Uspekhi Matematicheskikh Nauk
 RUS  ENG JOURNALS   PEOPLE   ORGANISATIONS   CONFERENCES   SEMINARS   VIDEO LIBRARY   PACKAGE AMSBIB
 General information Latest issue Archive Impact factor Subscription License agreement Submit a manuscript Search papers Search references RSS Latest issue Current issues Archive issues What is RSS

 Uspekhi Mat. Nauk: Year: Volume: Issue: Page: Find

 Uspekhi Mat. Nauk, 2020, Volume 75, Issue 6(456), Pages 107–152 (Mi umn9976)

The Dickman–Goncharov distribution

S. A. Molchanovab, V. A. Panovb

a University of North Carolina at Charlotte, Charlotte, NC, USA
b International Laboratory of Stochastic Analysis and Its Applications, National Research University Higher School of Economics, Moscow, Russia

Abstract: In the 1930s and 40s, one and the same delay differential equation appeared in papers by two mathematicians, Karl Dickman and Vasily Leonidovich Goncharov, who dealt with completely different problems. Dickman investigated the limit value of the number of natural numbers free of large prime factors, while Goncharov examined the asymptotics of the maximum cycle length in decompositions of random permutations. The equation obtained in these papers defines, under a certain initial condition, the density of a probability distribution now called the Dickman–Goncharov distribution (this term was first proposed by Vershik in 1986). Recently, a number of completely new applications of the Dickman–Goncharov distribution have appeared in mathematics (random walks on solvable groups, random graph theory, and so on) and also in biology (models of growth and evolution of unicellular populations), finance (theory of extreme phenomena in finance and insurance), physics (the model of random energy levels), and other fields. Despite the extensive scope of applications of this distribution and of more general but related models, all the mathematical aspects of this topic (for example, infinite divisibility and absolute continuity) are little known even to specialists in limit theorems. The present survey is intended to fill this gap. Both known and new results are given.
Bibliography: 62 titles.

Keywords: Dickman–Goncharov distribution, Vershik chain, Erdős problem, random energy model, cell growth model, random walks on solvable groups.

 Funding Agency Grant Number Russian Science Foundation 17-11-01098 The article was prepared within the framework of the HSE University Basic Research Programme. The results in §§ 4–6 have been obtained under support of the RSF grant no. 17-11-01098.

DOI: https://doi.org/10.4213/rm9976

Full text: PDF file (935 kB)
First page: PDF file
References: PDF file   HTML file

English version:
Russian Mathematical Surveys, 2020, 75:6, 1089–1132

Bibliographic databases:

UDC: 519.2+519.1+511.3
MSC: Primary 60E05, 60E07; Secondary 60Fxx, 60G50, 60G51, 60G70

Citation: S. A. Molchanov, V. A. Panov, “The Dickman–Goncharov distribution”, Uspekhi Mat. Nauk, 75:6(456) (2020), 107–152; Russian Math. Surveys, 75:6 (2020), 1089–1132

Citation in format AMSBIB
\Bibitem{MolPan20} \by S.~A.~Molchanov, V.~A.~Panov \paper The Dickman--Goncharov distribution \jour Uspekhi Mat. Nauk \yr 2020 \vol 75 \issue 6(456) \pages 107--152 \mathnet{http://mi.mathnet.ru/umn9976} \crossref{https://doi.org/10.4213/rm9976} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=4181059} \elib{https://elibrary.ru/item.asp?id=46769435} \transl \jour Russian Math. Surveys \yr 2020 \vol 75 \issue 6 \pages 1089--1132 \crossref{https://doi.org/10.1070/RM9976} \isi{http://gateway.isiknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&DestLinkType=FullRecord&DestApp=ALL_WOS&KeyUT=000626157600001} \scopus{https://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-85103061938}