Random walks in Dirichlet environment: an overview

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by  Christophe Sabot

RANDOM WALKS IN DIRICHLET ENVIRONMENT:
AN OVERVIEW
CHRISTOPHE SABOT AND LAURENT TOURNIER
Abstract: Random Walks in Dirichlet Environment (RWDE) correspond to Random
Walks in Random Environment (RWRE) on Zd where the transition probabilities are
i.i.d. at each site with a Dirichlet distribution. Hence, the model is parametrized by
a family of positive weights (α ) , one for each direction of Zd. In this case,
i i=1,...,2d
the annealed law is that of a reinforced random walk, with linear reinforcement on
6
1 directed edges. RWDE have a remarkable property of statistical invariance by time
0
reversal from which can be inferred several properties that are still inaccessible for
2
general environments, such as the equivalence of static and dynamic points of view
n
a and a description of the directionally transient and ballistic regimes. In this paper
J we give a state of the art on this model and several sketches of proofs presenting the

9
core of the arguments. We also present new computation of the large deviation rate
2
function for one dimensional RWDE.

]
R
1. Introduction
P
.
h
Multidimensional Random Walks in Random Environment (RWRE) have been the
t
a
object of intense investigation in the last fifteen years. Important progress has been
m
made but some central questions remain open. The ballistic case, i.e. the case where
[
an a priori ballistic condition (as the (T) condition of Sznitman, [43]) is assumed, is
γ
1
by far the best understood (cf e.g. [25, 45, 42, 43, 34, 5]). The non ballistic case is
v
9 more difficult and researches have concentrated on the perturbative regime, where the
1 environment is assumed to be a small perturbation of the simple random walk (see in
2
particular [44, 8]), or on two special cases, the balanced case ([29]) and the Dirichlet
8
0 case. The object of this paper is to give an overview of what is known in this last case:
.
1 Random Walks in Dirichlet Environment (RWDE) correspond to a special instance of
0 i.i.d. random environment where the environment at each site is chosen according to a
6
Dirichlet random variable. Note that compared to the balanced case, where the drift of
1
: the environment at each site is almost surely null, there is no almost sure restriction on
v
i the possible environments, more precisely the support of the law on the environment
X
is the whole set of environments. The main property that justifies the interest in this
r
a special case is a property of statistical invariance by time reversal (cf. Section 3) from
which several results can be inferred and which is the main focus of this paper.
For simplicity, in this paper we restrict ourselves to the case of RWRE on Zd to
nearest neighbors, (except for Sections 2 and 3), even if most of the results on RWDE
could be extended to more general settings. Denote (e ,...,e ) the canonical basis of
1 d
Zd and set e = e so that (e ,...,e ) is the set of unit vectors of Zd. Recall that
i+d i 1 2d
−
2010 Mathematics Subject Classification. primary 60K37, 60K35.
Key words and phrases. Random walk in random environment, Dirichlet distribution, Reinforced
random walks, invariant measure viewed from the particle.
This work was supported by the ANR project MEMEMO2.
1
2 C. SABOT AND L. TOURNIER
in this case the set of environments is the set
2d
(cid:110) (cid:88) (cid:111)
Ω = (ω(x,x+e )) : ω(x,x+e ) (0,1), ω(x,x+e ) = 1 .
i x∈Zd,i=1,...,2d i i
∈
i=1
The classical model of RWRE is the model where the transition probabilities at each
site (ω(x,x + )) are independent with a same law µ, which is a distribution on the
·
simplex
2d
(cid:110) (cid:88) (cid:111)
(1.1) ∆ = (ω ) (0,1)2d : ω = 1 .
2d i i=1,...,2d i
∈
i=1
We denote by P = µ⊗Zd, the law obtained on Ω. Traditionally, the quenched law,
which is the law of the Markov chain (X ) in a fixed environment ω, is denoted by
n n≥0
P , i.e. we have P (X = x ) = 1 and
x ,ω x ,ω 0 0
0 0
P (X = x+e X = x, (X ) ) = ω(x,x+e ).
x ,ω n+1 i n k k≤n i
0 |
Theannealedlawisthelawobtainedafterexpectationwithrespecttotheenvironment
(cid:90)
P ( ) = P ( ) P(dω).
x x ,ω
0 · 0 ·
To define Dirichlet environment, we fix some positive parameters (α ) , one for
i i=1,...,2d
each direction (e ) of Zd. The RWDE with parameters (α ) is the RWRE
i i=1,...,2d i i=1,...,2d
with the following specific choice for µ. We choose µ = ((α ) ), which is the
i i=1,...,2d
D
Dirichlet law with parameters (α ): it corresponds to the law on the simplex (1.1) with
i
distribution
(cid:32) (cid:33) (cid:32) (cid:33)
Γ((cid:80)2d α ) (cid:89)2d (cid:89)
i=1 i ωαi−1 1 (ω) dω ,
(cid:81)2d Γ(α ) i ∆2d i
i=1 i i=1 i(cid:54)=i
0
where Γ(α) = (cid:82)∞tα−1e−tdt is the usual Gamma function, and i is an irrelevant choice
0 (cid:81) 0 (cid:80)
of index in 1,...,2d (i.e., we integrate on dω with ω = 1 ω ). We
{ } i(cid:54)=i0 i i0 − i(cid:54)=i0 i
denote by P(α) the associated law on Ω and by P(α) the annealed law of RWDE.
x
0
TheDirichletlawisaclassicallawthatplaysanimportantroleinBayesianstatistics.
It is also intimately related to Pólya urns (cf. Section 2.1), and this relation implies
that the annealed law of RWDE is that of a directed edge reinforced random walk
(cf. Section 2.3). The important property that justifies that RWDE is an interesting
special case of RWRE, is the aforementioned property of statistical invariance by time
reversal. It asserts that on finite graphs, under a condition of zero divergence of the
weights, the time reversed environment is again a Dirichlet environment, in particular
time reversed transition probabilities are independent at each site. In this paper we
review what is known on Dirichlet environments (with a few new results) and give
sketches of proofs or new proofs of some of the results involving the time reversal
property. This property has been principally applied in the following directions:
Description of directionally transient/recurrent regimes in any dimension (im-
•
plying in particular a positive answer to directional 0-1 law).
Proof of transience in dimension d 3 for all parameters.
• ≥
Characterizationoftheparametersforwhichthereisequivalencebetweenstatic
•
and dynamic points of view in dimension d 3.
≥
RANDOM WALKS IN DIRICHLET ENVIRONMENT 3
Characterization of ballistic regimes in dimension d 3, which gives an answer
• ≥
in this context to the question of the equivalence between directional transience
and ballisticity.
Let us also mention, on a different but somehow related model of random walk in
space time Beta random environment, the recent work of Barraquand and Corwin, [3],
where Tracy-Widom distribution appears in the second order correction in the large
deviation principle. This model is closely related to the exactly solvable model of
log-gamma polymers introduced by Seppäläinen, [39].
Let us describe the organization of the paper. In Section 2, we give definition and
basic properties of Dirichlet laws, Pólya urns and RWDE. In Section 3, we state the
important property of statistical invariance by time reversal. We give a proof, slightly
shorter than that of [38]. In Section 4, we explain the role of traps of finite size and
define the important parameter κ. In Section 5 we state the main results which are
consequences of the time reversal property. In Section 6, we consider the question
of quenched central limit theorems in the case of ballistic RWDE. In Section 7, we
give sketches of proofs and some extensions of the results involving the time reversal
property. We do not optimize on the parameters, which makes the proofs more trans-
parent than the originals. Finally, in Section 8, we describe the case of one-dimensional
RWDE, for which special calculations can be made. In particular, we give an explicit
new computation of the rate function of one-dimensional RWDE.
2. Random Walk in Dirichlet environment and directed edge
reinforced random walk
2.1. Dirichlet laws and Pólya urns. Dirichlet distributions classically arise as the
limit distribution of colors in Pólya urns. Let us recall this result.
An urn contains balls of r different colors. Initially, α balls of color i are present,
i
for i = 1,...,r. After each draw, the ball is put back in the urn together with one
additional ball of the same color. In other words, if (X ) denotes the sequence
n n≥1
of colors drawn from the urn, and = σ(X ,...,X ), then we have for all n, for
n 1 n
F
i = 1,...,r,
N (n)
i
(2.1) P(X = i ) = ,
n+1 n
|F α + +α +n
1 r
···
where N (n) = α +# 1 k n : X = i is the number of balls of color i in the urn
i i k
{ ≤ ≤ }
after n draws. Such an urn is usually called reinforced as the chosen color becomes
more likely in the future draws. Let us underline that the formal definition of the
model doesn’t require the α ’s to be integers but merely positive real numbers.
i
One can check that the proportion of balls of color i after n draws, M (n) =
i
Ni(n) , is a bounded martingale and therefore converges almost surely to a random
α1+···+αr+n
variable U . Note that the vector (U ,...,U ) takes values in the simplex
i 1 r
(cid:8) (cid:9)
∆ = (u ,...,u ) (0,1)r : u + +u = 1 .
r 1 r 1 r
∈ ···
Before stating the main result about Pólya urns, let us give a central definition:
Definition 1. Given positive real numbers α ,...,α , the Dirichlet distribution with
1 r
parameters α ,...,α , is the distribution on ∆ given by
1 r r
Γ(α + +α )
(α ,...,α ) = 1 ··· r uα1−1 uαr−1dλ (u ,...,u ),
D 1 r Γ(α ) Γ(α ) 1 ··· r ∆r 1 r
1 r
···
4 C. SABOT AND L. TOURNIER
where dλ is the Lebesgue measure on ∆ , that is to say 1 (u ,...,u )(cid:81) du
∆r r ∆r 1 r i(cid:54)=i0 i
for an arbitrary choice of i 1,...,r .
0
∈ { }
A classical proof of the above normalization would consist in writing
(cid:90)
Γ(α ) Γ(α ) = xα1−1 xαr−1e−(x1+···+xr)dx dx ,
1 ··· r 1 ··· r 1··· r
(0,+∞)r
and letting u = x /(x + + x ) for i = 1,...,r 1, and v = x + + x .
i i 1 r 1 r
··· − ···
This is tightly related to Property 1 below. As a consequence of this normalization,
we immediately deduce joints moments of marginals of the Dirichlet distribution: if
(V ,...,V ) (α ,...,α ), then
1 r 1 r
∼ D
Γ(α +ξ ) Γ(α +ξ ) Γ(α + +α )
(2.2) E(cid:2)(V )ξ1 (V )ξr(cid:3) = 1 1 ··· r r 1 ··· r ,
1 r
··· Γ(α +ξ + +α +ξ ) Γ(α ) Γ(α )
1 1 r r 1 r
··· ···
for all real numbers ξ ,...,ξ R such that α + ξ > 0 for all i. If ξ + α 0 for
1 r i i i i
∈ ≤
somei,thentheexpectationisinfinite. Inthecasewhenξ ’sareintegers,thefunctional
i
equationofthegammafunctionreducesthepreviousformulatoanelementaryproduct
that has an interpretation in terms of Pólya urn and leads to the next lemma.
Lemma 1. The vector U = (U ,...,U ) of asymptotic proportions of colors in the
1 r
Pólya urn follows the Dirichlet distribution (α ,...,α ). Furthermore, conditional
1 r
D
on U, the sequence (X ) is independent and identically distributed with, for n 1
n n≥1
≥
and i = 1,...,r,
P(X = i U ,...,U ) = U .
n 1 r i
|
Proof. It is a simple matter to check that, for any x ,...,x 1,...,r , if we let
1 n
∈ { }
n = # 1 k n : x = i for i = 1,...,r,
i k
{ ≤ ≤ }
(cid:81)r α (α +1) (α +n 1)
P(X = x ,...,X = x ) = i=1 i i ··· i i −
1 1 n n
(α + +α ) (α + +α +n 1)
1 r 1 r
··· ··· ··· −
Γ(α +n ) Γ(α +n ) Γ(α + +α )
1 1 r r 1 r
= ···
Γ(α ) ··· Γ(α ) Γ(α + +α +n + +n )
1 r 1 r 1 r
··· ···
= E[(V )n1 (V )nr],
1 r
···
where V = (V ,...,V ) (α ,...,α ). Thus, (X ) has same law as i.i.d. variables
1 r 1 r n n
∼ D
(Y ) withcommonlawV δ + +V δ givenV, whereV (α ,...,α ). Bythelaw
n n 1 1 r r 1 r
··· ∼ D
oflargenumbersfortheY ’sgivenV,V isthevectorofalmostsurelimitingproportions
n
of colors in the sequence (Y ) , hence ((X ) ,U) has same law as ((Y ) ,V), which
n n n n n n
concludes. Notethatthisactuallyre-provesthealmostsureconvergenceofproportions
of colors toward U without a martingale convergence theorem. (cid:3)
This lemma can also be seen as an instance of de Finetti’s theorem (see for in-
stance [17, p.268]), since it is easily noticed that the sequence (X ) is exchangeable.
n n
In the usual two-color case, we have (U ,1 U ) (α ,α ), which reduces to
1 1 1 2
− ∼ D
1
U Beta(α ,α ) = uα1−1(1 u)α2−11 (u)du,
1 1 2 (0,1)
∼ B(α ,α ) −
1 2
where B(α,β) = Γ(α)Γ(β) is the Beta function.
Γ(α+β)
For later convenience, we will also consider more general index sets:
RANDOM WALKS IN DIRICHLET ENVIRONMENT 5
Definition 2. For a finite set I and (α ) (0,+ )I, the Dirichlet distribution
i i∈I
∈ ∞
((α ) ) on
i i∈I
D (cid:110) (cid:88) (cid:111)
∆ = (u ) (0,1)r : u = 1
I i i∈I i
∈
i∈I
is given by
(cid:80) (cid:18) (cid:19)
((α ) ) = Γ( i∈I αi) (cid:89)uαi−1 1 ((u ) )(cid:89) du ,
D i i∈I (cid:81) Γ(α ) i ∆I i i∈I i
i∈I i i∈I i(cid:54)=i
0
for an irrelevant choice of i .
0
NB. From Lemma 1 for instance, or continuity in distribution, it is natural to allow
someparametersofthedistribution,butnotall,tobezero,bysettingthesecoordinates
equal to 0 a.s. and viewing (α) as a distribution on ∆ .
{i:α (cid:54)=0}
D i
2.2. Properties of Dirichlet distributions. Let I be finite, and (α ) (0,+ )I.
i i∈I
∈ ∞
Dirichletdistributioncouldequivalentlyhavebeendefinedasthelawofanormalized
Gamma vector. By routine computation, one can indeed check that:
Property 1. Let (W ) be independent random variables such that, for i I,
i i∈I
∈
1
W Γ(α ,1) = wαi−1e−w1 (w)dw.
i i (0,∞)
∼ Γ(α )
i
We have
1
(cid:0) (cid:1)
(U ) := W ((α ) ),
i i∈I (cid:80) W i i∈I ∼ D i i∈I
i∈I i
(cid:80)
and (U ) is independent of W .
i i∈I i∈I i
Recall that, if X Γ(α,1) and Y Γ(β,1) are independent, then X + Y
∼ ∼ ∼
Γ(α+β,1). Together with the previous property, this gives:
Property 2. Assume (U ) has Dirichlet distribution ((α ) ).
i i∈I i i∈I
D
(cid:0)(cid:80) (cid:1)
(Agglomeration): LetI ,...,I beapartitionofI. Therandomvariable U
1 n i∈Ik i k∈{1,...,n}
(cid:80)
on ∆ follows the Dirichlet distribution (( α ) ).
n D i∈Ik i 1≤k≤n (cid:16) (cid:17)
(Restriction): LetJ beanonemptysubsetofI. Therandomvariable Ui
(cid:80) U
on∆ followstheDirichletdistribution ((α ) )andisindependentofj(cid:80)∈J j Ui∈J.
J D i i∈J j∈J j
In particular, from the agglomeration property, the marginal U of a Dirichlet vector
i
(cid:80)
(U ) ((α ) ) follows the law Beta(α , α ).
i i∈I ∼ D i i∈I i j(cid:54)=i j
One may notice that Property 2 can also be elementarily deduced from Lemma 1
by identifying together or disregarding some colors.
Property 1 also enables to derive the following degenerate large weights limit, which
means that the effect of reinforcement vanishes as the initial number of balls goes to
infinity:
(2.3) ((λ α ) ) δ .
D · i i∈I λ−→→∞ (cid:80)i1αi(αi)i
In the opposite direction, with small weights, the distribution concentrates on the
extreme points of the simplex, which means that the first draw from the urn becomes
6 C. SABOT AND L. TOURNIER
“overwhelming” as the initial weights go to 0: with (1 ) = δ for i,j I,
{i} j ij
∈
(cid:88) α
i
(2.4) ((λ α ) ) δ .
i i∈I (cid:80) 1
D · λ−→→0+ i∈I jαj {i}
(These asymptotics can be quickly obtained by taking the limit in the joint moments
given in the proof of Lemma 1)
2.3. RWDE on general graphs, and reinforcement. Let G = (V,E) be a locally
finite directed graph. Recall that directed means that edges e = (x,y) E V V
∈ ⊂ ×
have a tail e = x and a head e = y, while locally finite means that vertices have finite
degree.
Let α = (α ) (0,+ )E be positive weights on the edges.
e e∈E
∈ ∞
We denote by (X ) the canonical process on V.
n n≥0
Definition 3. Let x V. The directed edge linearly reinforced random walk on G
0
∈
with initial weights α and starting at x is the process on V with law P(α) defined by:
0 x0
P(α)-a.s., X = x and, for all n 0, for all edges e E,
x0 0 0 ≥ ∈
N (n)
P(α)((X ,X ) = e X ,...,X ) = e 1 ,
x0 n n+1 | 0 n (cid:80) N (n) {e=Xn}
f∈E,f=e f
where N (n) = α +# 0 k n 1 : (X ,X ) = e .
e e k k+1
{ ≤ ≤ − }
In other words, at time n, this walk jumps through a neighboring edge e chosen with
probability proportional to its current weight N (n), where this weight initially was
e
equal to α and then increased by 1 each time the edge e was chosen.
e
Since edges are oriented, the decisions of this process are ruled by independent Pólya
urns, one per vertex, where outgoing edges play the role of colors, and α is the initial
numbers of balls of each color. By Lemma 1, this reinforced walk may equivalently be
obtained by assigning a Dirichlet random variable ω to each vertex x, and sampling
(x,·)
i.i.d. edges according to this variable in order to define the next step of the walk every
timeitisatx: thisisthedescriptionofarandom walk in Dirichlet random environment
(RWDE), that we formalize now.
The set of environments on G is
(cid:89) (cid:110) (cid:88) (cid:111)
Ω = ∆ = (ω ) (0,1]E : for all x V, ω = 1 ,
G {e∈E:e=x} e e∈E e
∈ ∈
x∈E e∈E,e=x
and we shall denote by ω the canonical random variable on Ω .
G
Definition 4. Let x V. For ω Ω , the quenched random walk in environment ω
0 G
∈ ∈
starting at x is the Markov chain on V starting at x and with transition probabili-
0 0
ties ω. We denote its law by P . Thus, P -a.s., X = x and for all n, for all
x ,ω x ,ω 0 0
0 0
e E,
∈ P (cid:0)(X ,X ) = e(cid:12)(cid:12)X ,...,X (cid:1) = ω 1 .
x0,ω n n+1 0 n e {e=Xn}
The Dirichlet distribution on G with parameter α is the product distribution on Ω
G
(cid:89)
P(α) = ((α ) ).
e {e∈E:e=x}
D
x∈V
Thus, under P(α), the random variables ω , x V, are independent and follow
(x,·)
∈
Dirichlet distributions with parameters given by α , x V, respectively.
(x,·)
∈
RANDOM WALKS IN DIRICHLET ENVIRONMENT 7
Let us consider the joint law P(α) of (ω,X) on Ω VN such that ω P(α) and the
x0 G× ∼
conditional distribution of X given ω is P . Then, under P(α), X is the annealed
x0,ω x0
random walk in Dirichlet environment with parameter α starting at x . Its law is thus
0
P(α)(X ) = E(α)[P ( )].
x0 ∈ · x0,ω ·
Because of the previous remark, it follows from Lemma 1 that the notation P(α) is
x
0
unconsequently ambigous:
Lemma 2. Let x V. The directed edge linearly reinforced random walk on G with
0
∈
initial weights α starting at x , and the annealed random walk in Dirichlet environment
0
with parameter α starting at x , are equal in distribution.
0
Given the natural definition of directed edge reinforced random walk, this property
provides a first justification for the interest in RWDE.
Let us mention that this connection was first used in the context of (non oriented)
edge reinforced random walks on trees by Pemantle [33] where, due to the absence of
cycles, independence between the Pólya urns still holds. On other graphs, non oriented
edge reinforced random walks can be seen as random walks in a correlated, yet rather
explicit, random environment. This leads to very different behaviors and techniques,
see for instance [15, 26, 32, 37, 2]. RWDE were first considered for their own as a
special instance of RWRE in Zd by Enriquez and Sabot, [18].
For any vertex x, we let α be the sum of the weights of the edges exiting from x:
x
(cid:88)
α = α .
x e
e∈E,e=x
With this notation, when G is finite, the Dirichlet distribution may be written as
(cid:81)
(2.5) D(α) = Z1 (cid:89)ωeαe−1(cid:89) dωe, where Zα = (cid:81)e∈EΓΓ((ααe)),
α e∈E e∈E(cid:101) x∈V x
and E(cid:101) is obtained from E by removing arbitrarily, for each x V, one edge with
∈
origin x. From this we can infer the following formula for the moments of ω:
(cid:32) (cid:33)(cid:32) (cid:33)
(cid:20) (cid:21)
(cid:89) Z (cid:89) Γ(α +ξ ) (cid:89) Γ(α )
(2.6) E(α) ωξe = α+ξ = e e x ,
e Z Γ(α ) Γ(α +ξ )
α e x x
e∈E e∈E x∈V
for every function (ξ ) RE such that α + ξ > 0 for all e, and where as usual we
e e e
(cid:80) ∈
write ξ = ξ . When ξ +α 0 for some edge e, the expectation is infinite.
x e,e=x e e e ≤
In the following, we are mainly interested in the case of Zd with nearest-neighbor
edges. There we always assume that weights are translation invariant, therefore given
by 2d parameters α ,...,α , so that for any x Zd, and i = 1,...,2d,
1 2d
∈
α = α ,
(x,x+e ) i
i
where (e ,...,e ) is the canonical basis of Zd and we let (e ,...,e ) = (e ,...,e ).
1 d 1+d 2d 1 d
−
See figure 2.1.
8 C. SABOT AND L. TOURNIER
x+e
2
α
2
α α
1+d 1
x x+e
1
α
2+d
Figure 2.1. Weights on the edges starting at x Zd
∈
3. The property of statistical invariance by time reversal
3.1. The main lemma and a probabilistic proof. Consider a directed graph G =
(V,E) with a family of positive weights (α ) . Assume that G is finite and strongly
e e∈E
connected, i.e. that for any x and y, there is a directed path in G from x to y. Consider
ˇ ˇ
the dual graph G = (V,E) obtained by reversing all edges, i.e.
ˇ
E = (y,x) : (x,y) E .
{ ∈ }
ˇ
For an edge e E we denote eˇ E the associated reversed edge. We define the family
∈ ∈
of reversed weights (αˇ ) by
e e∈Eˇ
αˇ = α , e E.
eˇ e
∀ ∈
We define the divergence operator on the graph G as the linear operator div : RE
RV given by →
(cid:88) (cid:88)
(3.1) div(θ)(x) = θ θ , θ RE, x V.
e e
− ∀ ∈ ∀ ∈
e,e=x e,e=x
Let (ω ) be an environment on the graph G. Since G is finite and strongly
e e∈E
connected, there exists an invariant probability for the quenched Markov chain Pω( ).
·
Denote it by (πω(x)) . We define the time reversed environment (ωˇ ) , which is
x∈V e e∈Eˇ
ˇ
an environment on the dual graph G, by
πω(x)
ωˇ = ω , (x,y) E.
y,x πω(y) x,y ∀ ∈
The following lemma was stated in [35], Lemma 1, where it was first given an analytic
proof that will be discussed in Subsection 3.2 below. A much shorter probabilistic
proof was given in [38].
Lemma 3. Assume G is finite and divα = 0. Then
(3.2) (cid:0)ω P(α)(cid:1) (cid:0)ωˇ P(αˇ)(cid:1).
∼ ⇒ ∼
Proof. We give here a proof in the spirit of that of [38], but even shorter. We say that
σ = (x ,x ,...,x ,x ) is a directed path if (x ,x ) E for all i = 0,...,n 1. It
0 1 n−1 n i i+1
∈ −
is a directed cycle if moreover x = x . For a directed path we set
n 0
n−1
(cid:89)
(3.3) ω = ω .
σ x ,x
i i+1
i=0
If σ is a path we write σˇ = (x ,...,x ) the reversed path, which is a directed path in
n 0
ˇ
the dual graph G. If σ is a directed cycle, we clearly have
(3.4) ω = ωˇ .
σ σˇ
RANDOM WALKS IN DIRICHLET ENVIRONMENT 9
For a cycle σ = (x ,x ,...,x ,x = x ) we write
0 1 n−1 n 0
n−1 n−1
(cid:88) (cid:88)
N (e) = 1 , N (x) = 1 ,
σ {(x ,x )=e} σ {x =x}
i i+1 i
i=0 i=0
the number of visits of the directed edge e and of the vertex x by the cycle σ. If is
C
a finite family of cycles, we write
(cid:88) (cid:88)
N (e) = N (e), N (x) = N (x).
C σ C σ
σ∈C σ∈C
ˇ
We also write = σˇ : σ for the family of reversed cycles. We clearly have from
C { ∈ C}
(2.6)
(cid:20) (cid:21) (cid:18)(cid:81) (cid:19)(cid:18) (cid:81) (cid:19)
E(α) (cid:89)ω = e∈EΓ(αe +NC(e)) x∈EΓ(αx) .
σ (cid:81) (cid:81)
Γ(α ) Γ(α +N (x))
σ∈C e∈E e x∈E x C
(cid:80)
where we write α = α . The property div(α) = 0 is equivalent to the fact
x e,e=x e
that α = αˇ for all vertex x. Remark now that for a cycle N (e) = N (eˇ), and
x x σ σˇ
N (x) = N (x) for all edge e and vertex x. Hence from (3.4) and the previous remarks,
σ σˇ
changing e to reversed edges eˇ, we get
(cid:20) (cid:21) (cid:18)(cid:81) (cid:19)(cid:18) (cid:81) (cid:19)
E(α) (cid:89)ωˇ = e∈EˇΓ(αˇe +NCˇ(e)) x∈EΓ(αˇx)
σ (cid:81) (cid:81)
Γ(αˇ ) Γ(αˇ +N (x))
σ∈Cˇ e∈Eˇ e x∈E x Cˇ
(cid:20) (cid:21)
(cid:89)
= E(αˇ) ω .
σ
σ∈Cˇ
Byconsideringconcatenationsofcycleswiththemselves, thisexactlymeansthatunder
P(α) all joint moments of cycles of ωˇ coincide with the moments of cycles under P(αˇ).
It implies that the law of (ωˇ ) under P(α) coincides with the law (ω )
σ σ cycleofGˇ σ σ cycleofGˇ
under P(αˇ). But the law of cycle probabilities determine the law of the Markov chain.
Indeed, since G is finite and strongly connected, it is recurrent and thus
(cid:88)
ω = ω ,
(x,y) σ
σ
where the sum runs on the cycles that start by the edge (x,y) and come back only
once to x. Hence ωˇ under P(α) has distribution P(αˇ). (cid:3)
We will use several times the following corollary of this lemma. Assume that the
graph is finite and div(α) = 0, and let x V be a specified vertex, and (y,x) E.
Then, under P(α), ∈ ∈
(3.5)P (X comes back to x by the edge (y,x)) Beta(α ,α α ).
x,ω n (y,x) x (y,x)
∼ −
(cid:80)
Indeed, it comes from the fact that the left hand side term is the sum ω where the
σ σ
sum runs on all cycles starting from x and coming back only once to x and by the edge
(cid:80)
(y,x). By (3.4), it equals ωˇ = ωˇ , by Markov property. But ωˇ is distributed
σ σˇ (x,y)
according to the Dirichlet environment P(αˇ), which implies (3.5) by the agglomeration
property of Dirichlet distributions, cf. Property 2.
10 C. SABOT AND L. TOURNIER
3.2. Analytic approach. (This section is not necessary in the sequel and can be
skipped in first reading.) The original proof was analytic and based on a change of
variable (published only in the arXiv version of [35]). It is rather technical but gives
extrainformationonthedistributionoftheoccupationmeasureoftheRWDE.Weonly
give here the statement, the proof is available in the appendix of [35] (arXiv version).
Let e be a specified edge of the graph. Let be the affine space defined by
0 e
H 0
= (z ) RE : z = 1, div(z) 0 .
e e e∈E e
H 0 { ∈ 0 ≡ }
˜
and ∆ be the set defined by
e
0
∆˜ = (R∗)E.
e0 He0 ∩ +
We define
πωω
e e
Z = ,
e πωω
e0 e0
the occupation measure of the edges of the graph, normalized so that Z = 1. Clearly
e
0
˜
(Z ) ∆ . In the stationary regime, it is proportional to the expected number
e e∈E e
∈ 0
of traversals of the edge e. The proof was based on the explicit computation of the
distribution of the random variable (Z ) under P(α).
e e∈E
Let T be a spanning tree of the graph G such that e / T. (This is possible since
0
∈
the graph is strongly connected and thus e belongs to at least one directed cycle of
0
the graph.) We denote B = T e . Then (z z ) is a dual basis of , it
0 e e∈Bc e
˜ ∪ { } (cid:55)→ H 0
defines a natural measure on ∆ ,
e
0
(cid:89)
dλ = dz ,
∆˜e0 e
e∈Bc
which does not depend on the choice of T, e . Let x V be any vertex, and denote
0 0
∈
the set of directed spanning trees of the graph directed towards the vertex x .
x 0
T 0
Lemma 4. Under P(α), the random variable (Z ) has the following distribution
e e∈E
˜
on ∆ :
e
0  
(cid:18)(cid:81)(cid:81)xe∈∈VEΓΓ((ααex))(cid:19)(cid:18)(cid:81)(cid:81)e∈x∈EVzezαxeα−x1(cid:19)T(cid:88)∈Tx0e(cid:89)∈T ze dλ∆˜e0,
(cid:80)
where as usual z = z
x e,e=x e
Remark 1. We can remark that this formula is reminiscent of the distribution discov-
ered by Diaconis and Coppersmith ([15, 26]) which expresses edge-reinforced random
walk as a mixture of reversible Markov chains. Note that the sum on spanning trees
can also be expressed as a principal minor of the matrix with diagonal coefficients equal
to z and off diagonal coefficients equal to z . It does not depend on the choice
x (x,y)
−
of x .
0
We see that lemma 3 is a direct consequence of the previous result. Indeed we see
ˇ ˇ
that lemma 4 applied to the reversed graph (G,E), starting with the weights αˇ = α
eˇ e
(cid:80)
gives the same integrand with α replaced by αˇ = α . The two coincide after
x x e,e=x e
the change of variables exactly when div(α) 0.
≡