By building upon a Feynman-Kac formalism, we assess the distribution of the number of collisions in a given region for a broad class of discrete-time random walks in absorbing and nonabsorbing media. We derive the evolution equation for the generating function of the number of collisions, and we complete our analysis by examining the moments of the distribution and their relation to the walker ...

This paper provides a review of a new method of addressing problems in di usion Monte Carlo: the Green's function rst-passage method (GFFP). In particular, we address four new strands of thought and their interaction with the GFFP method: the use of angle-averaging methods to reduce vector or tensor Laplace equations to scalar Laplace equations; the use of the simulation-tabulation (ST) method ...

whereW is the Weyl group, ρ is the Weyl vector, and the product is over all positive roots α. This is a special case of Weyl’s character formula which says that the character of a finite dimensional representation is equal to a sum similar to the left hand side divided by the product on the right hand side; for the 1-dimensional representation the character is 1 so the sum is equal to the produ...