## 1 Introduction

A well-known classification of the infinite loaded graphs, or, which is the same, the infinite non-negative matrices, is very close to that of the Markov chains with infinitely many states: there are recurrent and transient loaded graphs, null recurrent and positive recurrent ones, while the latter can be stable positive and unstable positive. Distinctions between the last two classes (for details see [1]) justify their names. The aim of this note is to show by simple examples what happens if one deletes some edges of the graph, using a random mechanism. It turns out that some of the above classes are stable with respect to small random perturbations, while the others are not.

## 2 Basic definitions, description of the model

Consider a directed graph *G* = (*V*, *E*), where *V* = *V* (*G*) is the set of its vertices and *E* = *E*(*G*) ⊂ *V* × *V* the set of edges. *A path of length n* in *G* is a sequence *γ* = *e*_{1},...,*e _{n}* of edges such that the beginning of the edge

*e*

_{i+1}coincides with the end of the edge

*e*, 1 ≤

_{i}*i*<

*n*. Every function 𝒲 :

*E*→ (0, ∞) is called a weight function, and the pair (

*G*, 𝒲) is said to be a

*loaded graph*. The weight 𝒲 (

*γ*) of a path

*γ*is defined as the product of the weights over the edges forming

*γ*. The graph

*G*is assumed to be connected: for every pair of vertices

*u*,

*v*∈

*V*(

*G*) there exists a path from

*u*to

*v*.

Recall a classification of the loaded graphs (see [1], [2] for details). For every vertex *v* ∈ *V* (*G*), let Γ_{v,v} denote the set of all *v*-cycles, i.e., the paths *γ* from *v* to *v*, and Γ_{v, v |v} denote the set of those *γ* ∈ Γ_{v, v} in which *v* is only the first and last vertex. We refer to elements of Γ_{v, v|v} as *primitive v-cycles*.

*v*∈

*V*(

*G*), denote by

*q*=

_{n}*q*(

_{n}*G*,𝒲,

*v*) the sum of the weights over all primitive

*v*-cycles of length

*n*, and assume that

*q*< ∞ for all

_{n}*n*. Introduce the generating function

Let *R _{v}* =

*R*(

*G*, 𝒲,

*v*) be the radius of convergence of this power series; assume that

*R*> 0.

_{v}*G*, 𝒲) is said to be

*stable positive*,

*unstable positive*, and

*null recurrent*if there is a vertex

*v*∈

*V*(

*G*) such that

*positive recurrent*loaded graphs, while positive recurrent and null recurrent loaded graphs form the set of

*recurrent*ones. The remaining loaded graphs with

*R*> 0 are said to be

_{v}*transient*.

Since *G* is connected, the following is true: if *R _{v}* > 0 for some

*v*, then

*R*> 0 for all

_{v}*v*∈

*V*(

*G*), and if some of the conditions (2)–(4) holds for a given

*v*, then it does for all

*v*∈

*V*(

*G*).

In [1], §3 several characteristic properties of stable recurrent and unstable recurrent loaded graphs are collected and this terminology is justified.

Let us now describe the loaded graphs (*G*, 𝒲) that will be the subject of further studies in this paper. For *V* = *V* (*G*) we take ℕ, the set of positive integers, and for *E* = *E*(*G*) the union *E*_{1} ∪ *E*_{2} ∪ *E*_{3}, where *E*_{1} is the collection of all pairs (*i*, *i* + 1), *i* ∈ ℕ; *E*_{2} is the collection of the pairs (*i*, 1), *i* ∈ ℕ, and *E*_{3} is formed by some pairs of the form (*i*, *j*), where *i*, *j* ∈ ℕ, *i* < *j* − 1. We will refer to such a graph as *linear*. Suppose the following *bounded forward jump* (BFJ) condition is satisfied: there is *d* > 0 such that *j* − *i* ≤ *d* for all (*i*, *j*) ∈ *E*_{3}.

Notice that in [4], Example 5.6, a close class of loaded graphs was introduced in other terms and for another purpose.

Given an arbitrary linear graph *G*, one can chose, for each of the above classes of loaded graphs, a weight function 𝒲 in such a way that the loaded graphs (*G*, 𝒲) will fall into this class.

For a fixed loaded graph (*G*, 𝒲) we consider a random loaded graph (*G*, 𝒲, *ξ*) that differs from (*G*, 𝒲) in that the weight of every edge of the form (*i*, 1) ∈ *E*_{2} is multiplied by a random variable *ξ _{i}* with values in {0,1}, i.e., remains intact or disappears under the action of some stochastic mechanism. Let us assume that this mechanism is described by a stationary random sequence

*ξ*:= (

*ξ*,

_{i}*i*∈ ℤ) with decay of long-range dependence.

## 3 Classes of random linear loaded graphs

We begin with the following notion from ergodic theory.

*T*(automorphism) of a probability space (Ω, ℱ,

**P**) is said to be

*lightly mixing*(see [5]) if

*A*,

*B*∈ ℱ with

*P*(

*A*) > 0,

*P*(

*B*) > 0. The next lemma is due to Blum, Christiansen and Quiring [6], who used another terminology.

*If T is a lightly mixing automorphism of a probability space*(Ω, ℱ,

**P**),

*then for every infinite increasing sequence of positive integers n*,

_{k}*k*= 0, 1,...,

*and every set F*∈ ℱ

*with*

**P**(

**) > 0,**

*F**we have*

(see [6]) The automorphisms for which (5) holds are sometimes called sequence mixing or sweeping out. This property is known to be equivalent to the lightly mixing one.

If the shift in the sample space of a stationary process is lightly mixing, we say that the process itself is lightly mixing.

*If a stationary random process ξ* = {*ξ _{i}*,

*i*∈ ℤ

_{+}}

*with values*0

*and*1

*is lightly mixing, then for every increasing infinite sequence of positive integers r*∈ ℕ,

_{i}and every r*there exists with probability*1

*an i such that r*>

_{i}*r and ξ*= 1.

_{ri}Let Ω be the sample space of the process *ξ* (Ω consists of the double-sided infinite sequences of zeroes and ones), *T* the one step left shift in Ω, ℱ the cylinder *σ*-algebra in Ω and **P** the probability measure on (Ω, ℱ) induced by the process *ξ*. Let us assume the process *ξ* is defined on (Ω, ℱ **P**) as follows: if *ω* = (*ω _{i}*,

*i*∈ ℤ) ∈ Ω, then

*ξ*(

_{n}*ω*) =

*ω*,

_{n}*n*∈ ℤ. Put

*F*:= {

*ω*∈ Ω :

*ω*

_{0}= 1}. Suppose that the automorphism

*T*of the space (Ω, ℱ,

**P**) is lightly mixing. Then the desired statement follows from Lemma 1 applied to

*F*and the numbers

*n*:=

_{k}*r*

_{i0+k}, where

*i*

_{0}= min{

*i*:

*r*>

_{i}*r*}.

We now come back to the random linear loaded graph described in Section 2. For such a graph we will write
*q _{n}*(

*ξ*) instead of

**Φ**

_{v, v|v}(

*G*, 𝒲),

*R*and

_{v}*q*, respectively. Below it is assumed that

_{n}*v*= 1.

*Let 1) a linear graph G satisfies the BFJ condition, 2) a weight function 𝒲 is bounded, and 3) a stationary random process ξ with values* 0, 1 *induces a lightly mixing shift. Then for the random linear loaded graph* (*G*, 𝒲, *ξ*) *we have*

*G*, 𝒲). By the Cauchy-Hadamard formula there is a sequence of positive integers

*n*such that

_{k}*n*} instead of {

_{k}*n*} itself. By Corollary 2 there exists with probability one an infinite subsequence {

_{k}*k*. Hence, by (6), with probability one

But it is clear that the converse inequality holds, since *q _{n}*(

*ξ*) ≤

*q*for all

_{n}*n*. Hence

*ξ*can be treated as a small random perturbation of the process that identically equals one. Consider a sequence of stationary random processes

*k*= 1, 2,..., introduce the probability measure

**P**

^{(k)}, corresponding to

*ξ*

^{(k)}. Assume that there exists an increasing sequence of positive integers

*r*such that

_{k}This will be referred to as the *dominating 1* (D 1) condition. If
*k*, are independent random variables such that
*p _{k}* and

*p*, then the D 1 condition is satisfied when lim

_{k}_{k → ∞}

*p*=1. It is easy to define a sequence of stationary Markov chains for which it is also obeyed. But there are Markov chains, even mixing ones, without this property. On the other hand, it is obeyed by some non-Markovian stationary processes.

_{k}*Let* (*G*, 𝒲) *be a stable positive loaded graph, where G is a linear graph defined in § 2*, {*ξ*^{(n)}} *a sequence of lightly mixing stationary random processes with values 0 and 1 satisfying the D 1 condition, and* {(*G*, 𝒲, *ξ*^{(n)}‘)} *the corresponding sequence of random loaded graphs. Then the random loaded graph* (*G*, 𝒲, *ξ*^{(n)}) *is stable positive with probability one as n is large enough*.

*G*, 𝒲) is stable positive, we have

*n*

_{0}∈ ℕ such that

*k*with

*r*≥

_{k}*n*

_{0}(see (7)). Then

*q*(

_{n}*ξ*

^{(k)}) =

*q*for all

_{n}*n*≥

*n*

_{0}, and hence

*G*, 𝒲,

*ξ*

^{(k)}) is stable positive with probability one.

It is easy to see that if (*G*, 𝒲) is unstable positive, null recurrent, or transient, while *ξ* = (*ξ _{i}*,

*i*∈ ℤ) is a lightly mixing stationary random process with values 0 and 1 such that

**P**(

*ξ*= 0) > 0, then the random loaded graph (

_{i}*G*, 𝒲,

*ξ*) is transient with probability one (in view of Theorem 3). Thus among the linear loaded graphs we defined in Section 2 only the stable positive and transient classes are stable with respect to the small random perturbations of the above type.

The author is indebted to V.I. Oseledets and V.V. Ryzhikov for critical information relating to Lemma 1 and thanks the referee for useful suggetions.

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