A condition for weak disorder for directed polymers in random environment

Publikationstyp:
Zeitschriftenaufsatz
Metadaten:
Autoren
  • Matthias Birkner
DOI
10.1214/ecp.v9-1104
ISSN
1083-589X
Ausgabe der Veröffentlichung
none
Zeitschrift
Electronic Communications in Probability
Datum der Veröffentlichung
2004
Status
Published
Herausgeber
Institute of Mathematical Statistics
Herausgeber URL
http://dx.doi.org/10.1214/ecp.v9-1104
Datum der Datenerfassung
2021
Titel
A Condition for Weak Disorder for Directed Polymers in Random Environment
Ausgabe der Zeitschrift
9

Datenquelle: Crossref

Andere Metadatenquellen:
Abstract
We give a sufficient criterion for the weak disorder regime of directed polymers in random environment, which extends a well-known second moment criterion. We use a stochastic representation of the size-biased law of the partition function. We consider the so-called directed polymer in random environment, being defined as follows: Let p(x, y) = p(y-x), x, y is an element of Z(d) be a shift-invarient, irreducible transition kernel, (S-n)(nis an element ofNo) the corresponding random walk. Let xi(x, n), x is an element of Z(d), n is an element of N be i.i.d. random variables satisfying E[exp(betaxi(x,n))] < &INFIN; for all &beta; &ISIN; R, (1) We denote their cumulant generating function by &lambda;(&beta;) := log E[exp(&beta;&xi;(x, n))]. (2) We think of the graph of S-n as the (directed) polymer, which is influenced by the random environment generated by the &xi;(x, n) through a reweighting of paths with e(n) := e(n) (&xi;, S) :=exp (&USigma;(n)(j=1) betaxi(S-j,j) - gimel(beta)), that is, we are interested in the random probability measures on path space given by mu(n) (d(s)) = 1/Z(n)E[e(n)1(S is an element of ds) \textbackslash xi(.,.)], where the normalising constant (or partition function) is given by [GRAPHICS] Note that (Z(n)) is a martingale, and hence converges almost surely. This model has been studied by many authors, see e.g. [2] and the references given there. It is known that the behaviour of mu(n) as n –> infinity depends on whether lim(n) Z(n) > 0 or limn Z(n) = 0. One speaks of weak disorder in the first, and of strong disorder in the second case. Our aim here is to give a condition for the weak disorder regime. Let (S-n) and S’(n)) be two independent p-random walks starting from S-0 = S’(0) = 0, and let V := Sigma(n=1)(infinity) 1(sn = S’n) be the number of times the two paths meet. Define alpha(*) := sup alpha greater than or equal to 1 : E[alpha(V)\textbackslashS’] < &INFIN; almost surely.
Autoren
  • M Birkner
DOI
10.1214/ECP.v9-1104
ISSN
1083-589X
Zeitschrift
ELECTRONIC COMMUNICATIONS IN PROBABILITY
Notes
unique-id: WOS:000223128400003
Paginierung
22 - 25
Datum der Veröffentlichung
2004
Datum der Datenerfassung
2022
Titel
A condition for weak disorder for directed polymers in random environment
Sub types
  • article
Ausgabe der Zeitschrift
9

Datenquelle: Manual

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