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What happens to a random walk of chained particles when the chain forms a loop or is very long?

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Consider a random walk of a chain of K+1 particles at integer sites, where the chaining keeps each particle at distance 1 from its immediate neighbours. In dimension 1, we showed with Boissard, Espinasse and Norris that the effect of chaining is to slow down the walk by a factor of 2/(K+2). In this talk I will make some remarks for the cases when K is infinite and when the ends of the chain are joined.

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