Pierre Tarrès

Brownian polymer with repulsive interaction in dimension one


The Brownian polymer model was introduced by Durrett and Rogers (1992) in order to describe the shape of a growing polymer. It undergoes a drift which depends on its past trajectory, and a Brownian increment. I will discuss two conjectures of these authors on the model (1992), concerning repulsive interaction in dimension one. We showed the first one with T. Mountford (Ann. IHP 2008), for certain interaction functions with heavy tails, leading to transience to the right or to the left, both occuring with probability 1/2. We partially proved the second one with B. Toth and B. Valko (Ann. Prob. 2011), for rapidly decreasing interaction functions, through a study of the local time environment viewed from the particule. We explicitly display an associated invariant measure, which enables us to prove, under certain initial conditions, a strong law of large numbers with speed 0. We will also discuss the scaling of the walk, depending on the infrared spectrum of the interaction function. It is always at least diffusive and, in the case considered by Durrett and Rogers (1992), furthermore believed to be t^{2/3} at time t (Toth and Werner,1998): we show (same joint work with B. Toth and B. Valko, AP 2011) it to be between t^{5/8} and t^{3/4}.

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