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TL;DR

Spectral Gap and Markov Chain Mixing Time: A canonical quantitative trading interview question at olympiad difficulty. Commonly asked at Jane Street, Two Sigma, DE Shaw.

By Valenke Exam Prep Team·Last updated 2026-06-01
olympiadLinear Algebra & Optimization

Spectral Gap and Markov Chain Mixing Time

Asked at: Jane Street, Two Sigma, DE Shaw

Problem
A Markov chain on nn states has doubly stochastic transition matrix PP with eigenvalues 1=λ1>λ2≥⋯≥λn>−11 = \lambda_1 > \lambda_2 \geq \cdots \geq \lambda_n > -1. Show that the chain mixes to uniform, and prove that the mixing time satisfies tmix=O(ln⁡n1−λ2)t_{mix} = O\left(\frac{\ln n}{1 - \lambda_2}\right).
Related concepts
EigenvaluesMarkov Chains

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