Increasing subsequences of random walks

Abstract

Given a sequence of $n$ real numbers $\{S_i\}_{i\leq n}$, we consider the longest weakly increasing subsequence, namely $i_1<i_2<\dots <i_L$ with $S_{i_k} \leq S_{i_{k+1}}$ and $L$ maximal. When the elements $S_i$ are i.i.d. uniform random variables, Vershik and Kerov, and Logan and Shepp proved that $\mathbb{E} L=(2+o(1)) \sqrt{n}$. We consider the case when $\{S_i\}_{i\leq n}$ is a random walk on $\mathbb{R}$ with increments of mean zero and finite (positive) variance. In this case, it is well known (e.g., using record times) that the length of the longest increasing subsequence satisfies $\mathbb{E} L\geq c\sqrt{n}$. Our main result is an upper bound $\mathbb{E} L\leq n^{1/2 + o(1)}$, establishing the leading asymptotic behavior. If $\{S_i\}_{i\leq n}$ is a simple random walk on $\mathbb{Z}$, we improve the lower bound by showing that $\mathbb{E} L \geq c\sqrt{n} \log{n}$. We also show that if $\{\mathbf{S}_i\}$ is a simple random walk in $\mathbb{Z}^2$, then there is a subsequence of $\{\mathbf{S}_i\}_{i\leq n}$ of expected length at least $cn^{1/3}$ that is increasing in each coordinate. The above one-dimensional result yields an upper bound of $n^{1/2 + o(1)}$. The problem of determining the correct exponent remains open.