Subdominant eigenvalues for stochastic matrices with given column sums

Abstract

For any stochastic matrix A of order n, denote its eigenvalues as {λ1(A)}, . . . , {λn(A)}, ordered so that 1 = {{\textbar}λ1(A){\textbar}} ≥ {{\textbar}λ2(A){\textbar}} ≥ ... ≥ {{\textbar}λn(A){\textbar}.} Let {cT} be a row vector of order n whose entries are nonnegative numbers that sum to n. Define S(c), to be the set of n × n row-stochastic matrices with column sum vector {cT} . In this paper the quantity λ(c) = {max{{\textbar}λ2(A){\textbar}{\textbar}A} ∈ S(c)} is considered. The vectors {cT} such that λ(c) {\textless} 1 are identified and in those cases, nontrivial upper bounds on λ(c) and weak ergodicity results for forward products are provided. The results are obtained via a mix of analytic and combinatorial techniques.