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March 2007 When the law of large numbers fails for increasing subsequences of random permutations
Ross G. Pinsky
Ann. Probab. 35(2): 758-772 (March 2007). DOI: 10.1214/009117906000000728

## Abstract

Let the random variable Zn,k denote the number of increasing subsequences of length k in a random permutation from Sn, the symmetric group of permutations of {1, …, n}. In a recent paper [Random Structures Algorithms 29 (2006) 277–295] we showed that the weak law of large numbers holds for Zn,kn if kn=o(n2/5); that is, $$\lim_{n\to\infty}\frac{Z_{n,k_{n}}}{EZ_{n,k_{n}}}=1\qquad \mbox{in probability}.$$ The method of proof employed there used the second moment method and demonstrated that this method cannot work if the condition kn=o(n2/5) does not hold. It follows from results concerning the longest increasing subsequence of a random permutation that the law of large numbers cannot hold for Zn,kn if kncn1/2, with c>2. Presumably there is a critical exponent l0 such that the law of large numbers holds if kn=O(nl), with l<l0, and does not hold if $\limsup_{n\to\infty}\frac{k_{n}}{n^{l}}>0$, for some l>l0. Several phase transitions concerning increasing subsequences occur at l=½, and these would suggest that l0=½. However, in this paper, we show that the law of large numbers fails for Zn,kn if $\limsup_{n\to\infty}\frac{k_{n}}{n^{4/9}}=\infty$. Thus, the critical exponent, if it exists, must satisfy $l_{0}\in[\frac{2}{5},\frac{4}{9}]$.

## Citation

Ross G. Pinsky. "When the law of large numbers fails for increasing subsequences of random permutations." Ann. Probab. 35 (2) 758 - 772, March 2007. https://doi.org/10.1214/009117906000000728

## Information

Published: March 2007
First available in Project Euclid: 30 March 2007

zbMATH: 1124.60033
MathSciNet: MR2308596
Digital Object Identifier: 10.1214/009117906000000728

Subjects:
Primary: 60C05 , 60F05

Keywords: increasing subsequences in random permutations , Law of Large Numbers , Random permutations  