Speaker
Reimer Kuhn
(Kings College, London)
Description
We compute spectra of large random stochastic matrices,
i.e. Markov matrices defined on random graphs, where each
edge (ij) in a (sparse) random graph is given a positive
random weight W_{ij}>0 in such a fashion that the each
column sum of the matrix W is normalized to one,
\sum_i W_{ij}= 1. We use the replica method to compute
spectra in the thermodynamic limit, and the cavity method to
obtain results for very large single instances. The stucture
of the graphs and the distribution of the non-zeo weights
W_{ij} are largely arbitrary, as long as the mean degree
remains finite and the column sum constraint are satisfied.
Knowing the spectra of stochastic matrices is tantamount to
knowing the complete spectrum of relaxation times of
stochastic processes described by them, so our results
should have many interesting applications for the
description of relaxation in complex systems.
