[Paper Review] Asymptotic analysis of the cyclic structure of permutations

Robertas Petuchovas|arXiv (Cornell University)|Nov 9, 2016

Bayesian Methods and Mixture Models30 references3 citations

TL;DR

This doctoral dissertation presents asymptotic formulas for the density of permutations without long or short cycles in the symmetric group $\mathrm{S}_n$, using saddle-point analysis and Laplace transforms. It establishes precise asymptotic estimates for the total variation distance between the cycle count distribution of random permutations and a Poisson approximation, significantly improving prior results through refined analysis of the Dickman and Buchstab functions.

ABSTRACT

This is my dissertation. Its research object is a symmetric group of permutations acting on a finite set. The density of permutations with a given cycle structure pattern is explored when the group order tends to infinity. New and sharper asymptotic formulas are obtained. The latter are applied in approximations of the cycle vector distribution of a random permutation. The saddle-point method, Lagrange-Bürmann inversion formula, Laplace transformations, and other techniques of complex analysis are applied.

Motivation & Objective

• To derive asymptotic formulas for the proportion of permutations in $\mathrm{S}_n$ with no cycles of length $> r $, for $ r = r(n) \leq n $, covering the full range of $ r $.
• To obtain sharp asymptotic estimates for the density of permutations with no cycles of length $ \leq r $, extending known results via saddle-point techniques.
• To determine the asymptotic behavior of the total variation distance between the joint cycle count distribution of random permutations and a multivariate Poisson model with mean $ 1/j $ for cycle length $ j $.
• To refine existing bounds on the total variation distance by improving estimates for the densities $ \nu(n,r) $ and $ \kappa(n,r) $, which govern the distance.
• To demonstrate the applicability of number-theoretic techniques, such as those used in prime factorization, to the analysis of permutation cycle structures.

Proposed method

• Saddle-point analysis is employed to derive asymptotic expansions for the number of permutations avoiding cycles of certain lengths, based on generating functions and complex asymptotic methods.
• Laplace transforms of the Dickman and Buchstab functions are used to refine the asymptotic analysis of cycle count distributions.
• The method draws on techniques from analytic number theory, particularly those used in studying the distribution of prime factors in integers, and adapts them to permutation cycle structures.
• Key components include the use of the Dickman function $ \varrho(v) $, the Buchstab function $ \omega(v) $, and the function $ R(v) $, which captures the oscillatory behavior of $ \omega(v) - e^{-\gamma} $.
• The analysis involves splitting sums into ranges and applying integral approximations with error bounds derived from the variation and continuity properties of $ \varrho(v) $ and $ \omega(v) $.
• Estimates are glued together using bounds on the number of sign changes and oscillations of $ \omega(v) - e^{-\gamma} $, leveraging results on the distribution of zeros of related functions.

Experimental results

Research questions

• RQ1 What is the asymptotic density of permutations in $ \mathrm{S}_n $ with no cycles of length greater than $ r $, as $ n \to \infty $, for arbitrary $ r = r(n) \leq n $?
• RQ2 What is the asymptotic density of permutations in $ \mathrm{S}_n $ with no cycles of length at most $ r $, for $ r = r(n) \leq n $?
• RQ3 How does the total variation distance between the cycle count vector of a random permutation and a multivariate Poisson vector with means $ 1/j $ behave asymptotically as $ n \to \infty $?
• RQ4 Can the error terms in the asymptotic formula for the total variation distance be improved by refining the estimates of $ \nu(n,r) $ and $ \kappa(n,r) $?
• RQ5 To what extent can techniques from the analysis of prime factorizations be adapted to study the cyclic structure of permutations?

Key findings

• The asymptotic formula for $ \nu(n,r) $, the density of permutations without cycles of length $ > r $, is established for all $ r = r(n) \leq n $, improving on all prior results via saddle-point analysis.
• For $ \kappa(n,r) $, the density of permutations without cycles of length $ \leq r $, the thesis provides new asymptotic estimates that extend and refine previous work, again using saddle-point methods.
• The total variation distance $ d_{TV}(n,r) $ is shown to satisfy a precise asymptotic formula, with the main term involving integrals of $ \varrho(v) |\omega(u-v) - e^{-\gamma}| $, and error terms controlled by $ O\left( \frac{u^{3/2} \log^2(u+1)}{r} H(u) \right) $.
• The error in the asymptotic formula for $ d_{TV}(n,r) $ is shown to be $ O\left( \frac{u^{3/2} \log^2(u+1)}{r} H(u) \right) $, where $ H(u) \sim \int_1^u \varrho(u-v) R(v) dv $, with $ R(v) $ capturing the oscillatory behavior of the Buchstab function.
• The number of sign changes of $ \omega(v) - e^{-\gamma} $ in $ [1,u] $ is bounded by $ O(u) $, enabling the application of summation estimates via integral approximation.
• The results demonstrate that the saddle-point method, when combined with Laplace transforms and number-theoretic techniques, provides a powerful framework for analyzing cycle structures in permutations.