We propose a procedure (the first of its kind) for computing a fully
data-dependent interval that traps the mixing time t_mix of a finite
reversible ergodic Markov chain at a prescribed confidence level. The
interval is computed from a single finite-length sample path from the
Markov chain, and does not require the knowledge of any parameters of
the chain. This stands in contrast to previous approaches, which
either only provide point estimates, or require a reset mechanism, or
additional prior knowledge.

The interval is constructed around the relaxation time t_relax, which
is strongly related to the mixing time, and the width of the interval
converges to zero roughly
at a sqrt{n} rate, where n is the length of the sample path. Upper and
lower bounds are given on the number of samples required to achieve
constant-factor multiplicative accuracy. The lower bounds indicate
that, unless further restrictions are placed on the chain, no
procedure can achieve this accuracy level before seeing each state at
least \Omega(t_relax) times on the average. Finally, future
directions of research are identified.

Given a set A in a group, write $A^n$ for the set of all products $x_1…x_n$ with each $x_i$ belonging to A. Roughly speaking, a set A for which $A^2$ is “not much larger than” A is called an “approximate group”. The “growth” of A, on the other hand, refers to the behaviour of $

A^n

$ as n tends to infinity. Both of these have been fruitful areas of study, with applications in various branches of mathematics.

Remarkably, understanding the behaviour of approximate groups allows us to convert information $A^k$ for some single fixed k into information about the sequence $A^n$ as n tends to infinity, and in particular about the growth of A.

In this talk I will present various results describing the algebraic structure of approximate groups, and then explain how to use these to prove new results about growth. In particular, I will describe work with Romain Tessera in which we show that if $

A^k

$ is bounded by $Mk^D$ for some given M and D then, provided k is large enough,

A^n

is bounded by $M’n^D$ for every n > k, with M’ depending only on M and D. This verifies a conjecture of Itai Benjamini.