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Finally, using the link between VRJP and ERRW [14], it yields an answer to an old question of Diaconis about the direct computation of the normalizing constant of the ‘magic formula’ for the mixing measure of ERRW. Results of this paper are instrumental in [16], where the representation in terms of a random Schr¨odinger operator is extended to infinite graphs. Interesting new phenomena appear in the tran- sient case, where a generalized eigenfunction of the Schr¨odinger operator is involved in the represen- tation. Several consequences follow on the behavior of the VRJP and the ERRW in [16]: in particular a functional central limit theorem is proved for the VRJP and the ERRW in dimension d 3 at weak reinforcement, and recurrence of the 2-dimensional ERRW is shown, giving a full answer≥ to an old question of Diaconis. The paper is organized as follows. In Section 2, we define the new exponential family of distribu- tions and give its first properties. In section 3, we discuss the link between the exponential family and the Vertex reinforced jump processes. In Section 4 we consider the ERRW and answer the question of Diaconis. Sections 5 and 6 provide the proof of the two main results, namely Theorem 1 and Theorem 3.
2 A new exponential family
Let V = 1,...,n be a finite set, and let (W ) be a set of non-negative reals with W = { } i,j i6=j i,j W 0. Denote by E the edges associated to the positive W , i.e. consider the graph =(V, E) j,i ≥ i,j G with i, j E if and only if Wi,j > 0, and write i j if i, j E. Let dG be the graph distance on .{ } ∈ ∼ { } ∈ G When A is a symmetric operator on RV (also be considered as a V V matrix), write A> 0 if A is positive definite, and A for its determinant. × | |
Theorem 1. Let P =(Pi,j)1≤i,j≤n be the symmetric matrix given by
0 i = j, Pi,j = W i = j. ( i,j 6 For any θ Rn , we have ∈ +
n 2 n/2 −hθ,βi dβ 1 ( ) 1{2β−P>0}e = exp Wi,j θiθj (1) π 2β P − · √θi Z {i,j}∈E i=1 | − | X p Y p where dβ = dβ dβ , and 2β P is the operator on RV defined by 1 ··· n − [(2β P )f](i)=2β f(i) W f(j). − i − i,j j:j∼i X Definition 1. The exponential family of random probability measures νW,θ(dβ) is defined by
W,θ 2 n/2 i √θi ν (dβ)= 12β−P>0( ) exp θ, β + Wi,j θiθj dβ π −h i 2β P {i,j}∈E Q X p | − | p where θ, β = θ β . We will simply write νW for νW,1 in the case where θ =1 for all i V . h i i∈V i i i ∈ P 2 The proof of Theorem 1 is given in Section 5. We deduce the following simple but important properties of the measure νW,θ.
Proposition 1. The Laplace transform of νW,θ is
n −hλ,βi W,θ θi e ν (dβ) = exp Wi,j λi + θi λj + θj θiθj − − · λi + θi Z {i,j}∈E i=1 r X p p p Y Moreover, if β is a random vector with distribution νW,θ, then
1 The marginals βi are such that is an Inverse Gaussian distribution with parameters • 2βiθi ( 1 , 1) Pj∼i Wi,j √θiθj If V V , V V are two subsets of V such that d (V ,V ) 2, then (β ) and (β ) • 1 ⊂ 2 ⊂ G 1 2 ≥ i i∈V1 j j∈V2 are independent.
Proof. The Laplace transform of νW,θ can be computed directly from Theorem 1, from which we deduce independence at distance at least 2. We can also deduce, by identification of the Laplace transforms, that the marginals of this law are reciprocal inverse gaussian up to a multiplicative constant. The family can be reduced to the case θ =1 by changing W , as shown in the next corollary.
W,θ Corollary 1. Let (βj)j∈V be distributed according to ν . Then (θβ) is distributed according to W θ θ ν , where Wi,j = Wi,j θiθj. It is clear from the expressionp of the Laplace transform that if the graph has several connected components then the random field (βj)j∈V splits accordingly into independent random subvectors. Therefore, we will always assume in the sequel that the graph is connected. G 3 Link with the Vertex reinforced Jump process
3.1 Vertex Reinforced Jump Process: definition and main properties In this section we explain the link between the exponential family of Section 2 and the Vertex reinforced Jump Process (VRJP), which is a linearly reinforced process in continuous time, defined in [5], investigated on trees in [3], and on general graphs by the first two authors in [14]. Consider as in the previous section a conductance network (Wi,j) and the associated graph = (V, E). Fix also some positive parameters (φ ) on the vertices. Assume that the graph isG connected. i i∈V G We call VRJP with conductances (Wi,j) and initial local time (φi) the continuous-time process (Y ) on V , starting at time 0 at some vertex i V and such that, if Y is at a vertex i V at t t≥0 0 ∈ ∈ time t, then, conditionally on (Y ,s t), the process jumps to a neighbour j of i at rate s ≤
Wi,jLj(t),
where t 1 Lj(t) := φj + {Ys=j} ds. Z0
3 The following time change, introduced in [14], plays a central role. Let
D(t)= (L2(t) φ2), (2) i − i Xi∈V
define Zt as the time changed process Zt = YD−1(t). t 1 Let (ℓj(t)) be the local time of Z at time t (that is, ℓj(t)= 0 Zs=jds). Conditionally on the past, at time t, the process Z jumps from Z = i to a neighbour j at rate (c.f. [15], Lemma 3) t R
2 Wi,j φj + ℓj(t) 2 . 2 s φi + ℓi(t)
We state below one of the main results of [14], Proposition 1 and Theorem 2. The theorem was stated in [14] in the case φ = 1, this version of the theorem can be deduced by a simple change of time, details are given in Appendix B.
Theorem 2. Assume that is finite. Suppose that the VRJP starts at i . The limit G 0 2 2 1 ℓi(t)+ φi φi Ui = lim log log 2 t→∞ ℓ (t)+ φ2 − φ2 i0 i0 i0 exists a.s. and, conditionally on U, Z is a Markov jump processes with jump rate from i to j 1 W eUj −Ui . 2 i,j Moreover (U ) has the following distribution on (u ), u =0 j { i i0 }
φ 1 u −u u −u W,φ j6=i0 j − P u − P W (e i j φ2+e j i φ2−2φ φ ) (du)= e j∈V j e 2 {i,j}∈E i,j j i i j D(W, u) du, (3) i0 |V |−1 Q √Q2π p with du = du and j∈V \{i0} j
Q ui+uj D(W, u)= Wi,je XT {i,jY}∈T where the sum runs on the set of spanning trees T of . We simply write W for W,1 G Qi0 Qi0 W,φ The fact that the total mass of the measure i0 is 1 is both a non-trivial and a useful fact: in particular, it plays a central role in the delocalizationQ and localization results of [9, 10]. In [14] it is a consequence of the fact that it is the probability distribution of the limit random variables U. In [10] it is proved using a sophisticated supersymmetric argument, the so-called localization principle. Theorem 3 below provides a direct ’computational’ proof of that result, based on the identity (1) and on the change of variable in Proposition 2 that relates the field (uj) to the random vector (βj) in Definition 1.
4 3.2 Link with the random potential β The second main result of this paper enables us to construct the mixing field eu defined in the previous subsection from the random potential (βj) defined in Definition 1. It gives also a natural way to couple the mixing measure of VRJP starting from different points. Let us first state the following Proposition 2, which provides some elementary observations on the Green function. Define = (β ) (R 0 )V , 2β P > 0 . D { i i∈V ∈ + \{ } − } Proposition 2. Let β , and let G be the inverse of (2β P ). Then (G(i, j)) has positive ∈ D − coefficients. Define (u(i, j))i,j∈V by G(i, j) eu(i,j) = . G(i, i) Then for i V , the function j u(i , j) is the unique solution j u of the equation 0 ∈ → 0 7→ j 1 W euj −ui = β , i = i j∼i 2 i,j i 6 0 (4) u =0, (Pi0
In particular (u(i0, j))j∈V is (βj)j∈V \{i0} measurable. Moreover, at the site i0 we have 1 1 β = + W eu(i0,j). i0 2G(i , i ) 2 i0,j 0 0 j:j∼i X0 Theorem 3. Let β be a random potential with distribution νW,φ2 (dβ) as in Definition 1, and let (u(i, j))i,j∈V be defined as in Proposition 2. Then the following properties hold:
i) The random field (u(i , j)) has the distribution of the mixing measure W,φ(du) of the VRJP 0 j∈V Qi0 starting from i0 with initial local time (φi)i∈V .
ii) The random variable G(i0, i0) has the distribution of 1/(2γ), where γ is a gamma random variable 2 with parameters (1/2, 1/φi0 ). Moreover, G(i0, i0) is independent of (βj)j6=i0 , and thus also of the field (u(i0, j))j∈V . The proofs of Proposition 2 and Theorem 3 are given in Section 6. The next Corollary 2 describes how to construct the random potential β from the field u of Theorem 2.
Corollary 2. Consider a VRJP with edge weight (Wi,j) and initial local time (φi)i∈V , starting at i0. Let (u ) be distributed according to W,φ of Theorem 2. Let i i∈V Qi0 1 β˜ = W euj −ui . (5) i 2 i,j j:j∼i X 1 2 Let γ be a Gamma distributed random variable with parameters ( 2 , 1/φi0 ), independent of (uj), and let ˜ 1 βi = βi + i0 γ. (6) Then β has the law νW,φ2 of Definition 1.
5 Corollary 2 indeed follows directly from Theorem 3 and Proposition 2: the law of β in (6) is uniquely determined by the laws of (ui)i∈V and γ independent from (βi)i6=i0 , hence it is sufficient to W,φ2 show that, if β has distribution ν (dβ) and u is defined from (4) by Proposition 2, then (ui)i∈V W,φ ˜ 2 indeed has distribution i0 , and γ = βi0 βi0 = 1/(2G(i0, i0)) has distribution Γ(1/2, 1/φi0 ), which follows from TheoremQ 3. − As mentioned in the introduction, Theorem 3 has several consequences. Firstly it explicitly relates the VRJP to the random Schr¨odinger operator ∆W + V , where V is the random potential V = 2β W . Secondly it yields a natural coupling between− the random fields (u ) associated i i − i j j∈V with the VRJP starting from different sites, since the exponential family (βi)i∈V gives the same role to each vertex of the graph, and (u(i, j))i,j∈V arises from these random variables (βi)i∈V . Finally W,θ it also gives a computational proof of the identity i0 (du)=1, for any θ, as a consequence of 2 Q Theorem 1 that allows to define νW,φ (dβ) as a probability measure. R 4 Link with the Edge reinforced random walk and a question of Diaconis
4.1 Definition and magic formula The Edge Reinforced Random Walk (ERRW) is a famous discrete time process introduced in 1986 by Coppersmith and Diaconis, [4]. Let (ai,j){i,j}∈E be a set of positive weights on the edges of the graph . Let (Xn)n∈N be a random process that takes values in V , and let = σ(X ,...,X ) be the filtrationG of its past. For Fn 0 n any e E, n N, let ∈ ∈ n 1 Zn(e)= ae + {{Xk−1,Xk}=e} (7) Xk=1 be the number of crosses of the (non-directed) edge e up to time n plus the initial weight ae. Then (Xn)n∈N is called Edge Reinforced Random Walk (ERRW) with starting point i0 V and weights (a ) , if X = i and, for all n N, ∈ e e∈E 0 0 ∈ Z ( X , j ) P(X = j )= 1 n { n } . (8) n+1 | Fn {j∼Xn} Z ( X ,k ) k∼Xn n { n } PERRW,(a) P We denote by i0 the law of the ERRW starting from the initial vertex i0 and initial weights (a). A fundamental property of the ERRW, stated in the next theorem, is that on finite graphs the ERRW is a mixture of reversible Markov chains, and the mixing measure can be determined explicitly (the so-called Coppersmith-Diaconis measure, or ‘magic formula’). It is a consequence of a de Finetti theorem for Markov chains due to Diaconis and Freedman [7], and the explicit determination of the law is due to Diaconis and Coppersmith [4, 11, 12]. It has also applications in Bayesian statistics [2, 1, 8]. Theorem 4. [4, 11] Assume that =(V, E) is a finite graph and set a = a for all i V . Fix an edge G i j:{i,j}∈E i,j ∈ e incident to i , and define = y : e E, y > 0, y = 1 (similarly let y = y ). 0 0 He0 { ∀ ∈ e P e0 } i i∈e e Consider the following positive measure defined on e defined by its density H 0 P ae √yi y dy (a)(dy)= C(a, i ) 0 e∈E e D(y) e , (9) i0 0 1 (a +1) M 2 i ye Qy e6=e0 i∈V i p Y Q 6 with D(y)= ye, XT eY∈T where the sum runs on the set of spanning trees T of , and with G 1−|V |+ a 1 2 Pe∈E e Γ( (a +1 1 )) C(a, i )= i∈V 2 i − i=i0 0 |V |−1 · Γ(a ) √π Q e∈E e Q Then (a) is a probability measure on , and it is the mixing measure of the ERRW starting from Mi0 He0 i0, more precisely PERRW,(a) (y) (a) i0 ( )= Pi0 ( )d i0 (y), · H · M Z i0 (y) where Pi0 denotes the reversible Markov chain starting at i0 with conductances (y).
4.2 The question of Diaconis The fact that (a)(dy) is a probability measure is a consequence of the fact that it is the mixing Mi0 measure of the ERRW. In fact it is obtained as the limit distribution of the normalized occupation time of the edges [11]: Z (e) n law (a). Z (e ) −→ Mi0 n 0 e∈E One question raised by Diaconis is the following
(Q) Prove by direct computation that (a)(dy)=1. (10) Mi0 An answer was proposed by Diaconis and Stong [6] in the caseR of the triangle, using a subtle change of variables. Also note that Merkl and Rolles offered in [12] analytic tools for the computation of the ratio of the normalizing constants of the magic formula for two initial weights differing by integer values, which may possibly be extended to provide the normalizing constant. We provide below an answer to that question. A first simplification comes from [14], where an explicit link was made between the VRJP and the ERRW.
Theorem 5 (Theorem 1, [14]). Consider (Yn) the discrete time process associated with the VRJP (Yt) (i.e. taken at jump times) with conductances (Wi,j) and φ = 1. Take now the conductances (We)e∈E as independent random variables with gamma distribution with parameters (ae)e∈E. Then the ‘annealed’ law of Yn (i.e. the law after taking expectation with respect to the random (We)) is the law of the ERRW (Xn) with initial weights (ae)e∈E.
(a) W This immediately implies an identity between the mixing measures i0 and i0 : indeed, by M ui+uQj Theorem 2, (Yn) is a mixture of Markov jump processes with conductances Wi,je , which implies that for all 0-homogeneous bounded test functions φ (i.e. φ(λy)= φ(y), λ> 0), we have ∀
ae−1 −We (a) We e ui+uj W φ((ye)) i0 (dy)= φ((Wi,je )) i0 (du) dW. (11) E He M R Γ(ae) Q Z 0 Z eY∈E Z with dW = e∈E dWe. This identity was checked by direct computation in section 5 of [14]. Q
7 W Now, the fact that i0 (du)=1 is a consequence of the computation of the integral (1) in Theorem 1 and the changeQ of variables in Theorem 3, as explained at the end of Section 3. Therefore R a d i0 (y)=1. y =1 M Z e0 Note that this fact can be used to prove directly that a (dy) is the mixing measure of the Mi0 ERRW starting from initial condition (a) and initial vertex i0. Indeed, for any finite path σ : i0 i i , let N(i) (resp. N(e)) be the number of times vertex i (resp. edge e) is visited (resp.→ 1 →···→ n crossed): N(i)= k; 0 k n 1, i = i |{ ≤ ≤ − k }| N(e)= k; 0 k n 1, i , i = e . |{ ≤ ≤ − { k k+1} }| The probability of σ for the reversible Markov chain of conductance y is
N(e) ye py (σ)= e∈E i0 N(i) Qi∈V yi y a The integration of pi0 (σ) w.r.t. d i0 (y) can be computedQ by changing the constant Γ(ae) to Γ(ae + 1 1 M Ne) and Γ( 2 (ai + 1)) to Γ( 2 (ai +1)+ Ni). Using the property Γ(x +1) = xΓ(x) and the notation n−1 (a, n)= k=0(a + k), we deduce
Q (ae, N(e)) py (σ)d a (y)= e i0 Mi0 (a , N(i)) Z Q i i which is the probability of an ERRW to follow the pathQσ.
5 Proof of Theorem 1
Lemma 1. Let P =(Pi,j)1≤i,j≤n be a symmetric matrix with
0, i = j, Pi,j = W , R+ i = j, ( i,j ∈ 6 and let β be a diagonal matrix with entries β , i = 1,...,n, such that M = 2β P is positive i − definite. Let L be the lower triangular n n matrix and U be the upper unitary (with 1 on the diagonal) upper triangular matrix such that M× = LU (i.e. the LU decomposition of M), which exist and are unique. Then x1 H1,2 H1,n 0 −x ··· −H U = 2 2,n , ··· −H ··· − n−1,n 0 0 x ··· n where (xi)1≤i≤n and (Hi,j)1≤i H1,j = W1,j j > 1 i−1 Hk,iHk,j Hi,j = Wi,j + i 2, j>i k=1 xk 2 ≥ i−1 Hk,i xi =2βi i 1. − kP=1 xk ≥ P 8 Furthermore, M(1,...,i 1,...,i) x = | i M(1,...,i 1 1,...,i 1) − | − where M(I J) is the minor of matrix M that corresponds to the rows with index in I and columns with index in| J. The result follows directly from (2.6) of [17], but we prove it in Appendix A for completeness’ sake. Claim 1. For any θ > 0, θ 0, 1 2≥ ∞ θ x θ 1 2π exp( 1 2 ) dx = exp( θ θ ) . − 2 − 2x √x − 1 2 θ Z0 r 1 p 1 Proof. The case θ2 = 0 corresponds to the normalisation of the Γ( 2 ) variable. The case θ2 > 0 corresponds to the normalization of the Inverse Gaussian law IG( θ1 , 1 ). θ2 θ2 Let us now prove Theorem 1. In the sequel we take the convention, given any real sequence m (ak)k∈N, that k=n ak =0 if n > m. By Lemma 1, P n n k−1 2 n n xk Hl,k θlxl 1 2 θlβl = θk( + )= + ( θkHl,k) . 2 2xl " 2 2xl # Xk=1 Xk=1 Xl=1 Xl=1 kX=l+1 Define Ψ:(R 0 )n + \{ } −→ D i−1 2 xi Hk,i (xi)1≤i≤n (βi)1≤i≤n = + . 7−→ 2 xk ! Xk=1 1≤i≤n Then Ψ is a bijection, since a symmetric matrix is positive definite if and only if all of its diagonal minors are positive. Its Jacobian is 2−n, hence it is a diffeomorphism. Therefore n n 1 exp( θβ) θlxl 1 2 1 1 I := {2β−P>0} − dβ = exp + ( θkHl,k) n dx. Rn − 2 2x x x 2 2β P + " l #! √ 1 n Z | − | Z Xl=1 kX=l+1 ··· Let, for all 1 pl m n, ≤ ≤ ≤ n 2 m 2 Rl,m = Hl,j θj + θkHl,k j=m+1 ! k=l+1 X p X θlxl Rl,m Sl,m = + . 2 2xl Note that Rl,m (resp. Sl,m) only depends on x1, ... xl−1 (resp. x1, ... xl). Let, for all 1 m n, ≤ ≤ m dx1 dxm Im := exp Sl,m ··· . Rm − x x + ! √ 1 m Z Xl=1 ··· We will take the convention that, if m =0, the integral of dx1 dxm is 1, so that I0 =1. n ··· Note that I = In/2 . We also have the following lemma. 9 Lemma 2. For all 1 m n, we have ≤ ≤ 2π n I = exp W θ θ I . m θ − m,j m j m−1 r m j=m+1 ! X p Proof. Using Claim 1, we deduce m−1 θmxm Rm,m dx1 dxm Im = exp + + Sl,m ··· Rm − 2 2xm √x1 xm Z + " l=1 #! ··· m−1 X dx1 dxm−1 = exp Rm,mθm Sl,m ··· . (12) Rm−1 − − √x1 xm−1 Z + l=1 ! p X ··· 2 n Now Rm,m = j=m+1 Hl,j θj and P p m−1 Hl,mHl,j Hm,j = Wm,j + , xl Xl=1 so that n m−1 n Hl,m√θm Rm,mθm = Wm,j θmθj + Hl,j θj. xl j=m+1 l=1 j=m+1 p X p X X p On the other hand, for all 1 l m 1, ≤ ≤ − H √θ n S S = l,m m H θ . l,m − l,m−1 − x l,j j l j=m+1 X p Therefore m−1 n m−1 Rm,mθm + Sl,m = Wm,j θmθj + Sl,m−1, l=1 j=m+1 l=1 p X X p X which enables to conclude by (12). We deduce from Lemma 2, by induction, that n In 1 (2π) I = n = n exp Wi,j θiθj , 2 2 θn θ1 − s {i,j}∈E ··· X p which enables us to conclude. 6 Proof of Proposition 2 and Theorem 3 6.1 Proof of Proposition 2 Fix i V , and let β . Let us first justify the existence and uniqueness of u(i , i) defined by 0 ∈ ∈ D 0 the linear system (4). As (2β P ) is an M-matrix, its inverse G satisfies G(i, j) > 0 for any i, j. A − solution (u ) of equation (4) is necessarily of the form euj = 2γG(i , j) for some constant γ R. j 0 ∈ 10 The normalization u =0 implies γ = 1 . Hence the unique solution of the system (4) is given i0 2G(i0,i0) by uj = u(i0, j) defined in Theorem 3. Consider the following map: Φ: (u ) RV , u =0 (R 0 ) D →{ j j∈V ∈ i0 } × + \{ } (β) ((u ),γ), (13) 7→ j where (u ) is the unique solution of the system (4) and γ = 1 . j 2G(i0,i0) We first prove that Φ is a diffeomorphism. By the previous argument it is well-defined and injective. Reciprocally, starting from ((uj),γ) on the right hand side, we define (βi) by 1 β = W euj −ui + 1 γ. (14) i 2 i,j i=i0 j∼i X It is clear that with this definition, (uj) is the solution of (4) with (βj). It remains to prove that 2β P > 0: it is a consequence Theorem (2.3)- (J30) of [13]: − 1 Proposition 3. Let A Zn = M Mn(R), mi,j 0, if i = j . Then A is positive stable if and ∈ 2 { ∈ 3 ≤ 6 } only if there exists ξ 0 with Aξ > 0 and ≫ k ak,jξj > 0, k =1, . . . , n. (15) j=1 X We will choose a bijection σ between V and 1,..., V , and apply Proposition 3 with { | |} u −1 A = ((2β P ) −1 −1 ) , ξ =(e σ (i) ) . − σ (i),σ (j) 1≤i,j≤|V | 1≤i≤|V | u. Obviously, ξ 0, and Aξ > 0 follows from (2β P )e = δi0 /(2G(i0, i0)). Now fix any spanning tree of the graph≫ and its corresponding distance−d on V throughout the tree. Choose σ so that T σ(i ) = V , and σ(i) < σ(j) if d(i , i) > d(i , j): this implies that, for all k < V , there exists 0 | | 0 0 | | l>k such that W −1 −1 > 0 and therefore that (15) holds. We conclude that 2β P > 0. σ (k),σ (l) − 6.2 Proof of Theorem 3 We give two proofs. First proof: We make the change of variable given by Φ−1, in (13) and we now prove that if β has distribution νW,φ2 , then (u,γ) = Φ−1(β) has distribution W,φ Γ( 1 , 1 ). i0 2 φ2 Q ⊗ i0 −1 ∂βi ∂βi Let J be the Jacobian matrix of Φ (i.e. Ji,j = , j = i0 Ji,i = ), then ∂uj 6 0 ∂γ δi,i0 if j = i0, 1 uj −ui Ji,j = W e if i = j, j = i , 2 i,j 6 6 0 βi if i = j = i0. − 6 We can factorize the ith row of J bye−2ui for each i, then expand the resulting matrix according to the i0th column, and we find that 1 J = e−2 Pi ui D(W, u) | | 2|V |−1 1All of its eigenvalues have positive real part. 2 ξ η means for any coordinate i, ξi > ηi 3 ≫ ξ > 0 means ξi 0 and ξ =0 ≥ 6 11 On the other hand, by (14) we deduce 2β P =2γe−2 Pi ui D(W, u). | − | Let ψ be a positive test function. We have 2 ψ(u,γ)νW,φ (dβ) Z 2 φi exp( i βiφi + {i,j}∈E Wi,jφiφj) 1 |V |/2 i − −2 Pi ui = ψ(u,γ)2 |V |/2 |V |−1 e D(W, u)dudγ π −2 Pi ui 2 Z Q P 2γe PD(W, u) −φ2 γ u u u u i0 i φi − P u(pi ,i) − 1 P W (e i− j φ2+e j − i φ2−2φ φ ) e = ψ(u,γ) e i 0 e 2 i∼j i,j j i i j D(W, u) dudγ (2π)(|V |−1)/2 · √πγ Z Q −φ2 γ p φ e i0 = ψ(u,γ) W,φ(du) i0 dγ. Qi0 √πγ Z This concludes the proof of Theorem 3 and of Corollary 2. Second proof: This proof does not make use of the explicit expression of law W,φ of U in (3), Qi0 but rather deduces its Laplace transfom from direct computation of the probability of a path. Note that compared to the first proof, this one uses the representation of the VRJP as a mixture of Markov Jump Processes, cf Theorem 2 of [14] or Theorem 2 in section 3, and hence it uses implicitly that the measure W,φ is a probability measure. Qi0 We will show that, if (u,γ) has distribution W,φ Γ( 1 , 1 ), then β = Φ(u,γ) has distribution i0 2 φ2 Q ⊗ i0 νW,φ2 , which clearly implies the result. It follows by direct computation (see [15], proof of Theorem 3) that the probability that, at time t, the VRJP Z has followed a path Z0 = x0, x1, ..., Zt = xn with jump times respectively in [ti, ti + dti], i =1 ...n, where t0 =0 < t1 <... φ p = exp W φ2 + ℓ φ2 + ℓ φ φ i t i,j i i j j i j 2 − − φi + ℓi {i,jX}∈E q q iY6=i0 n 1 p dt = W dt , 2 xi−1xi i i=1 Y with (ℓi)i∈V =(ℓi(t))i∈V local time at time t. On the other hand, using that, conditionally on U =(Ui)i∈V in Theorem 2, Z is a Markov jump Uj −Ui process with jump rate Wije /2 from i to j, this probability of a path is also qtdt, where − P β˜ ℓ W,φ q = e i∈V i i (du) t Qi0 Z and β˜ is defined in (5). 1 1 Let Γ=Γ( 2 , φ2 ). By identification of pt and qt we deduce that i0 − P β ℓ W,φ − P β˜ ℓ W,φ −ℓ γ e i∈V i i (du)Γ(dγ)= e i∈V i i (du) e i0 Γ(dγ) Qi0 Qi0 Z Z Z 2 2 φi 1 = exp Wi,j φi + ℓi φj + ℓj φiφj , − − 2 2 {i,j}∈E i6=i φi + ℓi ! 1+ ℓ /φ X q q Y0 i0 i0 W,φ p q 2 which shows that the distribution Γ( 1 , 1 ) has the same Laplace transform as νW,φ in i0 2 φ2 Q ⊗ i0 Proposition 1. 12 A Proof of Lemma 1 Proof. We perform successive Gauss elimination on M to make it upper triangular. Denote by l ,...,l the n rows of any n n matrix. Firstly, let 1 n × (1) (1) (1) x1 H1,2 H1,n (1) − (1) ··· − (1) (1) H x H M = M = − 1,2 2 ··· − 2,n ············(1) (1) (1) H H xn − 1,n − n,2 ··· (1) (1) where we set, for any 1 i, j n, x =2β and H = W . ≤ ≤ i i i,j i,j We define a sequence of matrices M (k) recursively, such that (1) (1) (1) x1 H1,2 H1,n − (2) ···(2) ··· ··· ··· ··· − (2) 0 x2 H2,3 H2,n . −. . − . . 0 .. .. . . .. (k−1) (k−1) (k−1) . . xk−1 Hk−1,k Hk−1,n M (k) = . − ··· ··· − , . 0 x(k) H(k) H(k) k − k,k+1 ··· − k,n . . (k) . . . . . H .. .. . − k,k+1 . . . .. (k) . . . . Hn−1,n (k) (k) − (k) 0 0 0 H H xn ··· − k,n ··· − n−1,n − (k+1) (k) by the following rule: M is constructed from M by addition of columns lk+1 lk+1 + (k) (k) ← Hk,k+1 Hk,n (k) (k) lk,...,ln ln + (k) lk in M . In other words, xk ← xk 1 i = j (k) (k) (k+1) Hk,i TkM = M , where [Tk]i,j = i > j = k x(k) k 0 otherwise (k) (k) It is easy to check that (xi )i≥k, (Hi,j )i,j≥k satisfy the following induction rule: (k) (k) (k+1) (k) Hk,i Hk,j Hi,j = Hi,j + (k) , i, j k +1, xk ≥ (k) 2 (k+1) (k) (Hk,i ) xi = xi (k) , i k +1. − xk ≥ At step n, we have x(1) H(1) H(1) 1 − 1,2 ··· − 1,n 0 x(2) H(2) (n) 2 2,n Tn−1 T1M = M = . . ···. − (n−1) ··· . .. .. H − n−1,n (n) 0 0 xn ··· Hence, it gives the LU-decomposition of M where L−1 = T = T T T and U = M (n). It is n−1 n−2 ··· 1 easy to check that (i) xi = xi i =1,...,n (i) (Hi,j = Hi,j i < j satisfy the recursion in the statement, and that x = M(1,...,i 1,...,i)/M(1,...,i 1 1,...,i 1). i | − | − 13 B Time rescaling Let Ys be the VRJP with conductances (W ) and initial local time (φi)i∈V defined in Section 3. Recall t Li(s) that Li(t) = φi + 1Y =ids. Consider the increasing functional A(s) = ( 1), and the 0 s i φi − ˜ time-changed processR Ys˜ = YA−1(˜s). Let P t L˜ (˜s)=1+ 1 ds.˜ i {Y˜s˜=i} Z0 We always denote by s˜ the time scale of Y˜ , we can write ds 1 s˜ = A(s), ds˜ = , Li(˜s)= Li(s). φYs φi Obviously, Y˜ is a VRJP with edge weight Wi,jφiφj and initial local local time 1 : that is, conditionally on Y˜ , Y˜ jumps from i to j at rate Fs˜ Wi,jφiφjL˜j(˜s). Note for simplicity φ Wi,j = Wi,jφiφj. We can apply [14] Theorem 2 to Y˜ . Let D˜(˜s)= L˜ (˜s)2 1, i − i X ˜ ˜ ˜ ˜ t˜ 1 and set Zt˜ = YD˜ −1(t˜), with local time ℓi(t)= 0 X˜u=idu. By proposition 1 of [14] translated in time scale L (cf relation (2.1) of [14]), we have that log L˜ (˜s) 1 log L˜ (˜s) converges a.s. when R i − N j∈V j s˜ to a random vector with distribution given by (3.1) of theorem 1 of [14], where the weights (W→∞) are replaced by (W φ ). Changing to variables u u Pu , we deduce i,j i,j i → i − i0 lim log L˜i(˜s) log L˜i (˜s)= Ui s˜→∞ − 0 exists and has distribution φ 1 1 φ W − Pj∈V uj − 2 Pi∼j Wi,j (cosh(ui−uj )−1) φ i0 (du)= N−1 e e D(W ,u) du, Q √2π p ˜ 1 φ Uj −Ui and that Z is a mixture of Markov Jump Process with jumping rates 2 Wi,je . We now come back to (Zt). Recall that Zt = YD−1(t), where D(t) is defined in (2). ¿From this we have t˜ = D˜(A(D−1(t))), and 1 L˜ ˜ (˜s) 1 dt˜= Ys˜ dt = dt. φ L (s) φ2 Y˜s˜ Ys Zt 1 Uj +log φj −Ui−log φi This implies that (Zt) is a mixture of Markov Jump processes with jumping rates 2 Wi,je . 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