GEODESICS AND RECURRENCE OF RANDOM WALKS IN DISORDERED SYSTEMS

with stationary conductances; Geodesics in percolation model; Reversible Abstract In a (cid:12)rst-passage percolation model on the square lattice Z 2 , if the passage times are independent then the number of geodesics is either 0 or + 1 . If the passage times are stationary, ergodic and have a (cid:12)nite moment of order (cid:11) > 1 = 2 , then the number of geodesics is either 0 or + 1 . We construct a model with stationary passage times such that E [ t ( e ) (cid:11) ] < 1 , for every 0 < (cid:11) < 1 = 2 and with a unique geodesic. The recurrence/transience properties of reversible random walks in a random environment with stationary conductances ( a ( e ); e is an edge of Z 2 ) are considered.


Introduction.
In a first-passage percolation model, a sequence of non-negative stationary random variables (t(e); e is an edge of Z 2 ) is given. A finite path γ, in the square lattice Z 2 , from x to y is a finite sequence of neighboring vertices of Z 2 x = x 0 , x 1 , . . . , x n = y and the passage time of the path is defined by where {e 1 , e 2 } is the canonical basis of R 2 (see for instance [2, propositions 6.9 and 6.12] or [11, section 1.4]).
With this representation, we have that (S x ; x ∈ Z 2 ) is ergodic if and only if (t(e); e is an edge of Z 2 ) is ergodic [2, proposition 6.18].
For two vertices x, y ∈ Z 2 , y ∼ x means that x and y are neighbors in the square lattice. For A a finite subset of Z 2 , ∂A := {x / ∈ A ; there exists y ∼ x , y ∈ A} and A = A ∪ ∂A.
Proof of theorem 1. Suppose that N , the number of geodesics, is 0 < N < +∞.
By stationarity, η := P (x belongs to a geodesic) does not depend on the particular vertex x and η > 0 since N > 0.
Since the random variables t(e) are positive, there is δ > 0 such that P (t < δ) < η/9N for = 1, 2, where t 1 and t 2 are the random variables of the representation given above.
Let Q n be the set of vertices of Z 2 in the square ] − n, n] 2 and Q n be the set of edges with at least one vertex in Q n .
By the multidimensional pointwise ergodic theorem [11, p.205], for n large enough, the number of vertices x in Q n such that t(x, x + e 1 ) < δ or t(x, x + e 2 ) < δ is less than η(2n) 2 /9N . Thus, the number of edges e in Q n such that t(e) < δ is less than 2η(2n) 2 /9N . Furthermore, for n large enough, one of the geodesics, denoted by γ in the sequel, must contain at least η(2n) 2 /2N vertices of Q n . Thus, it must contain at least η(2n) 2 /4N edges of Q n . Let v and w be respectively the first and the last vertex of γ that are in ∂Q n−1 . Since the passage time between v and w along γ is less than the passage time along a portion of ∂Q n−1 , for n large enough and for all 0 < α ≤ 1 and ε > 0, we have and therefore lim inf • Since there is an equivalence between the existence of nonconstant ground states in a disordered Ising ferromagnetic model and the existence of geodesics in the corresponding first-passage percolation model (see [12], [14,Chapter 1] or [16]), it follows that if the interactions are stationary, ergodic and with finite moment of order α > 1/2 then P -a.s., there are either two ground states or an infinity of ground states. In dimension d > 2, the notion of geodesic can be replaced by the interface of a non constant ground state for an Ising ferromagnetic model in a random environment. In this case, with a similar argument, one proves that if the interactions are stationary, ergodic and with finite moment of order α > (d − 1)/d then P -a.s., there are either two or an infinity of ground states. In Licea and Newman [12], the non existence of (x,ŷ)-bigeodesics is proved for distributions without atoms but there are no moment conditions. Related arguments can be found in [3].
where C is the number of vertices in C. To prove theorem 2, we only need to know that θ(p) > 0 for some p < 1.
First we prove the theorem using the following two lemmas.
This lemma is used to prove the existence of a sequence of circuits with linearly growing passage times. Then let v n and w n be respectively the first and the last vertex of γ that are on π n . Since the passage time between v n and w n along γ is less than the passage time along a portion of π n , for infinitely many n we have that which leads to a contradiction. • Proof of lemma 1. As in [7, section 1.4], if C < ∞, consider the circuit π that surrounds C in IL 2 , the dual lattice with vertices in Z 2 + (1/2, 1/2). An edge of IL 2 is said to be closed if it intersects a closed edge of Z 2 . Suppose that π consists of n edges. Then n ≥ 4 and at least n/4 of these edges must be closed. To see this, for j ≥ 0, let −1 ≤ i 1 < ... < i k < ... < i v(j) be such that the horizontal edge from (i k , j) to (i k + 1, j) intersects a vertical edge of π. If k ≤ v(j) is even then this edge must be closed while if k is odd it might not be. Since π is a circuit, v(j) is even. Therefore v(j) vertical edges of π intersect the horizontal line (·, j) and at least v(j)/2 of them are closed. Now to count how many horizontal edges of π are closed, for i ≥ 0, let h(i) be the number of horizontal edges of π which intersect the vertical line (i, ·). Since π is a circuit, h(i) is even.
Therefore, if ρ(n) is the number of circuits in IL 2 which have length n and which contain 0,

and consider
A n (1), the event that there is an open path, consisting of less than 12 · 7 n−1 edges, inside the right-angle triangle A n (2), the event that there is an open path, consisting of less than 16 · 7 n−1 edges, inside the right-angle triangle (v n,1 ; v n, 4 ; v n,5 ) from v n,1 to the opposite side [v n,4 ; v n,5 ], A n (3), the event that there is an open path, consisting of less than 12 · 7 n−1 edges, inside the right-angle triangle (v n, 6 ; v n, 7 ; v n,4 ) from v n, 6 to the opposite side [v n,7 ; v n,4 ], A n (4), the event that there is an open path, consisting of less than 16 · 7 n−1 edges, inside the right-angle triangle (v n, 6 ; v n, 3 ; v n, 8 ) from v n, 6 to the opposite side [v n, 3 ; By symmetries, for each n ≥ 1 and 0 < p < 1, we have that P p (A n (1)) = P p (A n (3)) ≥ θ(p) and P p (A n (2)) = P p (A n (4)) ≥ θ(p) .
In this section, we define a stationary and ergodic sequence of positive random variables (t(e); e is an edge of Z 2 ) with finite moments of order α for any 0 < α < 1/2, such that, almost surely, there exists exactly one geodesic. For two integers a, b such that a ≤ b, denote ], then the translates, Γ n + 2 n+1 e 1 , Γ n + 2 n+1 e 2 and Γ n + 2 n+1 (e 1 + e 2 ) are the restrictions of Γ to the respective translated squares.

Representation of Γ 1 and Γ 2
For n ≥ 1, Γ n+1 is obtained from Γ n by connecting the four translated paths with paths similar to the star-studded in the representation of Γ 2 given above. These paths correspond to γ in the notations below. Some randomness will be introduced in this process. To do so we will use a sequence ( X n ; n ≥ 1) of independent random variables defined on a probability space (Ω, F , P ) such that X 1 has uniform distribution on the vectors {(i, j) ; 0 ≤ i, j ≤ 3} and, for n > 1, X n has uniform distribution on {(0, 0), (2 n , 0), (0, 2 n ), (2 n , 2 n )}.
The sequence of random passage times will be defined so that almost surely, there exists only one geodesic γ = γ(ω) and that it looks like Γ.
(2.1) Definition of (t(e) ; e is an edge of Z 2 ).
The passage times are defined on (Ω, F , P ) using the sequence ( X n ; n ≥ 1). Most of the edges not in γ(ω) constitute barriers that force the way along the geodesic. These edges correspond to with the notations below.
At stage n ≥ 1, for = 1, 2, set These edges will be part of the geodesic.
Then, for = 1, 2, set The values α > 0 such that E[t(e) α ] < ∞ depend on how large these passage times have to be.
The passage time of any other edge is set equal to 0. Actually, any other constant will do.
The remainder of this section is to show that γ(ω) is the unique geodesic.
(2.2) Almost surely, γ(ω) is a geodesic : For each n ≥ 1, the set of edges with passage time defined up to stage n and equal to 1, is a union of vertex-disjoint finite paths in Z 2 , each one being a translation of a fixed path Γ n contained in [2, 2 n+1 ] × [2, 2 n+1 ]. Therefore γ(ω) is a path in Z 2 . Let x and y be two vertices of γ(ω) and denote the finite path between x and y along γ(ω) by γ(x, y). Our goal is to show that for any path π(x, y) from x to y, T (γ(x, y)) ≤ T (π(x, y)). Let N ≥ 1 be the greatest integer such that at stage N , the passage time of at least one edge of γ(x, y) is not yet defined. First, since all passage times of γ(x, y) are defined at stage N + 1, Secondly, since the passage time of some edge of γ(x, y) is not defined at stage N , x and y cannot belong to Z 2 ∩ [2, 2 N ] × [2, 2 N ] + m · ( X 1 , . . . , X N ) for the same index m. Therefore, any finite path π(x, y) from x to y contains an edge whose passage time is not defined at stage N . But all passage times defined at a later stage are either 1 or are ≥ 9 · 2 2(N +1) . In particular, if π(x, y) is edge disjoint from γ(x, y), then T (π(x, y)) ≥ 9 · 2 2N +2 > T (γ(x, y)).

Recurrence of reversible random walks on Z d .
As in a first-passage percolation model, we are given a sequence of positive random variables (a(e) : e is an edge of Z d ) which are now interpreted as the electrical conductance of the edges.
a(e) −1 is called the resistance of the edge. Almost surely, there is an associated random walk, (ξ k ; k ≥ 0), on Z d whose transition probabilities are given by where a(x) := y∼x a(x, y).
(ξ k ; k ≥ 0) is a reversible random walk with a(x) as an invariant measure.
Recall that in a finite graph with conductances given by a sequence a(e), e an edge of the graph, the effective resistance, R a (x, V ), between a vertex x and a set of vertices V not containing x is the intensity of the electric current needed to maintain a unit potential difference between x and V (cf. [5] or [15,Chapters 8 and 9] for example). It has the following probabilistic interpretation : Therefore, a.s., the random walk on Z d is recurrent if and only if R a (0, ∂Q n ) → +∞ as n → ∞ a.s.
where Q n is the set of vertices of Z d in the cube ] − n, n] d .
It follows from the Rayleigh's monotonicity principle (see [15,Theorem 8.5]), that if the conductances are bounded, but not necessarily stationary, then the associated reversible random walk on Z 2 is recurrent while if the conductances are bounded below away from 0 then the associated random walk on Z d , d ≥ 3, is transient (for instance [5], [6] or [15]).
For the one-dimensional walk, it is simple to see, using Poincaré recurrence theorem for instance, that if the conductances (a(e); e is an edge of Z) form a stationary sequence of positive random variables then, a.s., the associated reversible random walk is recurrent.
The next four remarks gather together the recurrence/transience properties of the reversible random walks on Z d , d ≥ 2 with random conductances, independent or not. (a(e); e an edge of Z 2 ) be a sequence of non-negative conductances.

Remark 1 Let
where the sum is over the edges of Q n , that is the edges with at least one vertex in Q n . Using this variational principle, the proof of Peres [1, lemma 4.3] shows that if the conduc- where for two vertices x ∼ y, J(x, y) denotes the flow in the edge from x to y (see [15, example 9.4 and exercise 11.4] or [13]).
Thus, we obtain in this case • Finally, the independent case follows immediately from the transience of the reversible random walk on the infinite open cluster in the supercritical case proved in [8]. It suffices to take ε > 0 small enough so that P (a(e) < ε) < p c (Z d ), the critical probability of bond percolation on Z d . While example (3.3) shows that the integrability condition in remark 1 for the recurrence of the reversible random walk with stationary conductances is tight, example (3.4) below shows that the appropriate moment condition in remark 3 is in between 1/(d − 1) and d/ (2(d − 1)). The conductances are defined so that from every vertex of Z 2 there is a path to infinity whose edges have increasing conductances. One expects that if they increase fast enough, then the walk will be transient. The example shows that this can happen for conductances with finite moments of order α < 1. = {(2 n−2 , 2 n−1 ), (3 · 2 n−2 , 2 n−1 )} As in section 2, we use a sequence ( X n ; n ≥ 2) of independent random variables defined on a probability space (Ω, F , P ) such that X n has uniform distribution on

Further remarks
The random conductances, (a(e); e is an edge of Z 2 ), are defined on Ω as follows : for n ≥ 2 and = 1, 2, set and for all the other edges of Z 2 , set a(e) = 1.
Almost surely, in any square, the proportion of edges with conductance equal to n 2 2 n is less than 2 n · 2 −2n , for all n ≥ 2 and = 1, 2, E[a(e) α ] < ∞ whether e is a horizontal or a vertical edge.
For N ≥ 1, let R N be the effective resistance between the origin and ∂Q N along the path γ, that is, by setting the resistance of all edges of Z 2 equals to +∞ except those of γ whose values are left unchanged. The walk is transient since by the monotonicity principle, Two other sequences of positive real numbers are used to define the conductances of the edges: (a n ; n ≥ 0) and (r n ; n ≥ 0) such that, a 0 = 1, r 0 = 1 and, as n → ∞, a n decreases to 0 and r n increases to ∞. Then the conductance of the edge between two neighboring vertices y and z of Z d is a(y, z) = inf{ a n ; n = Z(x) for some x ∈ Z d such that where | · | is the euclidean norm in R d , or, a(y, z) = 1 if the infimum is over an empty set.
Consequently, if x is a vertex where Z(x) = n, the conductance of every edge near the boundary of the ball B n (x), of radius r n and centered at x, is at most a n . The idea is to choose the sequences (p n ), (a n ) and (r n ) such that, almost surely, the origin, and therefore every vertex, belongs to an increasing sequence of balls and the effective resistances between 0 and the boundary of these balls are unbounded. A first condition must be given to insure that the conductivities are well defined and positive.
To do so, let Since, for some constant And therefore, by Borel-Cantelli lemma, the infimum in (6) is over a finite set of positive numbers. The next step is to prove that, almost surely, the origin belongs to an increasing sequence of balls such that the conductances of the edges near the boundary decreases to 0. Introduce a new sequence (ρ n ; n ≥ 0) such that ρ n ↑ ∞ and consider the events A n = {there is a vertex x ; ρ n < |x| < ρ n+1 , Z(x) = n}.
And we can use the second Borel-Cantelli lemma to conclude We also need the following property : if (x n ; n ≥ 1) is a sequence of vertices such that ρ n < |x n | < ρ n+1 for all n ≥ 1, then 0 ∈ B n (x n ) ⊂ B n+1 (x n+1 ), for all n sufficiently large. This is the case if the next condition, a third one, is satisfied for all n sufficiently large : The next step is to verify that for almost all environments, the reversible random walk is recurrent. If conditions (7) to (9) are satisfied, then for almost all environments, there are sequences (x n k ; k ≥ 1) and (r n k ; k ≥ 1) such that for all y ∈ ∂B n k (x n k ) and for all z ∼ y, a(y, z) ≤ a n k . Consider the network Z d with the following conductances: For 1 ≤ i ≤ d,ã(y, y + e i , ω) = a n k , if y ∈ ∂B n k (x n k ) for some k ≥ 1; ∞ otherwise.
Then by Rayleigh's monotonicity principle, R a (0, ∂B n k ) ≥ Rã(0, ∂B n k ) where the latter is the effective resistance calculated with the conductancesã. The edges around each sphere are in parallel and the spheres are in series, therefore, there is a constant c 3 > 0 such that which diverges if the following condition is satisfied lim sup n r d−1 n a n < ∞.
Then by Nash-Williams criterion, the random walk is recurrent.