Two Examples of COM Bounds using Spectral Gaps: Length of the LIS in a Random Permutation and Lipschitz Functions of 1d Markov Chains

Abstract

We consider two examples for a well-known method for obtaining concentration of measure (COM) bounds for a given observable in a given measure. The method is to consider an auxiliary Markov chain for which the invariant distribution is the measure of interest. Then one obtains COM bounds involving two quantities. The first is the spectral gap of the Markov transition matrix. The second is an appropriate Lipschitz constant for the observable of interest with respect to 1 step of the Markov chain. We consider two examples of the basic method. The first is to obtain rough COM bounds for the length of the longest increasing subsequence (LIS) in a uniform random permutation. The bounds are similar to well-known bounds of Talagrand using his isoperimetric inequality. The second example is to consider a 1d Markov chain: $X_0,X_1,\dots,X_n$. We assume the invariant measure for the chain $μ$ is reversible, and let the initial distribution of $X_0$ be $μ$. Then the observable of interest is any function $f(X_0,X_1,\dots,X_n)$, which is Lipschitz with respect to replacement of single variables. One case of this is `target frequency analysis,' which is of interest in biostatistics. The auxiliary Markov chain is Glauber dynamics which is gapped in 1d.