Education
We consider a directed polymer on the unit circle, with a continuous direction (time) parameter , defined as a simple random walk subjected via a Gibbs measure to a Hamiltonian whose increments in time have either long memory () or semi-long memory (), and which also depends on a space parameter (position/state of the polymer). is interpreted as the Hurst parameter of an infinite-dimensional fractional Brownian motion. The partition function of this polymer is linked to stochastic PDEs via a long-memory parabolic Anderson model. We present a summary of the new techniques which are required to prove that, in the semi-long memory case, converges to a positive finite non-random constant, and in the long-memory case, this limit is blows up, while the correct exponential growth function in that case is sandwiched between and . These tools include an almost sub-additivity concept, usage of Malliavin derivatives for concentration estimates, and an adaptation to the long-memory case of some arguments from the case (no memory), which require a detailed study of the interaction between the long memory, the spatial covariance, and the simple random walk. This talk describes joint work with Dr. Tao Zhang. Frederi VIENS. Purdue University. Bande son disponible au format mp3 Durée : 46 mn