{"id":5920,"date":"2013-04-21T15:59:28","date_gmt":"2013-04-21T13:59:28","guid":{"rendered":"http:\/\/djalil.chafai.net\/blog\/?p=5920"},"modified":"2014-06-17T21:34:37","modified_gmt":"2014-06-17T19:34:37","slug":"average-characteristic-polynomial","status":"publish","type":"post","link":"https:\/\/djalil.chafai.net\/blog\/2013\/04\/21\/average-characteristic-polynomial\/","title":{"rendered":"Average Characteristic Polynomial"},"content":{"rendered":"

 <\/p>\n

\"Fran\u00e7ois<\/a>
Fran\u00e7ois Vi\u00e8te - Opera Mathematica.<\/figcaption><\/figure>\n

The characteristic polynomial \\( {\\chi} \\) of a sequence of complex numbers \\( {z_1,\\ldots,z_n} \\) is defined by<\/p>\n

\\[ \\chi(z):=\\prod_{k=1}^n(z-z_k). \\]<\/p>\n

If \\( {z_1,\\ldots,z_n} \\) are now random, then one may ask about the roots \\( {z_1',\\ldots,z_n'} \\) of the average characteristic polynomial<\/p>\n

\\[ \\overline{\\chi}(z):=\\mathbb{E}(\\chi(z))=\\mathbb{E}\\left(\\prod_{k=1}^n(z-z_k)\\right) =\\prod_{k=1}^n(z-z_k'). \\]<\/p>\n

Note that \\( {z_1',\\ldots,z_n'} \\) are not necessarily real or distinct even if \\( {z_1,\\ldots,z_n} \\) are real and distinct. By expanding we get<\/p>\n

\\[ \\overline{\\chi}(z)= z^n+\\sum_{k=1}^n\\frac{(-1)^k}{k!}z^{n-k} \\sum_{i_1\\neq\\cdots\\neq i_k}\\mathbb{E}(z_{i_1}\\cdots z_{i_k}). \\]<\/p>\n

(Vieta's formulas: the coefficient of a polynomial are elementary symmetric functions of the roots, in honor of Fran\u00e7ois Vi\u00e8te (1540-1603)). Now, if \\( {z_1,\\ldots,z_n} \\) are independent with common mean \\( {m} \\) then<\/p>\n

\\[ \\overline{\\chi}(z) = z^n+\\sum_{k=1}^n\\frac{(-1)^k}{k!}z^{n-k} \\frac{n!}{(n-k)!}m^k =(z-m)^n. \\]<\/p>\n

In this case \\( {z_1'=\\cdots=z_n'=m} \\), and<\/p>\n

\\[ \\frac{1}{n}\\sum_{k=1}^n\\delta_{z_k'}=\\delta_m. \\]<\/p>\n

In contrast, note that by the law of large numbers,<\/p>\n

\\[ \\frac{1}{n}\\sum_{k=1}^n\\delta_{z_k} \\underset{n\\rightarrow\\infty}{\\overset{a.s.}{\\longrightarrow}}\\mu \\]<\/p>\n

where \\( {\\mu} \\) is the common law of the \\( {z_k'} \\). How about the case where \\( {z_1,\\ldots,z_n} \\) are dependent? Let us consider for instance the case where \\( {z_1,\\ldots,z_n} \\) are from the Gaussian Unitary Ensemble i.e. the eigenvalues of a \\( {n\\times n} \\) Gaussian Hermitian random matrix with density proportional to \\( {H\\mapsto\\exp(-n\\mathrm{Tr}(H^2))} \\). In this case, it is well known that \\( {\\frac{1}{n}\\sum_{k=1}^n\\delta_{z_k}} \\) tend to the Wigner semi-circle law as \\( {n\\rightarrow\\infty} \\). On the other hand, it is well known that \\( {\\overline{\\chi}(z)} \\) is the \\( {n} \\)-th monic Hermite polynomial, and that \\( {\\frac{1}{n}\\sum_{k=1}^n\\delta_{z_k'}} \\) also tends to the semi-circle law as \\( {n\\rightarrow\\infty} \\). Beyond the GUE, there are very nice answers when \\( {z_1,\\ldots,z_n} \\) follow a determinental point process, explored by Adrien Hardy<\/a> in arXiv:1211.6564<\/a>. One may ask the same for permanental point processes.<\/p>\n

\"Leuven<\/a>
Leuven (Louvain)<\/figcaption><\/figure>\n

\n","protected":false},"excerpt":{"rendered":"

  The characteristic polynomial \\( {\\chi} \\) of a sequence of complex numbers \\( {z_1,\\ldots,z_n} \\) is defined by \\[ \\chi(z):=\\prod_{k=1}^n(z-z_k). \\] If \\( {z_1,\\ldots,z_n}…<\/p>\n

Continue readingAverage Characteristic Polynomial<\/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\/5920"}],"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=5920"}],"version-history":[{"count":23,"href":"https:\/\/djalil.chafai.net\/blog\/wp-json\/wp\/v2\/posts\/5920\/revisions"}],"predecessor-version":[{"id":7442,"href":"https:\/\/djalil.chafai.net\/blog\/wp-json\/wp\/v2\/posts\/5920\/revisions\/7442"}],"wp:attachment":[{"href":"https:\/\/djalil.chafai.net\/blog\/wp-json\/wp\/v2\/media?parent=5920"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/djalil.chafai.net\/blog\/wp-json\/wp\/v2\/categories?post=5920"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/djalil.chafai.net\/blog\/wp-json\/wp\/v2\/tags?post=5920"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}