{"product_id":"bessel-processes-schramm-loewner-evolution-and-the-dyson-model-von-makoto-katori","title":"Bessel Processes, Schramm–Loewner Evolution, and the Dyson Model","description":"The purpose of this book is to introduce two recent topics in mathematical physics and probability theory: the Schramm–Loewner evolution (SLE) and interacting particle systems related to random matrix theory. A typical example of the latter systems is Dyson's Brownian motion (BM) model. The SLE and Dyson's BM model may be considered as \"children\" of the Bessel process with parameter \u003ci\u003eD\u003c\/i\u003e, BES(\u003ci\u003eD\u003c\/i\u003e), and the SLE and Dyson's BM model as \"grandchildren\" of BM. In Chap. 1 the parenthood of BM in diffusion processes is clarified and BES(\u003ci\u003eD\u003c\/i\u003e) is defined for any \u003ci\u003eD\u003c\/i\u003e ≥ 1. Dependence of the BES(\u003ci\u003eD\u003c\/i\u003e) path on its initial value is represented by the Bessel flow. In Chap. 2 SLE is introduced as a complexification of BES(\u003ci\u003eD\u003c\/i\u003e). Rich mathematics and physics involved in SLE are due to the nontrivial dependence of the Bessel flow on \u003ci\u003eD\u003c\/i\u003e. From a result for the Bessel flow, Cardy's formula in Carleson's form is derived for SLE. In Chap. 3 Dyson's BM model with parameter β is introduced as a multivariate extension of BES(\u003ci\u003eD\u003c\/i\u003e) with the relation \u003ci\u003eD\u003c\/i\u003e = β + 1. The book concentrates on the case where β = 2 and calls this case simply the Dyson model.The Dyson model inherits the two aspects of BES(3); hence it has very strong solvability. That is, the process is proved to be determinantal in the sense that all spatio-temporal correlation functions are given by determinants, and all of them are controlled by a single function called the correlation kernel. From the determinantal structure of the Dyson model, the Tracy–Widom distribution is derived. \u003cdiv class=\"aw-variant-hidden-subtitle-div\" id=\"aw-variant-subtitle-9789811002748\"\u003e\u003ch3\u003e\u003c\/h3\u003e\u003c\/div\u003e","brand":"Libri","offers":[{"title":"Softcover - 9789811002748","offer_id":39452138930269,"sku":"9789811002748","price":64.19,"currency_code":"EUR","in_stock":true}],"thumbnail_url":"\/\/cdn.shopify.com\/s\/files\/1\/0940\/0622\/files\/941bc504-7623-48ba-a6f8-760fc4580698.jpg?v=1772175825","url":"https:\/\/shop.autorenwelt.de\/products\/bessel-processes-schramm-loewner-evolution-and-the-dyson-model-von-makoto-katori","provider":"Autorenwelt Shop","version":"1.0","type":"link"}