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Selfavoiding polygons and walks: counting, joining and closing.Seminar  November 8  3:104 p.m.  1011 Evans Hall Alan Hammond, U.C. Berkeley Selfavoiding walk of length n on the integer lattice Z^d is the uniform measure on nearestneighbour walks in Z^d that begin at the origin and are of length n. If such a walk closes, which is to say that the walk's endpoint neighbours the origin, it is natural to complete the missing edge connecting this endpoint and the origin. The result of doing so is a selfavoiding polygon. We investigate the numbers of selfavoiding walks, polygons, and in particular the "closing" probability that a length n selfavoiding walk is closing. Developing a method (the "snake method") employed in joint work with Hugo DuminilCopin, Alexander Glazman and Ioan Manolescu that provides closing probability upper bounds by constructing sequences of laws on selfavoiding walks conditioned on increasing severe avoidance constraints, we show that the closing probability is at most n^{1/2 + o(1)} in any dimension at least two. Developing a quite different method of polygon joining employed by Madras in 1995 to show a lower bound on the deviation exponent for polygon number, we also provide new bounds on this exponent. We further make use of the snake method and polygon joining technique at once to prove an upper bound, valid subsequentially, on the closing probability of the form n^{4/7 + o(1)} in the twodimensional setting. 5100000000 

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