{"id":2907,"date":"2011-09-16T22:30:47","date_gmt":"2011-09-16T20:30:47","guid":{"rendered":"http:\/\/djalil.chafai.net\/blog\/?p=2907"},"modified":"2011-09-20T22:57:48","modified_gmt":"2011-09-20T20:57:48","slug":"around-the-circular-law","status":"publish","type":"post","link":"https:\/\/djalil.chafai.net\/blog\/2011\/09\/16\/around-the-circular-law\/","title":{"rendered":"Around the circular law"},"content":{"rendered":"<p style=\"text-align: justify;\">I have  uploaded lecture notes entitled <em><strong>Around the circular law<\/strong><\/em> written with <a href=\"\/scripts\/search.php\/?q=Charles+Bordenave\">Charles Bordenave<\/a> (<a href=\"http:\/\/hal.archives-ouvertes.fr\/hal-00623894\/\">hal-00623894<\/a> & <a href=\"http:\/\/arxiv.org\/abs\/1109.3343\">arXiv:1109.3343<\/a>). These lecture notes are the expanded version of a joint course that we gave at the occasion of the <a href=\"\/blog\/2011\/07\/30\/changchun-jilin-province-china\/\">France-China summer school held in Changchun<\/a>. They incorporate some posts from this blog. It was a great pleasure for us to write down this synthesis, accumulating few years of thinking on the subject, sometimes with  our friend  <a href=\"\/scripts\/search.php\/?q=Pietro+Caputo+Probability\">Pietro Caputo<\/a> (I will remember this stimulating week in Rome!). I hope to reduce my time turning around <a title=\"Posts on the circular law on this blog\" href=\"http:\/\/djalil.chafai.net\/blog\/?s=circular+law\">the circular law<\/a> in 2012, even if I am still fascinated by few of the open problems \ud83d\ude09<\/p>\n<figure id=\"attachment_3015\" aria-describedby=\"caption-attachment-3015\" style=\"width: 300px\" class=\"wp-caption aligncenter\"><a href=\"\/blog\/wp-content\/uploads\/changchun-young.jpg\"><img loading=\"lazy\" src=\"\/blog\/wp-content\/uploads\/2011\/09\/changchun-young-small.jpg\" alt=\"Group photo in Jilin province, near Changchun\" title=\"Group photo in Jilin province, near Changchun\" width=\"300\" height=\"158\" class=\"size-full wp-image-3015\" \/><\/a><figcaption id=\"caption-attachment-3015\" class=\"wp-caption-text\">Jilin province, on the road for Changbaishan, China. Photo by Jamal Najim. From left to right: Charles Bordenave, Djalil Chafa\u00ef, Hayat Cheballah, Maxime F\u00e9vrier, Damien Passemier, Camille M\u00e2le, Adrien Hardy, Shurong Zheng, Ningning Zhao, Zhengdong Wang.<\/figcaption><\/figure>\n<p style=\"text-align: justify;\"><a href=\"http:\/\/www.ams.org\/mathscinet-getitem?mr=1393180\">Girko (1946 - )<\/a> contributed substantially to the solution of the circular law problem. His interest in random matrices came from his early works on random determinants, motivated at the origin by the <a href=\"http:\/\/en.wikipedia.org\/wiki\/Van_der_Waerden%27s_conjecture#Minimal_permanent\">van der Waerden conjecture<\/a>: among all \\(n\\times n\\) <a href=\"http:\/\/en.wikipedia.org\/wiki\/Doubly_stochastic_matrix\">doubly stochastic matrices<\/a>, the  matrix with all entries equal to \\(1\/n\\) has minimal <a href=\"http:\/\/en.wikipedia.org\/wiki\/Permanent\">permanent<\/a>. It turns out that the van der Waerden conjecture was solved around 1980 by another Ukrainian mathematician: <a href=\"http:\/\/www.ams.org\/mathscinet-getitem?mr=625097\">Falikman<\/a>. Personally (author of this blog), I have two favorites (still open) problems on doubly stochastic matrices:<\/p>\n<ul>\n<li>find an exact representation of the uniform law on the polytope of doubly stochastic matrices (similar to what we have for the \\(\\ell^1\\) ball with i.i.d. exponentials),<\/li>\n<li>show that the empirical spectral distribution of random doubly stochastic matrices (distributed according to the uniform law) tends to the circular law.\n<\/li>\n<\/ul>\n<p style=\"text-align: justify;\">This is why I came to random matrix theory... A proof of the van der Waerden conjecture can be found in the last chapter of the book  <a href=\"http:\/\/www.ams.org\/mathscinet-getitem?mr=1927396\"><em>Matrix inequalities<\/em><\/a> by Zhan. Coincidentally, this book contains also somewhere (proof of theorem 3.32) an inequality (for singular values and rows norms) which turned out to be a crucial ingredient in the <a href=\"\/blog\/2010\/06\/12\/spectrum-of-non-hermitian-heavy-tailed-random-matrices\/\">solution of the heavy tailed analogue of the circular law theorem<\/a> (obtained in collaboration with Bordenave and Caputo). I knew this book from my postdoc (by simple curiosity), but it became useful only ten years later!<\/p>\n<p style=\"text-align: justify;\"><a href=\"http:\/\/en.wikipedia.org\/wiki\/Van_der_Waerden\">Van der Waerden<\/a> is famous for <a href=\"http:\/\/en.wikipedia.org\/wiki\/Van_der_Waerden%27s_theorem\">his heorem<\/a> in <a href=\"http:\/\/en.wikipedia.org\/wiki\/Ramsey_theory\">Ramsey theory<\/a> about the structure of integers.<\/p>\n<figure id=\"attachment_2745\" aria-describedby=\"caption-attachment-2745\" style=\"width: 210px\" class=\"wp-caption aligncenter\"><a href=\"\/scripts\/search.php\/?q=Vyacheslav+Girko\"><img loading=\"lazy\" class=\"size-full wp-image-2745   \" title=\"Vyacheslav Girko (no, he never worked for Hollywood)\" src=\"\/blog\/wp-content\/uploads\/2011\/09\/Girko.jpg\" alt=\"Vyacheslav Girko\" width=\"210\" height=\"256\" \/><\/a><figcaption id=\"caption-attachment-2745\" class=\"wp-caption-text\">The Ukrainian mathematician V. L. Girko who spent many years around the circular law<\/figcaption><\/figure>\n","protected":false},"excerpt":{"rendered":"<p>I have uploaded lecture notes entitled Around the circular law written with Charles Bordenave (hal-00623894 &amp; arXiv:1109.3343). These lecture notes are the expanded version of&#8230;<\/p>\n<div class=\"more-link-wrapper\"><a class=\"more-link\" href=\"https:\/\/djalil.chafai.net\/blog\/2011\/09\/16\/around-the-circular-law\/\">Continue reading<span class=\"screen-reader-text\">Around the circular law<\/span><\/a><\/div>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"iawp_total_views":69},"categories":[1],"tags":[],"_links":{"self":[{"href":"https:\/\/djalil.chafai.net\/blog\/wp-json\/wp\/v2\/posts\/2907"}],"collection":[{"href":"https:\/\/djalil.chafai.net\/blog\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/djalil.chafai.net\/blog\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/djalil.chafai.net\/blog\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/djalil.chafai.net\/blog\/wp-json\/wp\/v2\/comments?post=2907"}],"version-history":[{"count":90,"href":"https:\/\/djalil.chafai.net\/blog\/wp-json\/wp\/v2\/posts\/2907\/revisions"}],"predecessor-version":[{"id":3032,"href":"https:\/\/djalil.chafai.net\/blog\/wp-json\/wp\/v2\/posts\/2907\/revisions\/3032"}],"wp:attachment":[{"href":"https:\/\/djalil.chafai.net\/blog\/wp-json\/wp\/v2\/media?parent=2907"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/djalil.chafai.net\/blog\/wp-json\/wp\/v2\/categories?post=2907"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/djalil.chafai.net\/blog\/wp-json\/wp\/v2\/tags?post=2907"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}