15:45
I talk about a recent article of mine that aims at giving an alternative proof to a formula by Carne on random walks. Consider a discrete, reversible random walk on a graph (not necessarily the simple walk); then one has a surprisingly simple formula bounding the probability of getting from a vertex x at time 0 to another vertex y at time t, where it appears a universal Gaussian factor essentially depending on the graph distance between x and y. While Carne proved that result in 1985, through‘miraculous’ (though very pretty!) spectral analysis reasoning, I will expose my own ‘natural' probabilistic proof of that fact. Its main interest is philosophical, but it also leads to a generalization of the original formula. The two main tools we shall use will be techniques of forward and backward martingales, and a tricky conditioning argument to prevent a random walk from being `’too transient'.