Spectral extremes under exact cycle conditioning

Abstract

Let $P_n$ be the matrix of a random permutation of $n$ symbols and let $M_n=\log\max_{|z|=1}|\det(I-zP_n)|$. Cook and Zeitouni proved that $M_n/\log n$ converges in probability to a constant $x_0$ for a uniform permutation. We show that the $\sqrt{\log n}$ fluctuations of $M_n$ are carried entirely by the number of cycles $K_n$. Write $\lambda(s)=\log\{\Gamma(1+s)/\Gamma(1+s/2)^2\}$, let $s_\kappa$ minimize $(1+\kappa\lambda(s))/s$ on $(0,\infty)$, and put $v(\kappa)=\kappa\lambda'(s_\kappa)$ and $a_\theta=\lambda(s_\theta)/s_\theta$. Under the Ewens measure with any fixed parameter $\theta>0$ we prove $M_n=v(\theta)\log n+a_\theta(K_n-\theta\log n)+O_P(\log\log n)$, so that the standardized pair $(K_n,M_n)$ converges jointly to $(G,G)$ with $G$ standard normal: the maximum and the cycle count are asymptotically perfectly aligned. This is deduced from a statement about the exact conditional law, which does not depend on $\theta$: for every compact $[\kappa_-,\kappa_+]\subset(0,\infty)$ there is a finite $C$ such that $P(|M_n-v(k/\log n)\log n|>C\log\log n \mid K_n=k)$ tends to $0$ uniformly over integers $k$ with $\kappa_-\log n\le k\le\kappa_+\log n$, that is, over exact and possibly atypical cycle counts. The proof keeps the size and the cycle count simultaneously in a two-variable coefficient extraction. Cycles longer than $n/(\log n)^4$ are reserved as an analytic factor whose coefficients are flat under every size shift produced by the shorter cycles; positivity then converts a scalar coefficient asymptotic into a relative comparison of the entire path-constrained measure, with an error that does not degrade with the number of constraints or with the rarity of the event. The constrained lower bound comes from pointwise saddle estimates for killed convolutions along a dyadic chain of endpoint boxes.