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A probabilistic approach to Carne's bound

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In this talk I will speak about a 1985 result from Carne & Varopoulos: consider a Markov chain on a graph (i.e. whose transitions always follow the edges) which is reversible with stationary measure $\mu$; then, denoting by $pt(x,y)$ the probability that a chain starting at $x$ is at $y$ at time $t$, one has the Gaussian bound \[ pt(x,y) \leq \sqrt{e} \big( \mu(y)/\mu(x) \big) \exp \big( -d(x,y)2 / 2t \big) ,\] where $d(x,y)$ is the graph distance between $x$ and $y$. My goal will be to explain this result by probabilistic arguments (which was not the case of the original proof), close to the forward/ backward martingale decomposition. This approach will lead to a generalization of Carne’s bound to the case where the particle can occasionally make a jump not following an edge. I will also explain how one can improve Carne’s bound by a spectral factor, by considering the chain “conditioned to be recurrent”.

This talk is part of the Probability series.

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