>
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<p>The homepage is hosted on two S3 buckets - one for resources,
and another one for the main content. The main content one
is updated via Travis from the GitHub repo.</p>
<h3 id="bucket-with-resources">Bucket with resources</h3>
<p>The resource bucket does not have its own private domain name,
and was simply using the S3 website name. The bucket used to be public.</p>
<p>Now, I created a cloudfront distribution with the cloudfront SSL from Amazon.
Replaced the main resources url in my jekyll website by
the cloudfront one (one change in config).
Closed the storage bucket from the public.
Now all resources are under SSL</p>
<h3 id="main-bucket--website">Main bucket / website</h3>
<p>For this, I created a cloudfront distribution
associated with the hostnames <code class="highlighter-rouge">lpetrov.cc</code> and <code class="highlighter-rouge">www.lpetrov.cc</code>.
The Route 53 DNS records had to be updated, too.
Since I am using a custom hostname
I need a certificate. I created one at Amazon,
with DNS verification (very smooth via Route 53),
and attached it to the cloudfront distribution.</p>
<p>By the way, the DNS verification will definitely not work for the UVA Math website
since I do not manage the DNS there. There is an alternate email way of verification which I’ll try
while working for that website.</p>
<h3 id="caveats-i-encountered">Caveats I encountered</h3>
<p><strong>1.</strong> The wildcard certificate for <code class="highlighter-rouge">*.lpetrov.cc</code> (and no other hostnames) was frowned
upon by browsers. I just recreated the certificate for only
the hostnames <code class="highlighter-rouge">lpetrov.cc</code> and <code class="highlighter-rouge">www.lpetrov.cc</code> I need.</p>
<p>Then, close up the main content bucket from the public, too.</p>
<p><strong>2.</strong> The cloudfront distribution is different from S3 website
in that the URL like <code class="highlighter-rouge">https://lpetrov.cc/research/</code> is not pointing at any object
(I like pretty URLs via jekyll). Therefore, <code class="highlighter-rouge">https://lpetrov.cc/research/</code> was not accessible,
and one needed <code class="highlighter-rouge">https://lpetrov.cc/research/index.html</code>. This is not great.
I solved this problem by using Lambda, and added a script for URL rewriting,
so that cloudfront now automagically adds <code class="highlighter-rouge">index.html</code> to the
URLs.</p>
<p><strong>3.</strong> Paginator broke down since the
paginator URLs were like
<code class="highlighter-rouge">https://lpetrov.cc/posts/page2</code> without <code class="highlighter-rouge">/</code> at the end,
so I had to fix it.</p>
<p><strong>4.</strong> One of my resource folders
had a wrong case in one of the letters in the name.
This broke a page, and I was not sure why this happened.
After a lot of invalidations at cloudfront I noticed the different case.</p>
<p><strong>5.</strong> The navbar toggle button (the one visible for narrow screens, e.g.,
on mobile) did not show - likely because the Bootstrap css references to an http url
for <code class="highlighter-rouge">xmlns</code> (I don’t know what that is). So a fix was to simply use
another way of drawing the navbar toggle button.</p>
<h3 id="overall">Overall</h3>
<p>So, overall it took me half a day but I learned some things.
The experience of doing it through AWS was nice overall.
Two more websites to go though.</p>Leonid PetrovAs Google Chrome will mark all HTTP websites unsafe later this year, it is time to figure out how to enable SSL on my websites. I currently have 3 websites under active management: this homepage lpetrov.cc the FRG website the UVA Math department website All three of them are hosted through AWS, but the homepage is by far the easiest as it only involves S3 and no EC2 instances. So at first I decided to turn on SSL at the homepage, which I succeeded with.Quenched Central Limit Theorem in a Corner Growth Setting2018-04-12T01:00:00-04:002018-04-12T01:00:00-04:00https://lpetrov.cc/2018/04/LPP<p>We consider point-to-point directed paths in a random environment on the
two-dimensional integer lattice. For a general independent environment under
mild assumptions we show that the quenched energy of a typical path satisfies a
central limit theorem as the mesh of the lattice goes to zero. Our proofs rely
on concentration of measure techniques and some combinatorial bounds on
families of paths.</p>Leonid PetrovWe consider point-to-point directed paths in a random environment on the two-dimensional integer lattice. For a general independent environment under mild assumptions we show that the quenched energy of a typical path satisfies a central limit theorem as the mesh of the lattice goes to zero. Our proofs rely on concentration of measure techniques and some combinatorial bounds on families of paths.