TY - JOUR
T1 - Collision local time of transient random walks and intermediate phases in interacting stochastic systems
AU - Birkner, M.
AU - Greven, A.
AU - Hollander, den, W.Th.F.
PY - 2011
Y1 - 2011
N2 - In a companion paper (M. Birkner, A. Greven, F. den Hollander, Quenched LDP for words in a letter sequence, Probab. Theory Relat. Fields 148, no. 3/4 (2010), 403-456), a quenched large deviation principle (LDP) has been established for the empirical process of words obtained by cutting an i.i.d. sequence of letters into words according to a renewal process. We apply this LDP to prove that the radius of convergence of the generating function of the collision local time of two independent copies of a symmetric and strongly transient random walk on Zd, d = 1, both starting from the origin, strictly increases when we condition on one of the random walks, both in discrete time and in continuous time. We conjecture that the same holds when the random walk is transient but not strongly transient. The presence of these gaps implies the existence of an intermediate phase for the long-time behaviour of a class of coupled branching processes, interacting diffusions, respectively, directed polymers in random environments.
AB - In a companion paper (M. Birkner, A. Greven, F. den Hollander, Quenched LDP for words in a letter sequence, Probab. Theory Relat. Fields 148, no. 3/4 (2010), 403-456), a quenched large deviation principle (LDP) has been established for the empirical process of words obtained by cutting an i.i.d. sequence of letters into words according to a renewal process. We apply this LDP to prove that the radius of convergence of the generating function of the collision local time of two independent copies of a symmetric and strongly transient random walk on Zd, d = 1, both starting from the origin, strictly increases when we condition on one of the random walks, both in discrete time and in continuous time. We conjecture that the same holds when the random walk is transient but not strongly transient. The presence of these gaps implies the existence of an intermediate phase for the long-time behaviour of a class of coupled branching processes, interacting diffusions, respectively, directed polymers in random environments.
M3 - Article
SN - 1083-6489
VL - 16
SP - 552
EP - 586
JO - Electronic Journal of Probability
JF - Electronic Journal of Probability
IS - 20
ER -