Mathematics Colloquia and Seminars

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Since 1960's it has been clear that statistics of density fluctuations in one-dimensional reaction-diffusion systems are not predicted correctly by the Smoluchowski-type rate equations. I will review the work of Glauber, Bramson, Lebowitz, Griffeath and Kesten which led to this understanding. As an extension of their results, I will show how to calculate multi-particle probability densities for systems of annihilating and coalescing Brownian motions on the real line. In the large time limit these densities exhibit scaling behaviour. The scaling exponent turns out to be a quadratic function of the number of particles. The exact calculation of scaling exponent is possible due to an interesting structure of correlation functions which we discovered recently: the law of coalescing/annihilating Brownian motions at any fixed moment of time is a Pfaffian point process.