Generalized crested products of Markov chains

# Generalized crested products of Markov chains

Daniele D’Angeli Daniele D’Angeli. Department of Mathematics, Technion–Israel Institute of Technology, Technion City, Haifa 32 000.  and  Alfredo Donno Alfredo Donno. Dipartimento di Matematica “G. Castelnuovo”, Sapienza Università di Roma, Piazzale A. Moro, 2, 00185 Roma, Italia.
July 9, 2019
###### 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 Figà-Talamanca 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

 ∑x∈Xp(x0,x)=1,

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

 π(x)p(x,y)=π(y)p(y,x),

for all .

If this is the case, we say that and are in detailed balance [1].

Define a scalar product on as

 ⟨f1,f2⟩π=∑x∈Xf1(x)¯¯¯¯¯¯¯¯¯¯¯¯f2(x)π(x),

for all , and the linear operator as

 (1) (Pf)(x)=∑y∈Xp(x,y)f(y).

It is easy to verify that and are in detailed balance if and only if is self-adjoint 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) ∑y∈Xp(x,y)u(y,z)=u(x,z)λz.

Moreover, we have , where is the diagonal matrix of coefficients of . This second equation gives, for all ,

 (3) ∑x∈Xu(x,y)u(x,z)π(x)=δy(z).

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) p(k)(x,y)=π(y)∑z∈Xu(x,z)λkzu(y,z),

for all .

###### Proof.

The proof is a consequence of (2) and (3). In fact, the matrix is the inverse of , so that . In formulæ, we have

 ∑y∈Xu(x,y)u(z,y)=1π(z)Δz(x).

From the equation we get , which gives

 p(x,y)=π(y)∑z∈Xu(x,z)λzu(y,z).

Iterating this argument we obtain , which is the assertion. ∎

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

1. ;

2. if and only if .

If is a weighted graph, it is possible to associate with a stochastic matrix on by setting

 p(x,y)=w(x,y)W(x),

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

 π(x)=W(x)W,

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:

1. is bipartite;

2. the spectrum is symmetric;

3. .

###### Definition 2.5.

Let be a stochastic matrix. is ergodic if there exists such that

 p(n0)(x,y)>0,   for all x,y∈X.
###### Proposition 2.6.

Let be a finite graph. Then the following conditions are equivalent:

1. is connected and not bipartite;

2. 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:

 Ji=1mi⎛⎜ ⎜ ⎜ ⎜ ⎜ ⎜⎝11⋯11⋱⋮⋮⋱⋮1⋯⋯1⎞⎟ ⎟ ⎟ ⎟ ⎟ ⎟⎠.

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) P=∑i∈Ip0i⎛⎝Pi⊗⎛⎝⨂j∈H(i)Uj⎞⎠⊗⎛⎝⨂j∈I∖H[i]Ij⎞⎠⎞⎠.
###### 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

 π(x1,…,xn)=∏i∈¯¯¯Sσi(xi)∏i∈I∖¯¯¯Smi.
###### 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

 π(x)P(x,y) = ∏i∈¯¯¯Sσi(xi)∏i∈I∖¯¯¯Smi⋅p0j⋅pj(xj,yj)∏i∈H(j)mi⋅∏i∈I∖H[j]δi(xi,yi) = σj(xj)pj(xj,yj)⋅p0j⋅∏i∈¯¯¯S∖{j}σi(xi)∏i∈I∖¯¯¯Smi⋅∏i∈I∖H[j]δi(xi,yi)∏i∈H(j)mi = σj(yj)pj(yj,xj)⋅p0j⋅∏i∈¯¯¯S∖{j}σi(yi)∏i∈I∖¯¯¯Smi⋅∏i∈I∖H[j]δi(yi,xi)∏i∈H(j)mi = ∏i∈¯¯¯Sσi(yi)∏i∈I∖¯¯¯Smi⋅p0j⋅pj(yj,xj)∏i∈H(j)mi⋅∏i∈I∖H[j]δi(yi,xi) = π(y)P(y,x).

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

 π(x)P(x,y) = ∏i∈¯¯¯Sσi(xi)∏i∈I∖¯¯¯Smi⋅⎛⎝∑i∈A(j)p0i⋅pi(xi,yi)∏h∈I∖H[i]δh(xh,yh)∏h∈H(i)mh + p0j⋅pj(xj,yj)∏h∈I∖H[j]δh(xh,yh)∏h∈H(j)mh) = ∏i∈¯¯¯Sσi(yi)∏i∈I∖¯¯¯Smi⋅⎛⎝∑i∈A(j)p0i⋅pi(yi,xi)∏h∈I∖H[i]δh(yh,xh)∏h∈H(i)mh + p0j⋅pj(yj,xj)∏h∈I∖H[j]δh(yh,xh)∏h∈H(j)mh) = π(y)P(y,x).

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

 π(x)P(x,y)=π(y)P(y,x)⟺π(x)pi(xi,yi)=π(y)pi(yi,xi).

This gives

 (6) π(x)π(y)=pi(yi,xi)pi(xi,yi)=σi(xi)σi(yi).

Let such that for each and for each . Proceeding as above we get

 (7) π(¯¯¯x)π(y)=pi(yi,xi)pi(xi,yi)=σi(xi)σi(yi),

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) P(x,x′)=P(x′,x),

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

 JS:={j–=(ji1,…,jik) : jih∈{1,…,rih}}

and, for and , we put

 VS,j–:=Vi1ji1⊗⋯⊗Vikjik.

Moreover, we set

 (9) WS,j–:=VS,j–⊗⎛⎝⨂i∈A(S)L(Xi)⎞⎠⊗⎛⎝⨂i∈I∖A[S]Vi0⎞⎠

and

 (10) λS,j–=k∑h=1p0ihλjih+∑i∈I∖A[S]p0i.
###### 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

 L(X)=⨁S∈S⎛⎝⨁j–∈JSWS,j–⎞⎠.

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:

 (Pφ)(x) = ∑(y1,…,yn)∈X∑i∈Ip0i⎛⎝pi(xi,yi)φi(yi)∏j∈I∖H[i]δj(xj,yj)∏j∈H(i)mj∏j≠iφj(yj)⎞⎠ = ∑i∈Ip0i∑yj: j∈H[i]pi(xi,yi)φi(yi)∏j∈H(i)mj∏j∈I∖H[i]φj(xj)∏j∈H(i)φj(yj).

Observe that, if , then and so is constant for every . Suppose , for some . Hence, the term of corresponding to is

 p0ih∏j∈H(ih)mj⋅∑yihpih(xih,yih)φih(yih)∏j∈H(ih)mj∏j≠ihφj(xj)=(p0ihλjih)φ(x).

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

 p0i∏j∈H(i)mj⋅∑yipi(xi,yi)φi(yi)∏j∈H(i)mj∏j≠iφj(xj) = p0i∏j∈H(i)mj⋅φi(yi)∏j∈H(i)mj∏j≠iφj(xj)=p0iφ(x).

Finally, if , then there exists such that . In particular, is orthogonal to , i.e. and so the term corresponding to the index is

 p0i∑yj: j∈H[i]pi(xi,yi)φi(yi)∏j∈H(i)mj∏j∈I∖H[i]φj(xj)∏j∈H(i)φj(yj) = p0i⎛⎜⎝∑yj: k≠j∈H[i]pi(xi,yi)φi(yi)∏k≠j∈H(i)mj∏j∈I∖H[i]φj(xj)∏k≠j∈H(i)φj(yj)⎞⎟⎠⋅1mk∑yk∈Xkφk(yk) = 0.

Hence

 (Pφ)(x)=⎛⎝k∑h=1p0ihλjih+∑i∈I∖A[S]p0i⎞⎠φ(x).

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

 Iσi−normi=⎛⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜⎝1√σi(0)1√σi(1)⋱1√σi(mi−1)⎞⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟⎠.

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) MS:=(⨂i∈S(Ui−Ai))⊗⎛⎝⨂i∈I∖A[S]Ai⎞⎠⊗⎛⎝⨂i∈A(S)Iσi−normi⎞⎠

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) {1,…,n}=C∐N

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

 P = ∑i∈Cp0i(I1⊗⋯⊗Ii−1⊗Pi⊗Ii+1⊗⋯⊗In) + ∑i∈Np0i(I1⊗⋯⊗Ii−1⊗Pi⊗Ji+1⊗⋯⊗Jn).

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:

 N={i : H(i)≠∅}and C={1,…,n}∖N.
###### Proof.

The partition , with

 N={i : H(i)≠∅}and C={1,…,n}∖N,

gives:

 P = ∑I∈Cp0iI1⊗⋯⊗Ii−1⊗Pi⊗Ii+1⋯⊗In + ∑I∈Np0iI1⊗⋯⊗Ii−1