The number of cycles in random permutations without long cycles is asymptotically Gaussian

Abstract

For uniform random permutations conditioned to have no long cycles, we prove that the total number of cycles satisfies a central limit theorem. Under additional assumptions on the asymptotic behavior of the set of allowed cycle lengths, we derive asymptotic expansions for the corresponding expected value and variance. In contrast to the case of uniform permutations, the asymptotic mean and variance are not logarithmic in the system size.

Random permutations without macroscopic cycles

“…When we choose t = 1 in Theorem 2.5, then b t (n) = α(n) and we recover the result in [6]. Furthermore, if we define…”

“…The first part of the lemma is a reformulation of Lemma 4.11 in [6], which in turn follows [21]. In the latter reference, the claims are actually shown for more general functions α.…”

“…be the number of cycles with lengths less than m. Since no cycle can be larger than α(n), the total number of cycles K α(n) is at least ≥ n/α(n). In [6] it is shown that K α(n) ∼ n α(n) , and so the random variable…”

“…Then (2) When t = 1 in Theorem 2.6, the variance of the limit is zero. However, it has been shown in [6] that there exists a different rescaling so that the Gaussian fluctuations persist in the limit:…”

Precise asymptotics of longest cycles in random permutations without macroscopic cycles

“…[26,27]); our interest lies in the second case. Using precise asymptotic results by Manstavi£ius and Petuchovas [20], in [3,4] we investigated the case where a permutation of length n is prevented from having any cycles above a threshold α(n) that grows strictly slower than volume order. While the results in these papers were reasonably detailed, some interesting questions and ne details have been left out.…”