Generalized crested products of Markov chains
Abstract.
We define a finite Markov chain, called generalized crested
product, which naturally appears as a generalization of the first
crested product of Markov chains. A complete spectral analysis is
developed and the step transition probability is given. It is
important to remark that this Markov chain describes a more
general version of the classical Ehrenfest diffusion model.
As a particular case, one gets a generalization of the
classical Insect Markov chain defined on the ultrametric space.
Finally, an interpretation in terms of representation group theory
is given, by showing the correspondence between the spectral
decomposition of the generalized crested product and the Gelfand
pairs associated with the generalized wreath product of
permutation groups.
Key words and phrases:
Reversible Markov chain, generalized crested product, Insect Markov chain, spectral theory, Gelfand pairs.Mathematics Subject Classification (2010): 60J10, 05C50, 06A07, 20B25.
1. Introduction
This paper deals with the study of a finite Markov chain, called
generalized crested product, which is defined on the product of
finite spaces. The generalized crested product is a generalization
of the Markov chains introduced and studied in [6]
and [7]. More precisely, in [6] the
first crested product of Markov chains is defined, inspired by the
analogous definition for association schemes [2], as a
sort of mixing between the classical crossed and nested products
of Markov chains and it contains, as a particular case, the so
called Insect Markov chain introduced by FigàTalamanca in
[10] in the context of Gelfand pairs theory. In
[7] a Markov chain on some special structures
called orthogonal block structures is introduced. If the
orthogonal block structure is a poset block structure, then that
Markov chain can also be defined starting from a finite poset
and it can be interpreted as a slightly different
generalization of the classical Insect Markov chain and of the
associated Gelfand pairs theory. We noticed that, for some
particular posets, the Markov chain in [7] has the
same spectral decomposition as the first crested product of Markov
chains despite the corresponding operators do not coincide. So it
is natural to ask if it is possible to define a Markov chain
containing the first crested product as a particular case, giving
rise to an Insect Markov chain for a particular choice of the
parameters involved. This is the reason why this paper aims at
introducing a new Markov chain which can be seen as a modification
of the Markov chain on the orthogonal block structure and the
natural generalization of the first crested product of Markov
chains. The idea is to take a finite poset and a family
of Markov operators defined on finite sets indexed by
the elements of the poset. Then we consider the sum, over , of
tensor products of Markov chains reflecting, in some sense, the
poset hierarchy structure (see Definition 3.2).
A necessary and sufficient condition to have reversibility of the
Markov chain is proven in Theorem 3.4. In Theorem
3.5, we give a complete spectral analysis of this
Markov chain and we show in Proposition 3.8
that it coincides with the first crested Markov chain when the
poset satisfies some particular properties. A formula
for the step probability is given in Section 3.4.
Moreover, we introduce in Section 4 an Insect Markov
chain on the product , naturally identified
with the last level of a graph which is the
generalization of the rooted tree. This Insect Markov chain is
obtained from the generalized crested product of Markov chains for
a particular choice of the operators , i.e. , where
is the uniform operator on the set . If the poset
is totally ordered, this Insect Markov chain coincides
with the classical Insect Markov chain [10]. In Section
5 we highlight the correspondence with the
Gelfand pairs theory (for a general theory and applications see
[8]): taking the generalized wreath product of
permutation groups [3] associated with
and the stabilizer of an element of under the action of this
group, one gets a Gelfand pair [7], and the
decomposition of the action of the group on the space into
irreducible submodules is the same as the spectral decomposition
of the Insect Markov chain associated with . This allows
to study many examples of Gelfand pairs only by using basic tools
of linear algebra.
It is important to remark that the
generalized crested product can be seen as a generalization of a
classical diffusion model, the Ehrenfest model, as well as of the
Ehrenfest model described in [6] (see
[9] and [11] for more examples and
details).
2. Preliminaries
We recall in this section some basic facts about finite Markov chains (see, for instance, [4]). Let be a finite set, with . Let be a stochastic matrix, so that
for every . Consider the Markov chain on with transition matrix . By abuse of notation, we will denote by this Markov chain as well as the associated Markov operator.
Definition 2.1.
The Markov chain is reversible if there exists a strict probability measure on such that
for all .
If this is the case, we say that and are in detailed balance [1].
Define a scalar product on as
for all , and the linear operator as
(1) 
It is easy to verify that and are in detailed balance if
and only if is selfadjoint with respect to the scalar product
. Under these hypotheses, it
is known that the matrix can be diagonalized over the reals.
Moreover, is always an eigenvalue of and for any
eigenvalue
one has .
Let be the eigenvalues of the
matrix , with . Then there exists an
invertible unitary real matrix such
that , where is the diagonal matrix whose entries are the eigenvalues
of . This equation gives, for all ,
(2) 
Moreover, we have , where is the diagonal matrix of coefficients of . This second equation gives, for all ,
(3) 
It follows from (2) that each column of is an eigenvector of , and from (3) that these columns are orthogonal with respect to the product .
Proposition 2.2.
The th step transition probability is given by
(4) 
for all .
Proof.
Recall that there exists a correspondence between reversible Markov chains and weighted graphs.
Definition 2.3.
A weight on a graph is a function such that

;

if and only if .
If is a weighted graph, it is possible to associate with a stochastic matrix on by setting
with . The corresponding Markov chain is called the random walk on . It is easy to prove that the matrix is in detailed balance with the distribution defined, for every , as
with . Moreover, is strictly
positive if does not contain isolated vertices. The inverse
construction can be done. So, if we have a transition matrix
on which is in detailed balance with the probability ,
then we can define a weight as . This
definition guarantees the symmetry of and, by setting , we get a weighted graph.
There are some important relations between the weighted
graph associated with a transition matrix and its spectrum
. In fact, it is easy to prove that the multiplicity of
the eigenvalue 1 of equals the number of connected components
of . Moreover, the following propositions hold.
Proposition 2.4.
Let be a finite connected weighted graph and denote the corresponding transition matrix. Then the following are equivalent:

is bipartite;

the spectrum is symmetric;

.
Definition 2.5.
Let be a stochastic matrix. is ergodic if there exists such that
Proposition 2.6.
Let be a finite graph. Then the following conditions are equivalent:

is connected and not bipartite;

for every weight function on , the associated transition matrix is ergodic.
So we can conclude that a reversible transition matrix is ergodic if and only if the eigenvalue 1 has multiplicity one and is not an eigenvalue. Note that the condition that 1 is an eigenvalue of of multiplicity one is equivalent to require that the probability is irreducible, according with the following definition.
Definition 2.7.
A stochastic matrix on a set is irreducible if, for every , there exists such that .
3. Generalized crested product
3.1. Definition
Let be a finite poset, with . The following definitions are given in [3].
Definition 3.1.
A subset is said

ancestral if, whenever and , then ;

hereditary if, whenever and , then ;

a chain if, whenever , then either or ;

an antichain if, whenever and , then neither nor .
Given an element , we set to be the ancestral set of and . Analogously we set to be the hereditary set of and . For a subset we put , , and . Moreover, we denote by the set of the antichains of and we set . It is clear that and the empty set belong to . Note that .
For each , let be a finite set, with , so that we can identify with the set . Moreover, let be an irreducible Markov chain on and let be the corresponding transition probability. We also denote by the associated Markov operator defined as in (1). Let be the identity matrix of size and set:
We also denote by and the associated Markov operator on , that we call the identity and the uniform operator, respectively. We are going to define a Markov chain on the set .
Definition 3.2.
Let be a finite poset and let be a probability distribution on , i.e. for every and . The generalized crested product of the Markov chains defined by and is the Markov chain on whose associated Markov operator is
(5) 
Remark 3.3.
The generalized crested product can be seen as a generalization of the classical diffusion Ehrenfest model. This classical model consists of two urns numbered 0, 1 and balls numbered . A configuration is given by a placement of the balls into the urns. Note that there is no ordering inside the urns. At each step, a ball is randomly chosen (with probability ) and it is moved to the other urn. In [6] we generalized it to the Ehrenfest model. Now put , for each : then we have the following interpretation of the generalized crested product. Suppose that we have balls numbered by and urns. Let be a finite poset with elements. At each step, we choose a ball according with a probability distribution : then we move it to another urn following a transition probability and all the other balls numbered by indices such that in the poset are moved uniformly to a new urn. The balls corresponding to all the other indices are not moved.
From now on, we suppose that each is in detailed balance with the probability measure .
Theorem 3.4.
The generalized crested product is reversible if and only if is symmetric for every , i.e. , for all . If this is the case, is in detailed balance with the strict probability measure on given by
Proof.
We start by proving that the condition , for each , is sufficient. Consider two elements and in . First, suppose that there exists such that and for all . In this case, we have
If we suppose that there exist such that , for , then and there is nothing to prove.
Suppose now that for every and there is such that . We have
On the other hand, we show that the condition , for each , is necessary. Suppose that the equality holds for all . Let : by irreducibility, we can choose such that and . Let for every . We have
This gives
(6) 
Let such that for each and for each . Proceeding as above we get
(7) 
so that (6) and (7) imply , i.e. does not depend on the coordinates corresponding to indices in . Let and let such that for each and for each . The condition reduces to
(8) 
since and differ only for indices in and so . Observe that for each and so the summands corresponding to these indices are equal in both sides of (8). Moreover, for each , one has and so the summands corresponding to these indices are 0 in both sides of (8), since . Hence, (8) reduces to , what implies and so the hypothesis of irreducibility guarantees that is uniform on . This completes the proof. ∎
3.2. Spectral analysis
The next step is to study the spectral decomposition of the
operator . Suppose that
is the decomposition
of into eigenspaces of and that is
the eigenvalue corresponding to . The eigenspace
is the space of the constant functions over
: under our hypothesis of irreducibility, we have .
For every antichain , define
and, for and , we put
Moreover, we set
(9) 
and
(10) 
Theorem 3.5.
Let be a finite poset and let be the generalized crested product defined in (5). The decomposition of into eigenspaces for is
Moreover, the eigenvalue associated with is .
Proof.
Let be a function in . We can represent as the tensor product , with if , if , and if . We have to show that for every . One has:
Observe that, if , then and so is constant for every . Suppose , for some . Hence, the term of corresponding to is
On the other hand, if , then and so the identity operator , for , acts on the space orthogonal to and acts on . Note that , so that the term corresponding to the index is
Finally, if , then there exists such that . In particular, is orthogonal to , i.e. and so the term corresponding to the index is
Hence
and the claim is proven. ∎
Corollary 3.6.
If is ergodic for each , then is ergodic.
Proof.
The expression of the eigenvalues of given in (10) ensures that the eigenvalue 1 is obtained with multiplicity one and the eigenvalue can never be obtained. ∎
We are able now to provide the matrices and associated with . For every , let , and be the matrices of eigenvectors, of the coefficients of and of eigenvalues for the probability , respectively. Recall the identification of with the set .
Proposition 3.7.
The matrices and have the following form:

, where
By we denote the matrix of size whose entries on the first column are all 1 and the remaining ones are 0.

.

, where is the diagonal matrix of size .
Proof.
Let us start by proving the statement for the matrix . By construction, each column of is an eigenvector of . Let us show that the rank of is maximal. Fix . Then the matrix
(11) 
has rank if . If , then has rank 1. Moreover, eigenvectors arising from different
are independent because they belong to subspaces of
which are orthogonal with respect to the scalar product .
First, let us show that, if , then the sets of
indices corresponding to the non zero columns of
and are disjoint. Note that implies
. Hence, we can assume
without loss of generality that there exists such that either or . Suppose , and put , where
or according with
(11). Under our assumption and
, so that our claim is true for
and . Then the same property
can be deduced for and . Suppose
now that . Then there is such that . We claim that . In fact if then , which is absurd. If then , a contradiction again. Hence, there exists an index such that and from (11)
we deduce that the claim is true for
and .
Hence, we deduce from Theorem 3.5 that
the rank of is and so it is
maximal.
In order to get the diagonal matrix , whose entries are the
coefficients of , it suffices to consider the tensor product
of the corresponding matrices associated with the probability
, for every .
Finally, to get the
matrix of eigenvalues of it suffices to
replace, in the expression of the matrix , the matrix
by and the matrix by the corresponding
diagonal matrix .
∎
3.3. The case of the first crested product
In [6] the definition of the first crested product of Markov chains is given. More precisely, considering the product and a partition
(12) 
of the set , given a probability distribution on , the first crested product of the Markov chains with respect to the partition (12) is defined as the Markov chain on whose transition matrix is
We want to show in this section that, if the poset satisfies some special conditions, then the generalized crested product defined in (5) reduces to the first crested product. We denote by the usual ordering of natural numbers.
Proposition 3.8.
Suppose that satisfies the following property: given such that , then if and only if . Then the first crested product of Markov chains is obtained by the operator defined in (5) by putting:
Proof.
The partition , with
gives: