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The probability of long cycles in interchange processes. (English) Zbl 1269.82041

Summary: We examine the number of cycles of length \(k\) in a permutation as a function on the symmetric group. We write it explicitly as a combination of the characters of irreducible representations. This allows us to study the formation of long cycles in the interchange process, including a precise formula for the probability that the permutation is one long cycle at a given time \(t\), and estimates for the cases of shorter cycles.

MSC:

82C22 Interacting particle systems in time-dependent statistical mechanics
60B15 Probability measures on groups or semigroups, Fourier transforms, factorization
43A65 Representations of groups, semigroups, etc. (aspects of abstract harmonic analysis)
20B30 Symmetric groups
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