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Scientific Report 2007-2009
Theoretical physics
T19. Markov chains on graphs
Let G = (V, E) be a connected finite graph with vertex
set V = {1, 2, . . . , n}. The Laplacian of G is the n × n
matrix ∆ G := D − A, where A is the adjacency matrix
of G, and D = diag(d 1 , . . . , d n ) with d i denoting the degree
of the vertex i, i.e. the number of edges originating
from i. Since ∆ G is symmetric and positive semidefinite,
its eigenvalues are real and nonnegative and can be
ordered as 0 = λ 1 ≤ λ 2 ≤ · · · ≤ λ n . There is an extensive
literature dealing with bounds on the distribution
of the eigenvalues and consequences of these bounds. Of
particular importance for several applications is the second
eigenvalue λ 2 which is strictly positive since G is
connected. The Laplacian ∆ G can be viewed as the generator
of a continuous-time random walk on V , whose
invariant measure is the uniform measure on V . In this
respect, λ 2 is the inverse of the “relaxation time” of the
random walk, a quantity related to the speed of convergence
to equilibrium. λ 2 is also called the spectral
gap of ∆ G . There are several results which estabilish
relationships between the spectral gap and various geometric
quantities associated with the graph. Among
these we should mention upper and lower bounds on λ 2
in terms of the Cheeger isoperimetric constant, a result
closely related to the Cheeger’s inequality dealing with
the first eigenvalue of the Laplace–Beltrami operator on
a Riemannian manifold.
One can consider, besides the simple random walk,
more complicated Markov chains on the same graph G.
We mention two widely used processes: the exclusion
process and the interchange process. In the interchange
process each vertex of the graph is occupied by a particle
of a different color (Fig. 1), and for each edge {i, j} ∈ E,
at rate 1, the particles at vertices i and j are exchanged.
The exclusion process is analogous but with only two colors,
say k red particles and n−k green particles (particles
with the same color are considered indistinguishable).
The interchange process on G can be considered as
a random walk on a larger graph with n! vertices corresponding
to the configurations of the process. This
graph is nothing but the Cayley graph of the symmetric
groups S n with generating set given by the edges of G,
where each edge {i, j} is interpreted as a transposition.
We denote this graph with Cay(G). It is easy to show
that the spectrum of ∆ G is a subset of the spectrum of
∆ Cay(G) . By consequence
λ 2 (∆ G ) ≥ λ 2 (∆ Cay(G) ) .
Being an n! × n! matrix, in general the Laplacian of
Cay(G) has many more eigenvalues than the Laplacian of
G. Nevertheless, a neat conjecture due to David Aldous
states, equivalently:
Aldous’s conjecture (v.2). If G is a finite connected
simple graph, then the random walk and the interchange
process on G have the same spectral gap.
Aldous’s conjecture has been proven for trees in 1996.
We have found a proof for complete multipartite graphs
using a tecnique based on the representation theory of
the symmetric group. This result will be published in
a forthcoming issue of the Journal of Algebraic Combinatorics.
Shortly after the appearence of our result, a
general proof of the Aldous’s conjecture was found by
Caputo, Liggett and Richthammer.
In [1] we prove a similar result for a different Markov
chain called initial reversals. Here the set of generators
is given by the permutations
{1, 2, . . . k} −→ {k, . . . , 2, 1} ,
where k is an integer between 2 and n. Again the interest
of this result lies in the fact that it allows to compute
the spectral gap of an n! × n! matrix, by considering a
suitable (much smaller) n × n matrix.
Figure 1: A configuration of the interchange process.
References
1. F. Cesi, Electron. J. Combin. 16, N29 (2009)
Authors
F. Cesi
Aldous’s conjecture (v.1). If G is a finite connected
simple graph, then
λ 2 (∆ G ) = λ 2 (∆ Cay(G) ) .
Sapienza Università di Roma 42 Dipartimento di Fisica