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Title : The simple random walk conditioned to visit atypically few sites in dimension three and higher
Abstract
We consider the simple random walk on the Euclidean lattice in transient dimensions. It is known that the number of distinct visited sites is asymptotically linear in time. The probability of visiting a smaller number of sites, with a difference of the order of the mean, was evaluated asymptotically by Phetpradap in 2010, taking up the seminal work of van den Berg, Bolthausen and den Hollander in 2001 concerning the volume of a Wiener sausage. We consider the random walk conditioned to such a rare event and prove that the occupation measure sampled along an appropriate mesoscopic scale (the « skeleton ») converges to a unique optimal profile modulo space shift, provided the deviation from the mean is large enough if dimension is four or higher. Our proof of this so-called « tube property » relies on the recent compactification of the space of measures introduced by Mukherjee and Varadhan. This result is a first step in the rigourous proof of the « Swiss cheese » picture suggested by van den Berg, Bolthausen and den Hollander. This talk is based on joint work with Dirk Erhard (Salvador de Bahia, Brazil).
