# Spectral gap of mixture of Markov chains

## Context

Let $P$ be the transition matrix of an irreducible, aperiodic, discrete-time Markov chain. The spectral gap is given by

$$\xi = 1 - \lambda_\max$$

where $\lambda_\max = \max\{\lambda_2, -\lambda_n\}$, i.e. the second largest eigenvalue of the transition matrix $P$.

This is related to the mixing time of the Markov chain; the bigger the spectral gap, the faster the convergence to the stationary distribution.

## Problem

Now suppose I have two such chains on the same state space, with transition matrices $P_1$ and $P_2$, and spectral gaps $\xi_1$ and $\xi_2$.

Furthermore, let me define a new Markov chain with transition matrix:

$$P' = \alpha P_1 + (1-\alpha) P_2$$

in other words, each transition is done according to $P_1$ with probability $\alpha$, and according to $P_2$ with probability $1 - \alpha$.

Now the question is, what can I say about the spectral gap of $P'$? Intuitively, I would guess that the spectral gap is concave, i.e.

$$\xi' \ge \alpha \xi_1 + (1-\alpha) \xi_2$$

But I have no idea how to show this... Any help is appreciated!