Jekyll2017-12-17T21:09:09-05:00http://lpetrov.cc/Leonid PetrovLeonid Petrov. Integrable ProbabilityLeonid PetrovSimulations of inhomogeneous multilayer pushTASEP2017-12-17T05:00:00-05:002017-12-17T05:00:00-05:00http://lpetrov.cc/2017/12/pushTASEP_simulations<p>I’ve done some simulations of a multilayer version of the pushTASEP in inhomogeneous space,
in my new <a href="http://lpetrov.cc/simulations/">simulations gallery</a>.</p>
<p>Technical details and more pictures are <a href="http://lpetrov.cc/simulations/2017-12-15-pushtasep-multilayer/">here</a>.</p>Leonid PetrovI’ve done some simulations of a multilayer version of the pushTASEP in inhomogeneous space, in my new simulations gallery. Technical details and more pictures are here.Yang-Baxter field for spin Hall-Littlewood symmetric functions2017-12-13T11:00:00-05:002017-12-13T11:00:00-05:00http://lpetrov.cc/2017/12/YBfieldEmploying bijectivisation of summation identities, we introduce local
stochastic moves based on the Yang-Baxter equation for
<script type="math/tex">U_q(\widehat{\mathfrak{sl}_2})</script>. Combining these moves leads to a new object
which we call the spin Hall-Littlewood Yang-Baxter field - a probability
distribution on two-dimensional arrays of particle configurations on the
discrete line. We identify joint distributions along down-right paths in the
Yang-Baxter field with spin Hall-Littlewood processes, a generalization of
Schur processes. We consider various degenerations of the Yang-Baxter field
leading to new dynamic versions of the stochastic six vertex model and of the
Asymmetric Simple Exclusion Process.Leonid PetrovEmploying bijectivisation of summation identities, we introduce local stochastic moves based on the Yang-Baxter equation for . Combining these moves leads to a new object which we call the spin Hall-Littlewood Yang-Baxter field - a probability distribution on two-dimensional arrays of particle configurations on the discrete line. We identify joint distributions along down-right paths in the Yang-Baxter field with spin Hall-Littlewood processes, a generalization of Schur processes. We consider various degenerations of the Yang-Baxter field leading to new dynamic versions of the stochastic six vertex model and of the Asymmetric Simple Exclusion Process.Gallery of simulations of integrable stochastic systems2017-12-12T21:00:00-05:002017-12-12T21:00:00-05:00http://lpetrov.cc/2017/12/gallery<div>Updated <a href="http://lpetrov.cc/simulations/">gallery of simulations</a> of integrable stochastic systems</div>
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<p>There will be more than just simulations:</p>
<ul>
<li>raw data with well-described format;</li>
<li>links to source code of programs producing the data;</li>
<li>visualizations themselves</li>
</ul>Leonid PetrovUpdated gallery of simulations of integrable stochastic systemsSwitching TeXLive distributions for arXiv2017-12-12T07:00:00-05:002017-12-12T07:00:00-05:00http://lpetrov.cc/2017/12/tex-distributions<p>Doing bibliography with <a href="https://ctan.org/pkg/biblatex">BiBLaTeX</a> (and having one huge <code class="highlighter-rouge">.bib</code> file - mine is <a href="https://github.com/lenis2000/BiBTeX">public</a>, by the way)
works great for me.</p>
<p>One downside is that <a href="https://www.arxiv.org">arXiv</a> uses a <a href="https://arxiv.org/help/faq/texlive">specific TeXLive
distribution</a> (2016 as of today), and the distribution on my machine is more up to date.
Also, arXiv wants <code class="highlighter-rouge">.bbl</code> files uploaded instead of huge <code class="highlighter-rouge">.bib</code> files
(<code class="highlighter-rouge">.bbl</code> contains only the references actually included in a given paper, and not all over 900 references which are in my <code class="highlighter-rouge">.bib</code> file).
The problem is that <code class="highlighter-rouge">.bbl</code> files produced by different versions of BiBLaTeX are incompatible (!).
So, to upload a paper to arXiv, I need to install a version of TeXLive identical to the arXiv’s one.</p>
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<p>I have a Mac.
To install an older TeXLive version, I use <a href="http://www.tug.org/mactex/">MacTeX</a>.
To find an appropriate version, one needs to go to the TUG’s ftp at
<a href="ftp://tug.org/historic/systems/mactex/"><code class="highlighter-rouge">ftp://tug.org/historic/systems/mactex/</code></a>
where they keep all the historic versions.
The download speed from this ftp might be quite slow.
When installed, the older version will coexist with the current one,
and one can switch between the versions.</p>
<p><a href="http://www.tug.org/mactex/multipletexdistributions.html">MacTeX’s page</a> has some things to
say about switching, but neither of these ways worked for me.
Another (manual) solution I found over the internet is the following:</p>
<figure class="highlight"><pre><code class="language-bash" data-lang="bash"><table class="rouge-table"><tbody><tr><td class="gutter gl"><pre class="lineno">1
2
3
</pre></td><td class="code"><pre><span class="nb">cd</span> /Library/TeX/Distributions/.DefaultTeX
<span class="nb">sudo </span>rm Contents
<span class="nb">sudo </span>ln <span class="nt">-s</span> ../TeXLive-2016.texdist/Contents Contents</pre></td></tr></tbody></table></code></pre></figure>
<p>Namely, <code class="highlighter-rouge">/Library/TeX/Distributions/.DefaultTeX/Contents</code>
is a symlink to the TeXLive distribution used throughout the system,
and redirecting this link will switch the version.</p>
<p>Bingo, a new paper is submitted to the arXiv! (will be in the next post when it appears online)</p>Leonid PetrovDoing bibliography with BiBLaTeX (and having one huge .bib file - mine is public, by the way) works great for me. One downside is that arXiv uses a specific TeXLive distribution (2016 as of today), and the distribution on my machine is more up to date. Also, arXiv wants .bbl files uploaded instead of huge .bib files (.bbl contains only the references actually included in a given paper, and not all over 900 references which are in my .bib file). The problem is that .bbl files produced by different versions of BiBLaTeX are incompatible (!). So, to upload a paper to arXiv, I need to install a version of TeXLive identical to the arXiv’s one.External Tikz pictures2017-12-12T03:00:00-05:002017-12-12T03:00:00-05:00http://lpetrov.cc/2017/12/tikz-external<p>I produce almost all pictures in my math <script type="math/tex">\mathrm{\LaTeX}</script> writing in <a href="http://www.texample.net/tikz/">TikZ</a>.
This is a nice library (and I’ve learned it over the years), which allows for-loops, effects, etc.
The downside for me always was that compiling inline TikZ pictures takes a lot of time.
For some months now, while writing a particularly figure-heavy paper,
I wondered how I can optimize this.</p>
<p>Following <a href="https://stackoverflow.com/questions/2701902/standalone-diagrams-with-tikz">this stackoverflow discussion</a>, I have now adopted
a great way of optimizing TikZ pictures by placing them into separate standalone tex files.</p>
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<p>For example, this file</p>
<figure class="highlight"><pre><code class="language-tex" data-lang="tex"><table class="rouge-table"><tbody><tr><td class="gutter gl"><pre class="lineno">1
2
3
4
5
6
7
</pre></td><td class="code"><pre><span class="k">\documentclass</span><span class="p">{</span>standalone<span class="p">}</span>
<span class="k">\usepackage</span><span class="p">{</span>tikz<span class="p">}</span>
<span class="nt">\begin{document}</span>
<span class="nt">\begin{tikzpicture}</span>
<span class="c">%tikz code</span>
<span class="nt">\end{tikzpicture}</span>
<span class="nt">\end{document}</span></pre></td></tr></tbody></table></code></pre></figure>
<p>can be fed into my usual <code class="highlighter-rouge">latexmk</code></p>
<figure class="highlight"><pre><code class="language-bash" data-lang="bash">latexmk <span class="nt">-pdf</span> <span class="nt">-pvc</span> <span class="nt">-f</span> <span class="nt">-silent</span> <span class="nt">-synctex</span><span class="o">=</span>1 FILENAME</code></pre></figure>
<p>for automatic constant compilation, and the pdf result can be included into the main <script type="math/tex">\mathrm{\TeX}</script> file
using another thread of automatic
<code class="highlighter-rouge">latexmk</code> compilation.</p>
<p><strong>Note</strong>.
For further possible optimization, <script type="math/tex">\mathrm{\TeX}</script>
class <code class="highlighter-rouge">standalone</code> can already produce <code class="highlighter-rouge">png</code> output, which (I think) cost less compile time
to be included into the main <script type="math/tex">\mathrm{\TeX}</script> file.</p>
<p><strong>PS.</strong> I enjoy turning on <code class="highlighter-rouge">latexmk</code> to constantly recompile the <script type="math/tex">\mathrm{\LaTeX}</script> sources for me.
I think this might be the reason why my laptop battery just died…</p>Leonid PetrovI produce almost all pictures in my math writing in TikZ. This is a nice library (and I’ve learned it over the years), which allows for-loops, effects, etc. The downside for me always was that compiling inline TikZ pictures takes a lot of time. For some months now, while writing a particularly figure-heavy paper, I wondered how I can optimize this. Following this stackoverflow discussion, I have now adopted a great way of optimizing TikZ pictures by placing them into separate standalone tex files.Integrable Probability Boston 20182017-11-23T08:30:00-05:002017-11-23T08:30:00-05:00http://lpetrov.cc/2017/11/mit-integrable-conference<div>The conference <a href="http://2018.int-prob.org">Integrable Probability Boston 2018</a> will be held on <b>May 14-18, 2018</b> at MIT, Cambridge, MA.<div>
</div></div>Leonid PetrovThe conference Integrable Probability Boston 2018 will be held on May 14-18, 2018 at MIT, Cambridge, MA.First FRG workshop, October 2017, Columbia University2017-09-23T09:30:00-04:002017-09-23T09:30:00-04:00http://lpetrov.cc/2017/09/columbia-frg-meeting<div>The first <a href="http://frg.int-prob.org/">Integrable Probability FRG</a> workshop was held on <b>October 27-29, 2017</b> at Columbia University in New York. <a href="http://frg.int-prob.org/meeting2017/">Some photos from the meeting</a>.<div>
</div></div>Leonid PetrovThe first Integrable Probability FRG workshop was held on October 27-29, 2017 at Columbia University in New York. Some photos from the meeting.Coarsening model on with biased zero-energy flips and an exponential large deviation bound for ASEP2017-08-17T22:00:00-04:002017-08-17T22:00:00-04:00http://lpetrov.cc/2017/08/coarsening-ASEP-LDP<p>We study the coarsening model (zero-temperature Ising Glauber dynamics) on
$\mathbb{Z}^d$ (for $d \geq 2$) with an asymmetric tie-breaking rule. This is a
Markov process on the state space ${-1,+1}^{\mathbb{Z}^d}$ of “spin
configurations” in which each vertex updates its spin to agree with a majority
of its neighbors at the arrival times of a Poisson process. If a vertex has
equally many $+1$ and $-1$ neighbors, then it updates its spin value to $+1$
with probability $q \in [0,1]$ and to $-1$ with probability $1-q$.
The initial state of this Markov chain
is distributed according to a product measure with probability $p$ for a spin to be $+1$.</p>
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<p>In this
paper, we show that for any given $p>0$, there exist $q$ close enough to 1 such
that a.s. every spin has a limit of $+1$. This is of particular interest for
small values of $p$, for which it is known that if $q=1/2$, a.s. all spins have
a limit of $-1$. For dimension $d=2$, we also obtain near-exponential
convergence rates for $q$ sufficiently large, and for general $d$, we obtain
stretched exponential rates independent of $d$. Two important ingredients in
our proofs are refinements of block arguments of Fontes-Schonmann-Sidoravicius
and a novel exponential large deviation bound for the Asymmetric Simple
Exclusion Process.</p>Leonid PetrovWe study the coarsening model (zero-temperature Ising Glauber dynamics) on $\mathbb{Z}^d$ (for $d \geq 2$) with an asymmetric tie-breaking rule. This is a Markov process on the state space ${-1,+1}^{\mathbb{Z}^d}$ of “spin configurations” in which each vertex updates its spin to agree with a majority of its neighbors at the arrival times of a Poisson process. If a vertex has equally many $+1$ and $-1$ neighbors, then it updates its spin value to $+1$ with probability $q \in [0,1]$ and to $-1$ with probability $1-q$. The initial state of this Markov chain is distributed according to a product measure with probability $p$ for a spin to be $+1$.2018 travel plans2017-07-22T09:30:00-04:002017-07-22T09:30:00-04:00http://lpetrov.cc/2017/07/travel-2018<h5 id="january">January</h5>
<p>1-12 • Moscow, Russia</p>
<p>13-27 • Creswick (Victoria, Australia) • <a href="https://www.matrix-inst.org.au/events/non-equilibrium-systems-and-special-functions/">“Non-equilibrium systems and special functions” program at MATRIX Institute</a></p>
<p>28-31 • Moscow, Russia</p>
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<h5 id="may">May</h5>
<p>14 • Durham, NC • <a href="https://sites.duke.edu/sepc/">SouthEastern Probability Conference hosted by Duke and UNC</a></p>
<h5 id="june">June</h5>
<p>11–15 • Gothenburg, Sweden • <a href="http://spa2018.org/">40th Conference on Stochastic Processes and their Applications – SPA 2018</a></p>
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<!-- ##### December -->Leonid PetrovJanuary 1-12 • Moscow, Russia 13-27 • Creswick (Victoria, Australia) • “Non-equilibrium systems and special functions” program at MATRIX Institute 28-31 • Moscow, RussiaPCMI Summer session 2017 “Random matrices”2017-07-17T09:30:00-04:002017-07-17T09:30:00-04:00http://lpetrov.cc/2017/07/utah<p>A three-week program which brought together numerous leading experts in random matrices and related
fields. For me personally this program has started a very exciting collaboration on probabilistic understanding of
combinatorial summation identities.</p>
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<p>From the <a href="https://pcmi.ias.edu/program-index/2017">program website</a>:</p>
<p><span class="highlighted-item">
Random matrix theory sits at the interface of many fields of mathematics such as probability, combinatorics, partial differential equations, algebraic geometry, and number theory. It also has many applications in physics (statistical mechanics, quantum mechanics, nuclear physics), as well as practical applications in areas of computer science, statistics, and data science. This research program will bring together a diverse group of researchers united by their study and use of random matrix theory. This will naturally lead to exchanges in ideas and methods between different areas of science and mathematics, and will kickstart new interdisciplinary collaborations.
</span></p>
<h2 class="mb-1 mt-5">Photos</h2>
<p><img src="http://lpetrov.cc.storage.s3-website-us-east-1.amazonaws.com/travel/PCMI-2017/Fireworks.jpg" alt="July 4 fireworks in the mountains" title="July 4 fireworks in the mountains" style="max-width:100%;max-height:800px;height:auto;width:auto;" class="mb-3 mt-3" />
<br />July 4 fireworks in the mountains</p>
<p><img src="http://lpetrov.cc.storage.s3-website-us-east-1.amazonaws.com/travel/PCMI-2017/Waterfall.jpg" alt="Waterfall on the way to Mirror Lake" title="Waterfall on the way to Mirror Lake" style="max-width:100%;max-height:800px;height:auto;width:auto;" class="mb-3 mt-3" />
<br />Waterfall on the way to Mirror Lake</p>
<p><img src="http://lpetrov.cc.storage.s3-website-us-east-1.amazonaws.com/travel/PCMI-2017/Mirror_Lake.jpg" alt="Mirror Lake" title="Mirror Lake" style="max-width:100%;max-height:800px;height:auto;width:auto;" class="mb-3 mt-3" />
<br />Mirror Lake</p>
<p><img src="http://lpetrov.cc.storage.s3-website-us-east-1.amazonaws.com/travel/PCMI-2017/Utah_clouds.jpg" alt="Clouds and rain over the Uinta Range" title="Clouds and rain over the Uinta Range" style="max-width:100%;max-height:800px;height:auto;width:auto;" class="mb-3 mt-3" />
<br />Clouds and rain over the Uinta Range</p>
<p><img src="http://lpetrov.cc.storage.s3-website-us-east-1.amazonaws.com/travel/PCMI-2017/Utah_Mountain_Snow.jpg" alt="Snow in on the Bald Mountain" title="Snow in on the Bald Mountain" style="max-width:100%;max-height:800px;height:auto;width:auto;" class="mb-3 mt-3" />
<br />Snow in on the Bald Mountain</p>Leonid PetrovA three-week program which brought together numerous leading experts in random matrices and related fields. For me personally this program has started a very exciting collaboration on probabilistic understanding of combinatorial summation identities.