Large Deviations for the One - Dimensional Edwards Model

ONE-DIMENSIONAL EDWARDS MODEL Mar h 6, 2002 Rem o van der Hofstad 1 2 Frank den Hollander 3 Wolfgang K onig 4 Abstra t: In this paper we prove a large deviation prin iple for the empiri al drift of a onedimensional Brownian motion with self-repellen e alled the Edwards model. Our results extend earlier work in whi h a law of large numbers, respe tively, a entral limit theorem were derived. In the Edwards model a path of length T re eives a penalty e HT , where HT is the self-interse tion lo al time of the path and 2 (0;1) is a parameter alled the strength of self-repellen e. We identify the rate fun tion in the large deviation prin iple for the endpoint of the path as 2 3 I( 13 ), with I( ) given in terms of the prin ipal eigenvalues of a one-parameter family of Sturm-Liouville operators. We show that there exist numbers 0 < b < b <1 su h that: (1) I is linearly de reasing on [0; b ℄; (2) I is real-analyti and stri tly onvex on (b ;1); (3) I is ontinuously di erentiable at b ; (4) I has a unique zero at b . (The latter fa t identi es b as the asymptoti drift of the endpoint.) The riti al drift b is asso iated with a rossover in the optimal strategy of the path: for b b the path assumes lo al drift b during the full time T , while for 0 b < b it assumes lo al drift b during time b +b 2b T and lo al drift b during the remaining time b b 2b T . Thus, in the se ond regime the path makes an overshoot of size b b 2 T in order to redu e its interse tion lo al time.