Mathematics Colloquia and Seminars

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Loop percolation, also known as the dense O(1) loop model, can be thought of as a natural variant of critical bond percolation in Z^2. The model has been studied extensively in a cylindrical geometry, where the resulting connectivity pattern of 2n points arranged around the "lid" of the cylinder is a random noncrossing matching (a.k.a. "link pattern") with fascinating properties that appears naturally in connection with the enumeration of Fully Packed Loops (the Cantini-Sportiello-Razumov-Stroganov theorem) and the ground state of the quantum XXZ spin chain. I will discuss the cylindrical geometry and also consider the model on a half-plane, which is the limit of the cylindrical models. A remarkable rationality phenomenon has been observed by Zuber and others whereby the probabilities of certain connectivity events are simple rational numbers (in the half-plane case) or simple rational functions of n (in the cylindrical case). For example, the probability for two given adjacent endpoints to be connected is 3/2*(n^2+1)/(4n^2-1), or 3/8 in the limiting case of the half-plane. This result and a few other instances of the rationality phenomenon have been proved by Fonseca and Zinn-Justin. One of the new results I will discuss is a much more general formula expressing the probabilities of arbitrary "submatching events" as constant terms of certain multivariate Laurent polynomials. This reduces the problem of proving the rationality phenomenon in the general case to that of proving a purely algebraic family of constant term identities.