ISOMORPHISMS OF β-DYSON’S WITH BROWNIAN Titus Lupu

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Titus Lupu. ISOMORPHISMS OF β-DYSON’S BROWNIAN MOTION WITH BROWNIAN LOCAL TIME. 2020. ￿hal-02996476￿

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TITUS LUPU

Abstract. We show that the Brydges-Fr¨ohlich-Spencer-Dynkin and the Le Jan’s isomor- phisms between the Gaussian free fields and the occupation times of symmetric Markov pro- cesses generalizes to β-Dyson’s Brownian motion. For β P t1, 2, 4u this is a consequence of the Gaussian case, however the relation holds for general β. We further raise the question whether there is an analogue of β-Dyson’s Brownian motion on general electrical networks, interpolating and extrapolating the fields of eigenvalues in matrix valued Gaussian free fields. In the case n “ 2 we give a simple construction.

1. Introduction There is a class of results, known as isomorphism theorems, relating the squares of Gaussian free fields (GFFs) to occupation times of symmetric Markov processes. They originate from the works in mathematical physics [Sym69, BFS82]. For a review, see [MR06, Szn12]. Here in particular we will be interested in the Brydges-Fr¨ohlich-Spencer-Dynkin isomorphism [BFS82, Dyn84a, Dyn84b] and in the Le Jan’s isomorphism [LJ10, LJ11]. The BFS-Dynkin isomorphism involves Markovian paths with fixed ends. Le Jan’s isomorphism involves a Poisson of Markovian loops, with an intensity parameter α 1 2 in the case of real scalar GFFs. For vector valued GFFs with d components, the intensity“ parameter{ is α d 2. We show that both Le Jan’s and BFS-Dynkin isomorphisms have a generalization to β-Dyson’s“ { Brownian motion. For β 1, 2, 4 , a β-Dyson’s Brownian motion is the diffusion of eigenvalues in a Brownian motion onP t the spaceu of real symmetric β 1 , complex Hermitian β 2 , respectively quaternionic Hermitian β 4 matrices.p Yet,“ theq β-Dyson’s Brownianp motion“ q is defined for every β 0. The one-dimensionalp “ q marginals of β-Dyson’s Brownian motion are Gaussian beta ensemblesě GβE. The generalization of Le Jan’s and BFS-Dynkin isomorphisms works for every β 0, and for β 1, 2, 4 is follows from the Gaussian case. The intensity parameter α appearingě in the LeP Jan’s t typeu isomorphism is given by β 2α d β,n n n n 1 ., “ p q“ ` p ´ q 2 where n is the number of ”eigenvalues”. In particular, α takes not only half-integer values, as in the Gaussian case, but a whole half-line of values. The BFS-Dynkin type isomorphism involves polynomials defined by a recurrence with a structure similar to that of the Schwinger- Dyson equation for GβE. These polynomials also give the symmetric moments of the β-Dyson’s Brownian motion. We further ask the question whether an analogue of GβE and β-Dyson’s Brownian motion could exist on electrical networks and interpolate and extrapolate the distributions of the eigen- values in matrix valued GFFs. Our motivation for this is that such analogues could be related to of loops, in particular to those of non half-integer intensity parameter. If the underlying graph is a tree, the construction of such analogues is straightfor- ward, by taking β-Dyson’s Brownian motions along each branch of the tree. However, if the graph contains cycles, this is not immediate, and one does not expect a for the obtained fields. However, in the simplest case n 2, we provide a construction working on any graph. “

Key words and phrases. Dyson’s Brownian motion, Gaussian beta ensembles, Gaussian free field, isomorphism theorems, local time, permanental fields, topological expansion . 1 Our article is organized as follows. In Section 2 we recall the BFS-Dynkin and the Le Jan’s isomorphisms in the particular case of 1D Brownian motion. In Section 3 we recall the definition of Gaussian beta ensembles and the corresponding Schwinger-Dyson equation. In Section 4 we give the recurrence on polynomials that will be used for the BFS-Dynkin type isomorphism. The Section 5 deals with β-Dyson’s Brownian motion and the corresponding isomoprhims. Section 6 deals with general electrical networks. We give our construction for n 2 and ask our questions for n 3. “ ě 2. Isomorphism theorems for 1D Brownian motion x Let Bt t 0 be the standard Brownian motion on R. L will denote the Brownian local times: p q ě t x 1 L Bs 0 s t lim 1 Bs x εds. pp q ď ď q“ ε 0 2ǫ | ´ |ă Ñ ż0 R R p t,x,y will denote the heat kernel on , and pR` t,x,y the heat kernel on with condition p q p q ` 0 in 0: 2 1 py´xq p t,x,y e´ 2t , pR t,x,y p t,x,y p t,x, y . p q“ ?2πt ` p q“ p q´ p ´ q t,x,y t,x,y P will denote the probability from x to y in time t, and PR (for p¨q ` p¨q x,y 0) the probability measures where one conditions Pt,x,y on that the bridge does not ą 1 d2 p¨q R R hit 0. G ` x,y x,y 0 be the Green’s function of 2 dx2 on with 0 condition in 0, and for ě 2 ` p p qq 1 d K 0, GK x,y x,y 0 the Green’s function of 2 K on R: ą p p qq ě 2 dx ´ `8 GR` x,y 2x y pR` t,x,y dt, p q “ ^ “ ż0 p q

1 ?2K y x `8 Kt GK x,y e´ | ´ | p t,x,y e´ dt. p q “ ?2K “ ż0 p q x,y x,y Let µR x,y 0, resp. µK x,y R be the following measures on finite duration paths: p ` q ą p q P x,y `8 t,x,y x,y `8 t,x,y Kt (2.1) µ : P pR t,x,y dt, µ : P p t,x,y e dt. R` R` ` K ´ p¨q “ ż0 p¨q p q p¨q “ ż0 p¨q p q x,y x,y x,y x,y The total mass of µR , resp. µ , is GR x,y , resp. GK x,y . The image of µR , resp. µ , ` K ` p q p q ` K by time reversal is µy,x, resp. µy,x. R` K Let Tx denote the first hitting time of a level x by the Brownian motion Bt t 0. γ will R x,y x,y p q ě denote a generic path of . Let µˇ x y R, resp. µˇK x y R be the following measures on paths from x to y: p p¨qq ă P p p¨qq ă P x,y E x,y E KTx µˇ F γ B0 y F BTx t 0 t Tx , µˇK F γ B0 y e´ F BTx t 0 t Tx . p p qq “ “ r pp ´ q ď ď qs p p qq “ “ pp ´ q ď ď q x,y “ x,y ‰ µˇ has total mass 1 (probability measure), whereas the total mass ofµ ˇK is

KTx ?2K y x GK x,y EB0 y e´ e´ | ´ | p q . “ “ “ GK x,x “ ‰ p q For 0 x y z, the measure µx,z can be obtained as the image of the product measure R` x,y ăy,z ď ă µR µˇ under the concatenation of two paths. Similarly, for x y z R, the measure ` b ď ă P µx,z is the image of µx,y µˇy,z under the concatenation of two paths. K K b K Let W x x R denote a two-sided Brownian motion, i.e. W x x 0 and W x x 0 being two independentp p qq P standard Brownian motion started from 0p (Wp 0qq ě 0). Notep p´ thatqq hereě x is rather a one-dimensional space variable then a time variable. dWp q “x is a on R. p q Let φR` x x 0 denote the process ?2W x x 0. The covariance function of φR` is GR` . Let p p qq ě p p qq ě φK x x R be the stationary Ornstein–Uhlenbeck process with invariant measure N 0, 1 ?2K . Itp isp aqq solutionP to the SDE p { q

dφK x ?2dW x ?2KφK x dx. p q“ p 2q´ p q The covariance function of φK is GK . What follows is the BFS-Dynkin isomorphism (Theorem 2.2 in [BFS82], Theorems 6.1 and 6.2 in [Dyn84a], Theorem 1 in [Dyn84b]) in the particular case of a 1D Brownian motion. In general, the BFS-Dynkin isomorphism relates the squares of Gaussian free fields to local times of symmetric Markov processes. Theorem 2.1 (Brydges-Fr¨ohlich-Spencer [BFS82], Dynkin [Dyn84a, Dyn84b]). Let F be a bounded measurable functional on C R , resp. on C R . Let k 1 and x1,x2,...,x2k in 0, , resp. in R. Then p `q p q ě p `8q 2k k 2 2 xai ,xbi E φR xi F φR 2 E F φR 2 L γ1 L γk µR dγi , ` p q p ` { q “ ż p ` { ` p q`¨¨¨` p qq ` p q ” źi 1 ı a ,bÿ1 ” ı źi 1 i i ďiďk γ1,...,γk “ partitionpt uq in pairs “ of J1,2kK resp. 2k k 2 2 xai ,xbi E φK xi F φ 2 E F φ 2 L γ1 L γk µ dγi , p q p K { q “ ż p K { ` p q`¨¨¨` p qq K p q ” źi 1 ı a ,bÿ1 ” ı źi 1 i i ďiďk γ1,...,γk “ partitionpt uq in pairs “ of J1,2kK k where the sum runs over the 2k ! 2 k! partitions in pairs, γi-s are Brownian paths and L γi p q {p q x p q are the corresponding occupation fields x L γi . ÞÑ p q x,y x,y x,x x,y Remark 2.2. Since for x y, the measure µR , resp. µ , can be decomposed as µR µˇ , ă ` K ` b resp. µx,x µˇx,y, Theorem 2.1 can be rewritten using only the measures of type µx,x and µˇx,y, K K R` x,x b x,y resp. µK and µˇK .

To a wide class of symmetric Markov processes one can associate in a natural way an infinite, σ-finite measure on loops [LW04, LTF07, LL10, LJ10, LJ11, LJMR15, FR14]. It originated from the works in mathematical physics [Sym65, Sym66, Sym69, BFS82]. Here we recall it in the setting of a 1D Brownian motion, which has been studied in [Lup18]. Of course, the range on a loop will be just a segment on the line, but it will carry a non-trivial Brownian local time process which will be of interest for us. Given a Brownian loop γ, T γ will denote its duration. The measures on (rooted) loops are p q loop 1 x,x loop 1 x,x (2.2) µR dγ : µR dγ dx, µK dγ µK dγ dx. ` p q “ T γ żR ` p q p q“ T γ żR p q p q ` p q Usually one considers unrooted loops, but this will not be important here. The 1D Brownian loop-soups are the Poisson point processes, denoted Lα , resp. Lα , of intensity αµloop, resp. R` K R` loop α α αµ , where α 0 is an intensity parameter. L LR , resp. L L , will denote the occupation K ą p ` q p K q field of Lα , resp. Lα : R` K x Lα x x Lα x L R` : L γ , L K : L γ . p q “ α p q p q “ α p q γ ÿLR γÿL P ` P K α α The following statement deals with the law of L LR , resp. L L . See Proposition 4.6, p ` q p K q Property 4.11 and Corollary 5.5 in [Lup18]. For the analogous statements in discrete space setting, see Corollary 5, Proposition 6, Theorem 13 in [LJ10] and Corollary 1, Section 4.1, Proposition 16, Section 4.2, Theorem 2, Section 5.1 in [LJ11] In general, one gets α-permanental fields (see also [LJMR15, FR14]). For α 1 in particular, one gets square Gaussians. We recall “ 2 that given a matrix M Mij 1 i,j k, its α-permanent is “ p q ď ď k # cycles of σ Permα M : α Miσ i . p q “ p q σ permutationÿ źi 1 of 1,2,...,k “ t u 3 Theorem 2.3 (Le Jan [LJ10, LJ11], Lupu [Lup18]). For every α 0 and x R , resp. x α x α ą P 1` x R, the r.v. L L , resp. L L , follows the distribution Gamma α, GR x,x , resp. R` K ` ´ P p 1 q p q x α px α p q q Gamma α, GK x,x ´ . Moreover, the process α L LR , resp. L L is a pure jump p p q q ÞÑ p ` q p Kq Gamma subordinator with L´evy measure

l GR x,x l GK x,x e´ { ` p q e´ { p q 1l 0 dl, resp. 1l 0 dl. ą l ą l Let x1,x2,...,xk R , resp. R. Then P ` k k E xi Lα E xi Lα L R Permα GR` xi,xj 1 i,j k , L K Permα GK xi,xj 1 i,j k . p ` q “ p q ď ď p q “ p q ď ď ” źi 1 ı ” źi 1 ı “ ` ˘ “ ` ˘ x α For x 0, x L LR is a solution to the SDE ě ÞÑ p ` q 1 x α x α 2 dL LR 2 L LR dW x 2αdx, p ` q“ p ` q p q` with initial condition L0 Lα 0. That` is to say˘ it is a square Bessel process of dimension R` R x pα q “ 2α. For x x L LK is a stationary solution to the SDE P ÞÑ p q 1 dLx Lα 2 Lx Lα 2 dW x 2?2KLx Lα 2αdx. p Kq“ p K q p q´ p Kq` In particular, for α 1 , one has` the following˘ identities in law between stochastic processes: “ 2 α law 1 2 α law 1 2 (2.3) L LR p q φR , L L p q φ . p ` q “ 2 ` p K q “ 2 K 3. Gaussian beta ensembles For references on Gaussian beta ensembles, see [DE02, For15], Section 1.2.2 in [EKR18], n and Section 4.5 in [AGZ09]. Fix n 2. For λ λ1, λ2, . . . , λn R , D λ will denote the Vandermonde determinant ě “ p q P p q D λ : λj1 λj . p q “ p ´ q 1 jźj1 n ď ă ď For q 1, pq λ will denote the q-th power sum polynomial ě p q n q pq λ : λ . p q “ j jÿ1 “ By convention, p0 λ n. p q“ A Gaussian beta ensemble GβE follows the distribution n 1 1 β p2 λ (3.1) D λ e´ 2 p q dλj, Z | p q| β,n jź1 “ where Zβ,n is given by (Formula (17.6.7) in [Meh04] and Formula (1.2.23) in [EKR18]) n β n Γ 1 j 2 Zβ,n 2π 2 ` . “ p q Γ` 1 β ˘ jź1 ` 2 “ ` ˘ The brackets β,n will denote the expectation with respect to (3.1). For β 0 one gets n i.i.d. N 0, 1 Gaussians.x¨y For β equal to 1, 2, resp. 4, one gets the eigenvalue distribution“ of GOE, GUE,p q resp. GSE random matrices [Meh04, EKR18]. Usually the GβE are studied for β 0 2 2 ą [DE02], but the distribution (3.1) is well defined for all β n . For β n , 0 there is an ą ´ P p´ q 2 attraction between the λj-s instead of a repulsion as for β 0. Moreover, as β n , λ under (3.1) converges in law to ą Ñ´ 1 1 1 ξ, ξ,..., ξ , ´?n ?n ?n ¯ where ξ follows N 0, 1 . p q 4 Let d β,n denote p q β d β,n n n n 1 . p q“ ` p ´ q 2 For β 1, 2, 4 , d β,n is the dimension of the corresponding space of matrices. P t u p q Let be ν ν1,ν2,...,νm , where m 1, and for all k 1, 2,...,m , νk N 0 . We will denote “ p q ě P t u P zt u m ν p q m ν m, ν νk. p q“ | |“ kÿ1 “ pν λ will denote p q m ν p q pν λ : pν λ . p q “ k p q kź1 “ By convention, we set p λ 1 and 0. Note that p λ p0 λ . We are interested in Hp q“ |H| “ Hp q‰ p q the expression of the moments pν λ β,n. These are 0 if ν is not even. For ν even, these mo- ments are given by a recurrencex knownp qy as loop equation or| | Schwinger-Dyson| equation| (Lemma 4.13 in [LC09], slide 3 15 in [LC13] and Section 4.1.1 in [EKR18]). For the combinatorial in- terpretation of the solutions{ in terms of ribbon graphs or surfaces, see [LC13, LC09]. See the Appendix for the expression of some moments. Proposition 3.1 (Schwinger-Dyson equation [LC09, LC13, EKR18]). For every β 2 n and every ν as above with ν even, ą´ { | | ν 1 β mpνq´ (3.2) pν λ β,n p νr λ pi 1 λ pνmpνq 1 i λ β,n x p qy “ 2 x p qr‰mpνq p q ´ p q ´ ´ p qy iÿ1 “ β 1 νm ν 1 p νr r‰mpνq λ pνmpνq 2 λ β,n `´ ´ 2 ¯p p q ´ qx p q p q ´ p qy m ν 1 p q´ νk p νr λ pνk νmpνq 2 λ β,n, ` x p qr‰k,mpνq p q ` ´ p qy kÿ1 “ where p0 λ n. In particular, for q even, p q“ q 1 β ´ β pq λ β,n pi 1 λ pq 1 i λ β,n 1 q 1 pq 2 λ β,n, x p qy “ 2 x ´ p q ´ ´ p qy ` ´ 2 p ´ qx ´ p qy iÿ1 ´ ¯ “ and for ν with νm ν 1, p q “ m ν 1 p q´ pν λ β,n νk p νr λ pνk 1 λ β,n. x p qy “ x p qr‰k,mpνq p q ´ p qy kÿ1 “ The recurrence (3.2) and the initial condition p0 λ n determine all the moments pν λ β,n. p q“ x p qy Proof. Note that (3.2) determines the moments pν λ β,n because on the left-hand side one has a degree ν , and on the right-hand side all thex termsp qy have a degree ν 2. For a proof of (3.2) for β |0,| see Lemma 4.13 in [LC09] and Section 4.1.1 in [EKR18].| For|´β 2 n, 0 , the proof worksą the same, with some care about the divergences in the density. Alternatively,P p´ { q on can use the analiticity in β to extend to β 2 , 0 .  P p´ n q Next are some elementary properties of GβE, which follow from the form of the density (3.1). Proposition 3.2. The following holds. 1 (1) For every β 2 n, p1 λ under GβE has for distribution N 0, 1 . ą´ { ?n p q p q (2) For every β 2 n, p2 λ 2 under GβE has for distribution Gamma d β,n 2, 1 . ą´1 { p q{ p p q{ q (3) p1 λ and λ n p1 λ under GβE are independent. 1 p q 1 ´ 2 p q 1 1 (4) p2 λ p1 λ p2 λ p1 λ under GβE has for distribution Gamma d β,n 2 p q´ n p q “ 2 ´ n p q pp p q´ 1` 2, 1 . ˘ ` ˘ q{ q 5 Next is an embryonic version of the BFS-Dynkin isomorphism (Theorem (2.1)) for the GβE. One should imagine that the state space is reduced to one vertex, and a particle on it gets killed at an exponential time. Proposition 3.3. Let β 2 n. The following holds. ą´ n{ (1) Let a 0. Let h : R R be a measurable function such that h λ β,n . Assumeě that h is a-homogeneous,Ñ that is to say h sλ sah λ forx| everyp q|ys ă0. `8 Let F : 0, R be a bounded measurable function.p q Let “ θ bep aq r.v. with distributioną Gammar `8qd β,n Ñ a 2, 1 . Then pp p q` q{ q (3.3) h λ F p2 λ 2 β,n h λ β,nE F θ . x p q p p q{ qy “x p qy r p qs (2) In particular, let ν be a finite family of positive integers such that ν is even. Let | | T1,..., T ν 2 be an i.i.d. family of exponential times of mean 1, independent of the GβE. Then| |{

pν λ F p2 λ 2 β,n pν β,nE F p2 λ 2 T1 T ν 2 β,n . x p q p p q{ qy “x y x p p q{ ` `¨¨¨` | |{ qy “ ‰ Kt Proof. (1) clearly implies (2). It is enough to check (3.3) for F of form F t e´ , with K 0. Then p q “ ą n 1 1 Kp2 λ 1 β K 1 p2 λ h λ e´ 2 p q β,n h λ D λ e´ 2 p ` q p q dλj x p q y “ Z Rn p q| p q| β,n ż jź1 “ n n 2 1 1 1 K 1 ´ β p2 λ˜ p ` q h K 1 ´ 2 λ˜ D K 1 ´ 2 λ˜ e´ 2 p q dλ˜j “ Z Rn pp ` q q| pp ` q q| β,n ż jź1 “ 1 β 2 n n n 1 2 a K 1 ´ ` p ´ q ` h λ˜ β,n, “ p ` q x p qy ` ˘ 1 where on the second line we used the change of variables λ˜ K 1 2 λ, and on the third line the homogeneity. Further, “ p ` q

1 n n n 1 β a Kθ K 1 ´ 2 ` p ´ q 2 ` E e´ .  p ` q ` ˘ “ r s 4. A recurrence on formal polynomials

We consider a family of formal commuting polynomial variables Ykk, Ykk 1 k 1. We will p ` q ě consider finite families of positive integers ν ν1,ν2,...,νm ν with ν even. The order of the “ p p qq | | q νk will matter. We want to construct a family of polynomials Pν,β,n with parameters ν,β and n, where Pν,β,n has for variables Ykk 1 k m ν and Yk 1 k 2 k m ν . To simplify the notations, we will drop the subscripts β,np andq ď justď p writeq Pp .´ Theq ď polynomialsď p q P will appear in the qν ν expression of the symmetric moments of β-Dyson’s Brownian motion and the corresponding BFS-Dynkin type isomorphism. We will give a recursive definition of the Pν-s. The solutions to the recurrence (3.2), which for β 2 n, are the moments pν λ β,n, will be denoted c ν,β,n . By convention, c 0 ,β,nP p´n{and`8qc ,β,n 1. For kx pk qy N, p q pp q q “ pH q “ ě 1 P Jk, k1K will denote the interval of integers

Jk, k1K k, k 1,...,k1 . “ t ` u k For k 1 and P a polynomial, P Ð will denote the polynomial in the variables Yk1k1 1 k1 k and ě p q ď ď Yk1 1 k1 2 k1 k, obtained from P by replacing each variable Yk1k1 with k1 k 1 by the variable p ´ q ď ď ě ` m ν Ykk, and each variable Yk1 1 k1 with k1 k 1 by the constant 1. Note that Pν p qÐ Pν and q 1 ´ ě ` “ that P is an univariate polynomial in Y . For Y a formal polynomial variable, degY will ν Ð q 11 denote the partial degree in Y.

Definition 4.1. The family of polynomials Pν ν even is defined by the following. p q| | 1 Y ν 2 (1) Pν Ð c ν,β,n 11| |{ . “ p q 6 (2) If m ν 2, then for every k J2,m ν K, p qě P p q ν 1 2 β ν k k ´ P k 1 P k (4.1) B ν Ð p q Ðνr 1 ,i 1,ν 1 1 i Ykk “ 2 2 pp qr‰k ´ k ´ ´ q k kÿ1 m ν iÿ2 B “ ďν ď1 2p q k ą β n ν k P k 1 Ðνr 1 ,ν 1 2 ` 2 p q pp qr‰k k ´ q k kÿ1 m ν ďν ď1 2p q k ą β n2 P k νÐr 1 ` 2 p qr‰k k kÿ1 m ν ďν ď1 2p q k “ β ν 1 ν 1 1 k k P k 1 p ´ q Ðνr 1 ,ν 1 2 ` ´ 2 2 pp qr‰k k ´ q ´ ¯ k kÿ1 m ν ďν ď1 2p q k ą β n P k 1 νÐr 1 ` ´ 2 p qr‰k ´ ¯ k kÿ1 m ν ďν ď1 2p q k “ k ν 1 ν 2 P k k Ðνr 1 2 ,ν 1 ν 2 2 ` pp qr‰k ,k k ` k ´ q k k1 ÿk2 m ν ďν 1ă ν ď2 2p q k ` k ą n P k . νÐr 1 2 ` p qr‰k ,k k k1 ÿk2 m ν ďν 1ă ν ď2 1p q k “ k “ If k m ν , then the last two lines of (4.1) are zero. “ p q k Y (3) If m ν 2, then for every k J2,m ν K, the polynomial Pν Ð kk 0 (i.e. the part of k p qě P pY q p “ q Pν Ð that does not contain terms in kk) is such that each of its monomials Q satisfies

degY Q νr. k´1 k “ k rÿm ν q ď ď p q

Proposition 4.2. Definition 4.1 uniquely defines a family of polynomials Pν ν even. Moreover, the following properties hold. p q| |

(1) For every Q monomial of Pν and every k J2,m ν K, P p q degY Q 2 degY Q νk1 , k´1 k ` k1k1 “ k k1ÿm ν k kÿ1 m ν q ď ď p q ď ď p q and 2 degY Q ν . k1k1 “ | | 1 k1ÿm ν ď ď p q (2) For every k J1,m ν K and every permutation σ of Jk,m ν K, P p q p q P k P k . νÐr 1 r k 1, ν ν Ð p q ď ď ´ p σprqqkďrďmpνq “ Proof. The fact that the polynomials Pν are well defined can be proved by induction on ν 2. | |{ For ν 2 1, there are only two polynomials, P 2 and P 1,1 . According to the condition | |{ “ p q p q (1), β 2 β P 2 c 2 ,β,n Y11 d β,n Y11 n 1 n Y11. p q “ pp q q “ p q “ ´ 2 ` ´ ´ 2 ¯ ¯ The conditions (2) and (3) do not apply for P 2 . For P 1,1 , according to the condition (2), p q p q

B P 1,1 0. Y22 p q “ B 7 Thus, P 1,1 contains no terms in Y22. According to the condition (3), P 1,1 P Y11 Y12. From p q p q “ p q the condition (1) we further get r q Y Y Y Y P 1,1 c 1, 1 ,β,n 11 12 n 11 12. p q “ pp q q “ The induction step works as follows. Assume ν 2q 2. The rightq hand side of (4.1) involves | |{ ě k only families of integersν ˜ with ν˜ ν 2. According to the induction hypotheses, B Pν Ð is | | “ | |´ Ykk k kB Y uniquely determined for every k J2,m ν K. Thus, for every k J2,m ν K, Pν Ð Pν Ð kk 0 P p q 1 P p q ´ p “ q is uniquely determined. By the condition (1), Pν Ð is also uniquely determined. By the condition (3), for every k J2,m ν K, P p q k 1 νr k Y k 1 k k Y Y|p qkďrďmpνq| Pν Ð kk 0 Pν ´ Ð Pν Ð Pν Ð kk 0 ´ Ð k 1 k . p “ q“ ´ ´ p “ q ´ ` k ` ˘ ˘ Thus, all the polynomials Pν Ð 1 k m ν are uniquely determined,q with consistency by the p q ď ď p q k m ν P P Ð operations. Finally, Pν Pν p qÐ. ÞÑThe properties (1) and (2) again“ follow easily by induction on ν 2.  | |{ Next are the expressions for P 1,1,...,1 and P 2,2,...,2 that can be proved by induction. p q p q Proposition 4.3. Let m N 0 . Let M Mkk1 1 k,k1 m be the formal symmetric matrix with entries given by P zt u “ p q ď ď

(4.2) Mkk Ykk, for k k1, Mkk1 Mk1k Ykk Yr 1 r. “ ă “ “ ´ k 1źr k1 ` ď ď q The following holds.

(1) Assume m is even, and let ν 1, 1,..., 1 , where 1 appears m times. Then P 1,1,...,1 satisfies the Wick’s rule for Gaussians:“ p q p q m 2 M P 1,1,...,1 n aibi , p q “ ai,bi ÿ1ďiďm{2 partitionpt uq in pairs of J1,mK

m where the sum runs over the m! 2 2 m 2 ! partitions in pairs. (2) Let ν 2, 2,..., 2 , where 2 appears{p pm{times.q q Then “ p q m P 2,2,...,2 2 Permd β,n 2 M . p q “ p q{ p q For other examples of Pν, see the Appendix. 2 Next we observe that for β , the polynomials Pν give the moments of the stochastic “ ´ n processes φR` x x 0 and φK x x R introduced in Section 2, which are Gaussian. p p qq ě p p qq P Proposition 4.4. Let n 1. Let K 0. Let ν be a finite family of positive integers with ν ě ą | | even. Let x1 xm ν be m ν points in 0, , resp. in R. Then 﨨¨ď p q p q p `8q m ν 2 p q 2 Y Y m ν ν E νk Pν,β ,n kk 2xk 1 k m ν , k 1 k 1 2 k m ν n p q´| |{ φR` xk , n pp “ q ď ď p q p ´ “ q ď ď p qq“ p q “´ ” kź1 ı q “ resp.

?2K x x 1 2 Y ? Y k k´ Pν,β ,n kk 1 2K 1 k m ν , k 1 k e´ p ´ q 2 k m ν “´ n pp “ { q ď ď p q p ´ “ q ď ď p qq m ν q p q m ν ν 2 νk n p q´| |{ E φK xk . “ p q ” kź1 ı “ Y That is to say, the variables kk are replaced by GR` xk,xk , resp. GK xk,xk , and the variables Y p q p q k 1 k by GR` xk 1,xk GR` xk 1,xk 1 , resp. GK xk 1,xk GK xk 1,xk 1 . ´ ´ ´ ´ ´ ´ ´ p q{ p q 8 p q{ p q q Proof. First, one can check that

2 m ν ν 2 ν ! (4.3) c ν, β ,n n p q´| |{ | | . “´n “ 2 ν 2 ν 2 ! ´ ¯ | |{ p| |{ q This follows from Proposition (3.2). The key point is that 2 d β ,n 1. ´ “´n ¯ “ Given ν a finite family of positive integers, let kν : J1, ν K J1,m ν K be the function such that | | ÞÑ p q 1 1 (4.4) kν´ 1 J1,ν1K, for k1 J2,m ν K, kν´ k1 Jν1 νk1 1 1,ν1 νk1 K. p q“ P p q p q“ `¨¨¨` ´ ` `¨¨¨`

Further, let Pν ν even be the following formal polynomials: p q| | r m ν ν 2 M Pν n p q´| |{ kν ai kν bi , “ p q p q ai,bi ÿ1ďiď|ν|{2 r ptpartitionuq in pairs of J1, ν K | | M where the kk1 are given by (4.2). To conclude, we need only to check that Pν Pν,β 2 ,n for “ “´ n all ν with ν even. From (4.3) follows that the Pν satisfy the condition (1)r in Definition 4.1. | | The condition (3) is immediate since P k Y 0 corresponds to the partitions in pairs where ν Ð kk r each element of k 1 Jk,m ν K is pairedp with“ anq element of k 1 J1, k 1K . One can further ν´ r ν´ check the recurrencep (4.1),p andq q this amounts to counting the pairsp in k´1 Jk,mq ν K .  ν´ p p q q

5. Isomorphisms for β-Dyson’s Brownian motion 5.1. β-Dyson’s Brownian motions and the occupation fields of 1D Brownian loop- soups. For references on β-Dyson’s Brownian motion, see [Dys62, Cha92, RS93, CL97, CL07], Chapter 9 in [Meh04] and Section 4.3 in [AGZ09]. Let β 0 and n 2. The β-Dyson’s ě ě Brownian motion is the process λ x λ1 x , . . . , λn x x 0 with λ1 x λn x , satisfying the SDE p p q “ p p q p qqq ě p q ě ¨¨¨ ě p q dx (5.1) dλj x ?2dWj x β , p q“ p q` λ x λ 1 x jÿ1 j j j ‰ p q´ p q with initial condition λ 0 0. dWj x 1 j n are independent white noises. Since we will be interested in isomorphismsp q “ with Brownianp p qq ď localď times, the variable x corresponds here to a one- dimensional spatial variable rather then a time variable. For every x 0, λ x GR x,x ą p q{ ` p q“ λ x ?2x, is distributed, up to a reordering of the λj x -s, as a GβE (3.1). Fora β equal to p q{ p q 1, 2 resp. 4, λ x x 0 is the diffusion of eigenvalues in a Brownian motion on the space of real symmetric,p p complexqq ě Hermitian, resp. quaternionic Hermitian matrices. For β 1, there ě is no collision between the λj x -s, and for β 0, 1 two consecutive λj x -s can collide, but there is no collision of threep orq more particlesP [CL07].r q Note that for βp q 0 and j J2,nK, ą P λj x λj 1 x 2 behaves near level 0 like a Bessel process of dimension β 1 reflected at plevelp 0,q´ and´ sincep qq{β 1 1, the complication with the principal value and the local` time at zero ` ą does not occur; see Chapter 10 in [Yor97]. In particular, each λj x x 0 is a . p p qq ě For β 0, λ x ?2 x 0 is just a reordered family of n i.i.d. standard Brownian motions. “ p p q{ q ě Remark 5.1. We restrict to β 0 because the case β 0 has not been considered in the ě ă literature. The problem is the extension of the process after a collision of λj x -s. The collision 2 n 3 p q of three or more particles, including all the n together for β npn´1q , is no longer excluded. ă ´ p ´ q 2 However, we believe that the β-Dyson’s Brownian motion can be defined for all β n . This 9 ą ´ is indeed the case if n 2. One can use the reflected Bessel processes for that. Let ρ x x 0 be the Bessel process of“ dimension β 1, reflected at level 0, satisfying away from 0 thep p SDEqq ě ` β dρ x dW x dx, p q“ p q` 2ρ x p q 2 with ρ 0 0. The reflected version is precisely defined for β 1 ´ ; see Section XI.1 in p q “ ą ´ “ 2 [RY99] and Section 3 in [Law19]. Let W x x 0 be a standard Brownian motion started from p p qq ě 0, independent from W x x 0 Then, for n 2 one can construct the β-Dyson’s Brownian ě Ă motion as p p qq “

(5.2) λ1 x W x ρ x , λ2 x W x ρ x . p q“ p q` p q p q“ p q´ p q Next are some simple propertiesĂ of the β-Dyson’s BrownianĂ motion. Proposition 5.2. The following holds. 1 (1) The process p1 λ x has same law as φR . ?n p p qq x 0 ` `1 ˘ ě (2) The process 2 p2 λ x x 0 is a square Bessel process of dimension d β,n started from 0. p p p qqq ě p q 1 (3) The processes p1 λ x x 0 and λ x n p1 λ x x 0 are independent. p1 p p qqq ě 1 p q´2 p p qq ě (4) The process p2 λ x p1 λ` x is a square˘ Bessel process of dimension 2 p p qq ´ n p p qq x 0 d β,n 1 started` ` from 0. ˘˘ ě p q´ Proof. With Itˆo’s formula, we get n dp1 λ x ?2 dWj x , p p qq “ p q jÿ1 “ n 1 λj x d p2 λ x 2 p qdWj x d β,n dx, 2 p p qq “ ? p q` p q jÿ1 2 “ n 1 1 1 2 λj x n p1 λ x (5.3) d p2 λ x p1 λ x 2 p q´ p p qq dWj x d β,n 1 dx, 2 p p qq ´ n p p qq “ ? p q ` p p q´ q ´ ¯ jÿ1 2 “ where the points x R for which λj x λj 1 x for some j J2,nK can be neglected. This gives (1), (2) and (4)P since` the processesp q “ ´ p q P n λj x dW x p q dWj x , W 0 0, p q“ p2 λ x p q p q“ jÿ1 p p qq Ă “ a Ă and n 1 λj x n p1 λ x dW x p q´ p p qq dWj x , W 0 0, p q“ 1 2 p q p q“ jÿ1 p2 λ x n p1 λ x | “ b p p qq ´ p p qq | are both standard Brownian motions. Again, one can neglect the points x R where p2 λ x 1 2 P ` p p qq´ n p1 λ x 0, which only occur for n 2. Forp p (3),qq we“ have that “ 1 1 d λj x p1 λ x ?2d Wj x p1 W x ´ p q´ n p p qq¯ “ ´ p q´ n p p qq¯ dx β , ` 1 1 jÿ1 j λj x p1 λ x λj1 x p1 λ x ‰ p q´ n p p qq ´ p q´ n p p qq ` ˘ ` ˘ where n p1 W x Wj1 x . p p qq “ p q jÿ1 1 “ 10 The Brownian motion p W 1 p λ is independent from the family of Brownian motions 1 ?2 1 1 p q “ p q 1 Wj n p1 W 1 j n. Further, the measurability of λj n p1 λ 1 j n with respect to Wj 1 ´ p q ď ď ´ p q ď ď ´ ` p1 W ˘follows from the pathwise uniqueness` of the solution˘ to (5.1); see Theorem` 3.1 n p q 1 j n in [CL97].˘ ď ď  By combining Proposition 5.2 with Theorem 2.3, we get a first relation between the β-Dyson’s Brownian motion and 1D Brownian local time. Compare it with Le Jan’s isomorphism (2.3). 1 x Lα Corollary 5.3. The process 2 p2 λ x has same law as the occupation field L R x 0 p p qq x 0 p p ` qq ě of a 1D Brownian loop-soup Lα , with theě correspondence ` R` ˘ β (5.4) 2α d β,n n n n 1 . “ p q“ ` p ´ q 2 α 1 1 Further, let L ´ 2 and L 2 be two independent 1D Brownian loop-soups, α still given by (5.4). R` R` Then, one has the following identity in law between pairs of processes: r 1 1 1 1 2 1 2 (law) x α 2 x 2 p2 λ x p1 λ x , p1 λ x L LR´ ,L LR x 0. 2 p p qq ´ n p p qq 2n p p qq x 0 “ p p ` q p ` qq ě ´ ´ ¯ ¯ ě r R 5.2. Symmetric moments of β-Dyson’s Brownian motion. We will denote by ` the x¨yβ,n expectation with respect the β-Dyson’s Brownian motion (5.1).

Proposition 5.4. Let ν be a finite family of positive integers, with ν even. Let Pν Pν,β,n | | “ be the polynomial given by Definition 4.1. Let x1 x2 xm ν R . Then, ď 﨨¨ď p q P ` m ν R p q ` Y Y pνk λ xk Pν kk 2xk 1 k m ν , k 1 k 1 2 k m ν . p p qq β,n “ pp “ q ď ď p q p ´ “ q ď ď p qq A kź1 E “ q We start by some lemmas. Lemma 5.5. Let q 3. Then ě n q 2 q 1 β ´ dpq λ x q?2 λj x ´ dWj x q pi 1 λ x pq 1 i λ x dx p p qq “ p q p q` 2 ´ p p qq ´ ´ p p qq jÿ1 iÿ2 “ “ β β 2 nqpq 2 λ x dx 1 q q 1 pq 2 λ x dx. ` 2 ´ p p qq ` ´ ´ 2 ¯ p ´ q ´ p p qq Proof. By Itˆo’s formula, n q 1 dpq λ x q?2 λj x ´ dWj x q q 1 pq 2 λ x dx p p qq “ p q p q` p ´ q ´ p p qq jÿ1 “ q 1 q 1 λj x λj1 x βq p q ´ ´ p q ´ dx. ` λ x λ 1 x 1 jÿj1 n j j ď ă ď p q´ p q But

q 1 q 1 q 2 λj x ´ λj1 x ´ ´ r q 2 r p q ´ p q λj x λj1 x ´ ´ λ x λ 1 x “ p q p q “ 1 jÿj1 n j j 1 jÿj1 n rÿ0 ď ă ď p q´ p q ď ă ď “ q 2 q 1 1 ´ n ´ pq 2 λ x pi 1 λ x pq 1 i λ x .  ´ 2 ´ p p qq ` 2 ´ p p qq ´ ´ p p qq ´ ¯ iÿ2 “ Lemma 5.6. Let q,q 1 with q q 2. Then 1 ě ` 1 ą d pq λ x ,pq1 λ x 2qq1pq q1 2 λ x dx. x p p qq p p qqy “ ` ´ p p qq Moreover, d p1 λ x ,p1 λ x 2ndx. x p p qq p11 p qqy “ Proof. This is a straightforward computation.  Lemma 5.7. Let ν be a finite family of positive integers and let q 0. Then the process ě x n q (5.5) pν λ y λj y dWj y p p qq p q p q ż0 jÿ1 “ is a martingale. Proof. (5.5) is a . It actually has a locally L1 bounded : x R x 2 ` 2 ν q pν λ y p2q λ y dy pν λ p2q λ β,n 2y | |` dy A ż0 p p qq p p qq Eβ,n “x p q p qy ż0 p q ă `8 So it is a true martingale.  Proof of Proposition 5.4. The proof is done by induction on ν 2. The case ν 2 1 corresponds to ν 1, 1 or ν 2 . These| |{ are treated by Proposition 4.3 and Proposition| |{ 5.2,“ and taking into account“ p q that the“ p squareq Bessel processes are permanental fields. Now consider the induction step. Assume ν 2 2. Let x1 x2 xm ν R . For | |{ ě ď 﨨¨ď p q P ` k J1,m ν K, fk x will be the function P p q p q k 1 m ν R ´ p q ` fk x : pν 1 λ xk1 pν 1 λ x . p q “ k p p qq k p p qq β,n A kź1 1 kź1 k E “ “ We have that ν 2 1 (5.6) f1 x1 c ν,β,n 2x1 | |{ P Ð Y11 2x1 , p q“ p qp q “ ν p “ q where for the second equality we applied the condition (1) in Definition 4.1. If m ν 1, there is nothing more to check. In the case m ν 2, we need only to check that for everyp q“k J2,m ν K p qě P p q and every x xk 1, ą ´ d k Y Y Y (5.7) fk x B Pν Ð k1k1 2xk1 1 k1 k 1, kk 2x, k1 1 k1 1 2 k1 k dx p q “ x pp “ q ď ď ´ “ p ´ “ q ď ď q B k Y Y q Y 2 B Pν Ð k1k1 2xk1 1 k1 k 1, kk 2x, k1 1 k1 1 2 k1 k . “ Ykk pp “ q ď ď ´ “ p ´ “ q ď ď q ´B ¯ Indeed, given (5.6), by applying (5.7) to k 2, we further get q “ 2 f2 x2 P Ð Y11 2x1, Y22 2x2, Y12 1 , p q“ ν p “ “ “ q and by successively applying (5.7) to k 3,...,k m ν , weq at the end get “ “ p q m ν Y Y fm ν xm ν Pν p qÐ k1k1 2xk1 1 k1 m ν , k1 1 k1 1 2 k1 m ν , p qp p qq“ pp “ q ď ď p q p ´ “ q ď ď p qq which is exactly what we want. To show (5.7), we proceedq as follows. Let Fx x 0 be the p q ě filtration of the Brownian motions Wj x 1 j n x 0. Then, for x xk 1, pp p qq ď ď q ě ą ´ k 1 m ν R R ´ p q ` ` 1 F fk x pν 1 λ xk pν 1 λ x xk 1 , p q“ k p p qq k p p qq ´ β,n β,n A kź1 1 A kź1 k ˇ E E “ “ ˇ R` ˇ where Fx 1 denotes the conditional expectation. To express x¨| k´ yβ,n m ν p q R` F pν 1 λ x xk 1 , k p p qq ´ β,n A kź1 k ˇ E “ ˇ we apply Itˆo’s formula to ˇ m ν m ν p q p q R` pν 1 λ x pν 1 λ xk 1 . k p p qq ´ k p p ´ qq β,n kź1 k A kź1 k E “ 12“ The local martingale part is, according to Lemma 5.7 a true martingale, and thus gives a 0 conditional expectation. The bounded variation part is a linear combination of terms of form pν˜ λ x dx, with p p qq m ν p q ν˜ νk1 2, | |“ ´ ´ kÿ1 k ¯ “ the exact expressions being given by Lemma 5.5 and Lemma 5.6. By comparing these expres- sions with the recurrence (4.1), and using the induction hypothesis at the step ν 2 1, we get (5.7). | |{ ´  Note that in the proof above we did not use the condition (3) in Definition 4.1. It will be needed only later. 5.3. BFS-Dynkin isomorphism for β-Dyson’s Brownian motion. We will denote by Υ a generic finite family of continuous paths on R,Υ γ1, . . . , γJ , and J Υ will denote the size J of the family. We will consider finite Brownian“ measures p onq Υ wherep Jq Υ is not fixed but may take several values under the measure. Given x R, Lx Υ will denotep q the sum of Brownian local times in x: P p q J Υ x p q x L Υ L γi . p q“ p q iÿ1 “ L Υ will denote the occupation field x Lx Υ . p q ÞÑ p q Given ν a finite family of positive integers with ν even and 0 x1 x2 xm ν , ν,x1,...,xmpνq | | ă ă ă ¨¨¨ ă p q µR dΥ (also depending on β and n) will be the measure on finite families of continuous ` p q paths obtained by substituting in the polynomial Pν Pν,β,n for each variable Ykk the measure xk,xk xk´“1,xk µR , and for each variable Yk 1 k the measureµ ˇR ; see Section 2. Since we will deal with ` ´ ` ν,x1,...,xmpνq the functional L Υ under µR dΥ , the order of the Brownian measures in a product p q q` p q will not matter. For instance, for ν 2, 1, 1 (see Appendix), “ p q β 3 β 2 Y Y Y Y2 Y2 Y P 2,1,1 n 1 n 11 22 23 2n 11 12 23, p q “ ´ 2 ` ´ ´ 2 ¯ ¯ ` and q q q

2,1,1 ,x1,x2,x3,x4 β 3 β 2 x1,x1 x2,x2 x2,x3 µpR q n 1 n µR µR µˇR ` “ 2 ` ´ 2 ` b ` b ` ´ x1,x´1 x1¯,x1 ¯ x1,x2 x1,x2 x2,x3 2nµR µR µˇR µˇR µˇR . ` ` b ` b ` b ` b ` Next is a version of BFS-Dynkin isomorphism (Theorem (2.1)) for β-Dyson’s Brownian mo- tion.

Proposition 5.8. Let ν be a finite family of positive integers, with ν even and let 0 x1 | | ă ă x2 xm ν . Let F be a bounded measurable functional on C R . Then 㨨¨ă p q p `q m ν R` R` p q 1 1 ν,x1,...,xmpνq (5.8) pν λ xk F p2 λ F p2 λ L Υ µR dΥ . k p p qq 2 p q β,n “ ż 2 p q` p q β,n ` p q A kź1 ´ ¯E Υ A ´ ¯E “ Remark 5.9. In the limiting case when xk xk 1 for some k J2,m ν K, Yk 1 k in Pν has to be replaced by the constant 1 instead of a measure“ ´ on BrownianP paths.p q ´ q Remark 5.10. For β 0, 1, 2, 4 , (5.8) reduces to the Gaussian case of Theorem (2.1). P t u We start by some intermediate lemmas. Recall that Fx x 0 denotes the filtration of the p q ě Brownian motions Wj x 1 j n x 0 in (5.1). χ x will be a continuous non-negative function pp p qq ď ď q ě p q with compact support in 0, . uχ will denote the unique solution to p `8q Ó 1 d2 u χu 2 dx “ 13 which is positive non-increasing on R , with uχ 0 1. See Section 2.1 in [Lup18] for details. Then ` Óp q“ uχ lim uχ x 0. Ó x Ó p`8q “ Ñ`8 p qą Lemma 5.11. Let Dχ be the positive r.v. p`8q 1 d β,n 1 `8 (5.9) Dχ : uχ ´ 2 p q exp p2 λ y χ y dy . p`8q “ Óp`8q ´ ´ 2 ż0 p p qq p q ¯ R` Then Dχ 1. Moreover, x p`8qyβ,n “ R` (5.10) Dχ x : Dχ Fx p q “x p`8q| yβ,n x 1 d β,n 1 1 uχ1 x uχ x ´ 2 p q exp p2 λ y χ y dy exp p2 λ x Óp q . “ Óp q ´ 2 ż0 p p qq p q 4 p p qquχ x ´ ¯ ´ Óp q¯ Let x n 1 uχ1 y Mχ x : Óp q λj y dWj y . p q “ ? u y p q p q 2 ż0 χ jÿ1 Óp q “ Then Mχ x x 0 is a martingale with respect the filtration Fx x 0 and for all x 0, p p qq ě p q ě ě 1 Dχ x exp Mχ x Mχ, Mχ x . p q“ ´ p q´ 2x yp q¯ Proof. (5.9) and (5.10) follow from the properties of square Bessel processes. See Theorem (1.7), Section XI.1 in [RY99]. Mχ x x 0 is obviously a (true) martingale, as can be seen with the quadratic variation. Further,p p qq ě 2 1 uχ1 x 1 1 uχ1 x 1 uχ1 x Óp q M Óp q Óp q d p2 λ x d χ x p2 λ x χ x dx p2 λ x 2 dx d β,n dx, 4 p p qquχ x “ p q` 2 p p qq p q ´ 4 p p qquχ x ` 2 p quχ x ´ Óp q¯ Óp q Óp q and 2 1 1 uχ1 x M M Óp q d χ, χ x p2 λ x 2 dx. 2x yp q“ 4 p p qquχ x Óp q Thus 1 d Mχ x Mχ, Mχ x d log Dχ x .  ´ p q´ 2x yp q¯ “ p p qq Lemma 5.12. Let be λ˜ x λ˜1 x ,..., λ˜n x x 0 with λ˜1 x λ˜n x , satisfying the SDE p p q “ p p q p qqq ě p q쨨¨ě p q

uχ1 x dx (5.11) dλ˜j x ?2dWj x Óp qλ˜j x dx β , p q“ p q` u x p q ` ˜ ˜ 1 χ jÿ1 j λj x λj x Óp q ‰ p q´ p q with initial condition λ˜ 0 0. Further consider a change of measure with density Dχ p q “ p`8q (5.9) on the filtered probability space with filtration Fx x 0. Then λ after the change of measure p q ě and λ˜ before the change of measure have same law. Proof. The existence and uniqueness of strong solutions to (5.11) is given by Theorem 3.1 in [CL97]. The rest is a consequence of Girsanov’s theorem; see Theorems (1.7) and (1.12), Section VIII.1, in [RY99]. Indeed,

1 uχ1 x d Wj x , Mχ x Óp qλj x dx. x p q p qy “ ?2 uχ x p q Óp q Thus, after the change of measure, the x 1 uχ1 y Wj x Óp qλj y dy p q´ ?2 ż0 uχ y p q Óp q for j J1,nK are n i.i.d. standard Brownian motions.  P 14 Let ψχ denote the following diffeomorphism of R : ` x dy ψχ x 2 . p q“ ż0 uχ y Óp q 1 ψχ´ will denote the inverse diffeomorphism. Lemma 5.13. If λ˜ is a solution to the SDE (5.11), then the process

1 ˜ 1 1 λ ψχ´ x ´uχ ψχ´ x p p qq¯x 0 Óp p qq ě satisfies the SDE (5.1). 1 Proof. The process λ˜ x satisfies uχ x p q x 0 ´ Óp q ¯ ě ? 1 ˜ 2 1 dx d λj x dWj x β 2 . u x p q “ u x p q` 1˜ 1 ˜ 1 u x ´ χ ¯ χ jÿ1 j uχ x ´ λj x uχ x ´ λj x χ Óp q Óp q ‰ Óp q p q´ Óp q p q Óp q By further performing the change of variable given by ψχ, one gets (5.1). 

1 d2 R 2 R In the sequel G `,χ x,y x,y 0 will denote the Green’s function of 2 dx χ on with condition 0 in 0.p Then forp 0qq xě y, ´ ` ď ď

(5.12) GR`,χ x,y 2uχ x ψχ x uχ y . p q“ Óp q p q Óp q Indeed, 1 2 B 2uχ x ψχ x uχ y χ y 2uχ x ψχ x uχ y , 2 y2 Óp q p q Óp q “ p q Óp q p q Óp q B ´ ¯ ´ ¯ 2 1 1 uχ y Ó B 2 2uχ x ψχ x uχ y B 2uχ1 x ψχ x uχ y 2 p q 2 x Óp q p q Óp q “ 2 x Óp q p q Óp q` uχ x B ´ ¯ B ´ Óp q¯ χ x 2uχ x ψχ x uχ y 0, “ p q´ Óp q p q Óp q¯ ` and 1 B B 2uχ x ψχ x uχ y 1. 2´ xˇx y ´ y ˇy x¯´ Óp q p q Óp q¯ “ B ˇ “ B ˇ “ ˇ ˇ Lemma 5.14. Let λ˜ x x 0 be the solution to (5.11) with λ˜ 0 0. Let ν be a finite family p p qq ě p q “ of positive integers, with ν even. Let x1 x2 xm ν R . Then, | | ď 﨨¨ď p q P ` m ν R p q ˜ ` Y Y pνk λ xk Pν kk GR`,χ xk,xk 1 k m ν , k 1 k uχ xk uχ xk 1 2 k m ν . p p qq β,n “ pp “ p qq ď ď p q p ´ “ Óp q{ Óp ´ qq ď ď p qq A kź1 E “ q Proof. From Lemma 5.13 and Proposition 5.4 it follows that

m ν m ν p q R` p q ˜ νk Y Y pνk λ xk uχ xk Pν kk 2ψχ xk 1 k m ν , k 1 k 1 2 k m ν . p p qq β,n “ Óp q pp “ p qq ď ď p q p ´ “ q ď ď p qq A kź1 E ´ kź1 ¯ “ “ q Further, let Q be a monomial of Pν . One has to check that

m ν p q νk Y Y uχ xk Q kk 2ψχ xk 1 k m ν , k 1 k 1 2 k m ν Óp q pp “ p qq ď ď p q p ´ “ q ď ď p qq ´ kź1 ¯ “ q Y Y Q kk GR`,χ xk,xk 1 k m ν , k 1 k uχ xk uχ xk 1 2 k m ν . “ pp “ p qq ď ď p q p ´ “ Óp q{ Óp ´ qq ď ď p qq This follows from (5.12) and the point (1) in Propositionq 4.2.  15 Proof of Proposition 5.8. It is enough to show (5.8) for functionals F of form

F ℓ x x 0 exp ℓ x χ x dx , ě R pp p qq q“ ´ ´ ż ` p q p q ¯ where χ is a continuous non-negative function with compact support in 0, . For such a χ, p `8q

m ν R p q 1 ` pνk λ xk exp p2 λ x χ x dx p p qq ´ 2 R p p qq p q β,n “ A kź1 ´ ż ` ¯E “ R m ν R 1 ` p q ˜ ` exp p2 λ x χ x dx pνk λ xk , ´ 2 R p p qq p q β,n p p qq β,n A ´ ż ` ¯E A kź1 E “ where λ˜ is given by (5.11), with λ˜ 0 0. The symmetric moments of λ˜ are given by Lemma 5.14. To conclude, we use that p q “

z x,x exp L γ χ z dz µR dγ GR`,χ x,x , R ` żγ ´ ´ ż ` p q p q ¯ p q“ p q and for 0 x y, ă ă

z x,y GR`,χ x,y uχ y exp L γ χ z dz µˇ dγ p q Óp q ; żγ ´ żR p q p q p q“ GR`,χ x,x “ uχ x ´ ` ¯ p q Óp q see Section 3.2 in [Lup18]. 

5.4. The stationary case. In this section we consider the stationary β-Dyson’s Brownian motion on the whole line and state the analogues of Propositions 5.2, 5.4 and 5.8 for it. The proofs are omitted, as they are similar to the previous ones. As previously, n 2 and β 0. ě ě Let K 0. We consider the process the process λ x λ1 x , . . . , λn x x R with λ1 x ą p p q “ p p q p qqq P p q ě λn x , satisfying the SDE ¨¨¨ě p q dx (5.13) dλj x ?2dWj x ?2K λj x β?2K , p q“ p q´ p q` λ x λ 1 x jÿ1 j j j ‰ p q´ p q 1 the dWj, 1 j n being n i.i.d. white noises on R, and λ being stationary, with 2K 4 λ x ď ď p q p q being distributed according to (3.1) (up to reordering of the λj x ). p q Proposition 5.15. The following holds. 1 (1) The process p1 λ x has same law as φK . ?n p p qq x R ` ˘ P Lα 1 (2) Consider a 1D Brownian loop-soup K, with α given by (5.4). The process 2 p2 λ x x R x α p p p qqq P has same law as the occupation field L LK x R. p p 1 qq P R (3) The processes p1 λ x x and λ x n p1 λ x x R are independent. 1 1 P α p p p qqq p q´ p p qq P ´ 2 2 ` ˘ (4) Let LK and LK be two independent 1D Brownian loop-soups, α given by (5.4). Then, one has the following identity in law between pairs of processes: r 1 1 1 1 1 (law) α 2 2 x ´ 2 x 2 p2 λ x p1 λ x , p1 λ x L LK ,L LK x R. 2 p p qq ´ n p p qq 2n p p qq x R “ p p q p qq P ´ ´ ¯ ¯ P r K will denote the expectation with respect the stationary β-Dyson’s Brownian motion. x¨yβ,n Given ν a finite family of positive integers with ν even and x1 x2 xm ν R, ν,x1,...,x | | ă ă ¨¨¨ ă p q P µ mpνq dΥ (also depending on β and n) will be the measure on finite families of continuous K p q paths obtained by substituting in the polynomial Pν Pν,β,n for each variable Ykk the measure “ xk,xk Y xk´1,xk µK , and for each variable k 1 k the measureµ ˇK . ´ 16 q Proposition 5.16. Let ν a finite family of positive integers with ν even. Let x1 x2 | | ď 﨨¨ď xm ν R. Then, p q P m ν K p q ? Y ? Y 2K xk xk´1 pνk λ xk Pν kk 1 2K 1 k m ν , k 1 k e´ p ´ q 2 k m ν . p p qq β,n “ pp “ { q ď ď p q p ´ “ q ď ď p qq A kź1 E “ q Further, let F be a bounded measurable functional on C R . For x1 x2 xm ν R, p q ă 㨨¨ă p q P m ν K K p q 1 1 ν,x1,...,xmpνq pνk λ xk F p2 λ F p2 λ L Υ µK dΥ . p p qq 2 p q β,n “ 2 p q` p q β,n p q A kź1 ´ ¯E żΥ A ´ ¯E “

6. The case of general electrical networks: a construction for n 2 and further questions “ 6.1. Formal polynomials for n 2. In this section n 2, and β is arbitrary, considered as a formal parameter. Note that d “β,n 2 β 2. In“ Section 4 we introduced the formal p “ q “ ` commuting polynomial variables Ykk k 1. Here we further consider the commuting variables p q ě Ykk1 1 k k1 , and by convention set Ykk1 Yk1k for k1 k. Givenν ˜ ν˜1,..., ν˜m with p q ď ă “ ă “ p q ν˜k N (value 0 allowed), Pν,β˜ will be the following multivariate polynomial in the variables P Ykk1 1 k k1 m: p q ď ď ď 1 Y Pν,β˜ : Perm β` f i f j 1 i,j ν˜1 ν˜m , “ 2 pp p q p qq ď ď `¨¨¨` q 1 where f is a map f : J1, ν˜1 ν˜mK J1,mK, such that for every k J1,mK, f k ν˜k. `¨¨¨` Ñ P | ´ p q| “ It is clear that Pν,β˜ does not depend on the particular choice of f. In caseν ˜1 ν˜m 0, “¨¨¨“ “ by convention we set Pν,β˜ 1. Given ν a finite family of positive integers with ν even, let “ | | kν : J1, ν K J1,m ν K be the map given by (4.4). Let Iν be the following set of subsets of J1, ν K: | | ÞÑ p q | | 1 Iν : I J1, ν K k J1,m ν K, k´ k I is even , “ t Ď | | |@ P p q | ν p qz | u where denotes the cardinal. Note that necessarily, for every I Iν, the cardinal I is even. | ¨ | P | | Let Pν,β be the following multivariate polynomial in the variables Ykk1 1 k k1 m ν : p q ď ď ď p q p I 2 m ν I 2 | |{ P : 2 p q´| |{ Yk k P 1 ´1 . ν,β ν ai ν bi kν k I 1 ,β “ p q p q p 2 | p qz |q ďkďmpνq IÿIν ´ a ,b ÿ1 2 źi 1 ¯ P i i ďiď|I|{ “ p partitionpt uq in pairs of I

By construction, for every Q monomial of Pν,β and every k J1,m ν K, P p q

(6.1) 2 degY Q p degY Q νk. kk ` kk1 “ 1 k1ÿm ν ď k1ď k p q ‰ Proposition 6.1. Let ν be finite family of positive integers with ν even. Pν,β,n 2 is obtained Y | | Y “ Y from Pν,β by replacing each variable kk1 with 1 k k1 m ν by kk k 1 r k1 r 1 r: ď ă ď p q ´ ś ` ď ď p Y Y Y q Pν,β,n 2 Pν,β kk1 kk r 1 r 1 . “ “ “ ´ 1 k k m ν `` k 1źr k1 ˘ ď ă ď p q˘ p ` ď ď q Proof. Let be Y Y Y Pν,β : Pν,β kk1 kk r 1 r 1 . “ “ ´ 1 k k m ν `` k 1źr k1 ˘ ď ă ď p q˘ r p ` ď ď q We want to show the equality Pν,β Pν,β,n 2. Since a direct combinatorial proof would be a bit “ “ lengthy, we proceed differently. Let β 0 and let λ x λ1 x , λ2 x x 0 be the β-Dyson’s ě r ě 17 p p q “ p p q p qqq Brownian motion (5.1) in the case n 2. We use its construction through (5.2). We claim that “ for x1,x2,...,xm ν R , p q P ` m ν p q R` Y 1 pνk λ xk Pν,β kk GR` xk 1,xk 1 . p p qq β,n 2 “ “ p ´ q 1 k k m ν A kź1 E “ `` ˘ ď ď ď p q˘ “ p Indeed, in the expansion of

νk νk W xk ρ xk W xk ρ xk ´ p q` p q¯ ` ´ p q´ p q¯ Ă Ă only enter the even powers of ρ xk , which is how Iν appears. Then one uses that the square p q Bessel process ρ x x 0 is a β 1 2-permanental field with kernel GR` x,y x,y R` . Because p p qq ě p ` q{ p Rp qq P of the particular form of GR` , we have that for x1 x2 xm ν , ď 﨨¨ď p q P ` m ν R p q ` Y Y pνk λ xk Pν,β kk 2xk 1 k m ν , k 1 k 1 2 k m ν . p p qq β,n 2 “ pp “ q ď ď p q p ´ “ q ď ď p qq A kź1 E “ “ r q By combining with Proposition 5.4, we get that the following multivariate polynomials in the Y variables kk 1 k m ν are equal for β 0: p q ď ď p q ě

Pν,β Yk 1 k 1 2 k m ν Pν,β,n 2 Yk 1 k 1 2 k m ν . pp ´ “ q ď ď p qq“ “ pp ´ “ q ď ď p qq Since the coefficientsr ofq both are polynomials in β, the equalityq above holds for general β. To conclude the equality Pν,β Pν,β,n 2, we have to deal with the variables Yk 1 k 2 k m ν . For “ “ p ´ q ď ď p q this we use that both rPν,β,n 2 and Pν,β satisfy the point (1) of Propositionq 4.2. For Pν,β this follows from (6.1). “  r r 6.2. A construction on discrete electrical networks for n 2. Let G V, E be an undirected connected graph, with V finite. We do not allow multiple“ edges or“ self-loops. p q The edges x,y E are endowed with conductances C x,y C y,x 0. There is also a not t u P p q “ p q ą uniformly zero killing measure K x x V , with K x 0. We see G as an electrical network. p p qq P p q ě Let ∆G denote the discrete Laplacian

∆Gf x C x,y f y f x . p qp q“ p qp p q´ p qq yÿx „ 1 Let GG,K x,y x,y V be the massive Green’s function GG,K ∆G K ´ . The (massive) realp scalarp Gaussianqq P free field (GFF) is the centered random Gaussian“ p´ field` onq V with covariance GG,K, or equivalently with density

1 1 2 1 2 (6.2) 1 exp K x ϕ x C x,y ϕ y ϕ x . V 2 ´ 2 p q p q ´ 2 p qp p q´ p qq 2π | | det G ´ xÿV x,yÿ E ¯ pp q q P t uP Let Xt be the continuous time Markov to nearest neighbors with jump rates given by the conductances. Xt is also killed by K. Let ζ 0, be the first time Xt gets P p `8s killed by K. Let pG,K t,x,y be the transition probabilities of Xt 0 t ζ . Then pG,K t,x,y p q p q ď ă p q“ pG,K t,y,x and p q `8 GG,K x,y pG,K t,x,y dt. p q“ ż0 p q Let Pt,x,y be the bridge probability measure from x to y, where one conditions by t ζ. For G,K ă x,y V , let µx,y be the following measure on paths: P G x,y `8 Pt,x,y µG,K : G,K pG,K t,x,y dt. p¨q “ ż0 p¨q p q 18 x,y x,y It is the analogue of (2.1). The total mass of µG is GG,K x,y , and the image of µG by time y,x p q reversal is µG . Similarly, one defines the measure on (rooted) loops by 1 µloop dγ : µx,x dγ , G,K p q “ T γ G,Kp q xÿV p q P where T γ denotes the duration of the loop γ. It is the analogue of (2.2). µloop has an infinite p q G,K total mass because it puts an infinite mass on trivial ”loops” that stay in one vertex. For α 0, α loop ą one considers Poisson point processes LG,K of intensity αµG,K . These are (continuous time) random walk loop-soups. For details, see [LTF07, LL10, LJ10, LJ11]. For a continuous time path γ on G of duration T γ and x V , we denote p q P T γ x p q L γ : 1γ s xds. p q “ ż0 p q“ Further, Lx Lα : Lx γ . p G,Kq “ p q γ ÿLα P G,K 1 x L 2 1 2 One has equality in law between L G,K x V and 2 φG,K x x V , where φG,K is the GFF p p qq P p p q q P distributed according to (6.2) [LJ10, LJ11]. This is the analogue of (2.3). For general α 0, the x α ą occupation field L LG,K x V the α-permanental field with kernel GG,K [LJ10, LJ11, LJMR15]. p p qq P V In this sense it is analogous to squared Bessel processes. If χ x x V R is such that p p qq P P ∆G K χ is positive definite, then ´ ` ´ α E x α det ∆G K (6.3) exp χ x L LG,K p´ ` q . p q p q “ ˆdet ∆G K χ ˙ ” ´ xÿV ¯ı P p´ ` ´ q 1 β 2 1 Now we proceed with our construction. Fix β 1. Let α 2 d β,n 2 `2 2 . ą ´ α 1 “ p “ q “ ą ´ 2 Let φG,K be a GFF distributed according to (6.2), and LG,K an independent random walk loop-soup. For x V we set P 1 1 1 x α 2 1 x α 2 λ1 x : φG,K x L L ´ , λ2 x : φG,K x L L ´ , p q “ ?2 p q` c p G,K q p q “ ?2 p q´ c p G,K q G,K and λ : λ1 x , λ2 x x V . β,n 2 will denote the expectation with respect to λ. As in Section “ p p q p qq P x¨y “ 5.3, Υ γ1, . . . , γJ Υ will denote a generic family of continuous time paths, this time on the graph G“. p For x V ,p qq P J Υ x p q x L Υ : L γi , p q “ p q iÿ1 “ and L Υ will denote the occupation field of Υ, x Lx Υ . Given ν a finite family of positive p q ÞÑ ν,β,xp 1,...,xq mpνq integers with ν even, and x1,x2,...,xm ν V ,µ ˆG,K will denote the measure on | | p q P families of ν 2 paths on G obtained by substituting in the polynomial Pν,β for each variable | |{ xk,xk1 Ykk1 , 1 k k m ν , the measure µ . The order of the paths will not matter. ď ď 1 ď p q G,K p Proposition 6.2. The following holds.

(1) For every x V , λ1 x GG,K x,x , λ2 x GG,K x,x is distributed, up to reorder- P p p q{ p q p q{ p qq ing, according to (3.1) foran 2. a (2) Let x,y V . Let “ P GG,K x,x GG,K y,y (6.4) η p q 2p q 1. “ GG,K x,y ě p q Then the couple ?2λ x GG,K x,x , ?2ηλ y GG,K y,y is distributed like the β- p p q{ p q p q{ p qq Dyson’s Brownian motiona(5.1) at points 1 and ηa, for n 2. 19 “ (3) Let ν be finite family of positive integers with ν even and x1,x2,...,xm ν V . Then | | p q P m ν p q G,K Y 1 1 pνk λ xk Pν,β kk GG,K xk,xk 1 k k1 m ν . p p qq β,n 2 “ pp “ p qq ď ď ď p qq A kź1 E “ “ p (4) (BFS-Dynkin’s isomorphism) Moreover, given F a measurable bounded function on RV ,

m ν G,K G,K p q 1 1 ν,β,x1,...,xmpνq (6.5) pνk λ xk F p2 λ F p2 λ L Υ µˆG,K dΥ . p p qq 2 p q β,n 2 “ ż 2 p q` p q β,n 2 p q A kź1 ´ ¯E “ Υ A ´ ¯E “ “ (5) For β 1, 2, 4 , λ1 x , λ2 x x V is distributed like the ordered family of eigenvalues in a GFFP t with valuesu p p inq 2 p2 qqrealP symmetric β 1 , complex Hermitian β 2 , resp. quaternionic Hermitian βˆ 4 matrices, withp density“ q proportional to p “ q p “ q 1 1 (6.6) exp K x Tr M x 2 C x,y Tr M y M x 2 . ´ 2 p q p p q q´ 2 p q pp p q´ p qq q ´ xÿV x,yÿ E ¯ P t uP

(6) Assume that β 0. Let φ1 and φ2 be two independent scalar GFFs distributed according α 1 ą β 2 to (6.2). L ´ be a random walk loop-soup independent from φ1, φ2 , with still α ` . G,K p q “ 2 Then λ1 x , λ2 x x V is the ordered family of eigenvalues in the matrix valued field p p q p qq P x α 1 φ1 x L LG´,K (6.7) ¨ p q b p q ˛ , x V. x α 1 P L LG´,K φ2 x ˝ b p q p q ‚ V (7) Given another killing measure K R , non uniformly zero, and λ˜ λ˜1, λ˜2 the field P ` “ p q K K λ˜ λ obtained by using instead of r, the density of the law of with respect to that of is β`2 r 2 det ∆G K 1 p´ ` q exp K x K x p2 λ x . ˆdet ∆G K ˙ ´ 2 p p q´ p qq p p qq r ´ xÿV ¯ p´ ` q P r Proof. (1) This follows from Proposition 3.2 and the fact that φG,K x GG,K x,x is dis- 1 p q{ p q x α 2 a 1 tributed according to N 0, 1 , and L L ´ GG,K x,x according to Gamma α , 1 . p q p G,K q{ p q ´ 2 (2) One uses (5.2). Indeed, ?2φG,K x GaG,K x,x , ?2ηφG,K y GG,K y,y ` is distributed˘ p 1 p q{ p q p1 q{ p qq x α 2 a y α 2 a as φR 1 , φR η , and ?2L L ´ GG,K x,x , ?2ηL L ´ GG,K y,y is distributed p ` p q ` p qq p p G,K q{ p q p G,K q{ p qq as ρ 1 , ρ η . The latter can be seen usinga the moments, that characterizea the finite-dimensional marginalsp p q p ofqq the Bessel process ρ. (3) This follows by expanding

1 1 νk νk 1 α 2 1 α 2 φ x Lxk L ´ φ x Lxk L ´ . G,K k c G,K G,K k c G,K ´?2 p q` p q¯ ` ´?2 p q´ p q¯

(4) The GFF φG,K satisfies the BFS-Dynkin isomorphism; see Theorem 2.2 in [BFS82], Theorems 6.1 and 6.2 in [Dyn84a], Theorem 1 in [Dyn84b]. Moreover, there is a version of BFS- α 1 Dynkin isomorphism for the occupation field L L ´ 2 obtained by applying Palm’s identity to p G,K q Poisson point processes; see Theorem 1.3 in [LJMR15] and Sections 3.4 and 4.3 in [Lup18]. Together, this implies (6.5). (5) Recall that for all three matrix spaces considered, β 2 is the dimension. Given M x x V ` 1 p p qq P an matrix field distributed according to (6.6), M0 x will denoted M x Tr M x I2, where p q p q´ 2 p p qq I2 is the 2 2 identity matrix, so that Tr M0 x 0. Since the hyperplane of zero trace matrices ˆ p p qq “ is orthogonal to I2 for the inner product A, B Re Tr AB , we get that M0 x x V and p 1 q ÞÑ p p qq p p qq P Tr M x x V are independent. Moreover, Tr M x x V is distributed as the scalar GFF p p p qqq P p ?2 p p qqq P 20 2 (6.2). As for Tr M x x V , on one hand it is the sum of β 2 i.i.d. squares of scalar GFFs (6.2) correspondingp p p toq theqq P entries of the matrices. On the other` hand,

2 2 1 2 Tr M x Tr M0 x Tr M x . p p q q“ p p q q` 2 p p qq 2 So Tr M0 x x V is distributed as the sum of β 1 i.i.d. squares of scalar GFFs (6.2). So p p p q qq P ` β`1 x 2 in particular, this is the same distributions as for 2L LG,K x V . Finally, the eigenvalues of p p qq P M x are p q 1 1 Tr M x Tr M x 2 . 2 ? 0 p p qq ˘ 2a p p q q (6) The eigenvalues of the matrix (6.7) are

φ1 x φ2 x x α 1 2 p q` p q L LG´,K φ2 x φ1 x 4. 2 ˘ b p q ` p p q´ p qq { φ1 φ2 ?2 and φ2 φ1 ?2 are two independent scalar GFFs. Moreover, p ` q{ p ´ q{ α 1 1 2 L L ´ φ2 φ1 p G,K q` 4p ´ q α 1 has same distribution as L L ´ 2 . p G,K q (7) The density of the GFF φG,K with respect to φG,K is 1 r det ∆G K 2 1 p´ ` q exp K x K x ϕ x 2 . ˆdet ∆G K ˙ ´ 2 p p q´ p qq p q r ´ xÿV ¯ p´ ` q P r α 1 α 1 The density of L L ´ 2 with respect to L L ´ 2 is p G,K q p G,K q 1 r α 1 det ∆G K ´ 2 α x ´ 2 p´ ` q exp K x K x L LG,K , ˆdet ∆G K ˙ ´ p p q´ p qq p q r ´ xÿV ¯ p´ ` q P r as can be seen from the Laplace transform (6.3).  6.3. Further questions. Here we present our questions that motivated this paper. The first question is combinatorial. We would like to have the polynomials Pν,β,n given by Definition 4.1 under a more explicit form. The recurrence on polynomials (4.1) is closely related to the Schwinger-Dyson equation (3.2). Its very form suggests that the polynomials Pν,β,n might be ex- pressible as weighted sums over maps drawn on 2D compact surfaces (not necessarily connected), where the maps associated to ν have m ν vertices with degrees given by ν1,ν2,...,νm ν , with p q p q powers of n corresponding to the number of faces. This is indeed the case for β 1, 2, 4 , and this corresponds to the topological expansion of matrix integrals [BIPZ78, IZ80, MW03,P t Lup19].u

Question 6.3. Is there a more explicit expression for the polynomials Pν,β,n? Can they be expressed as weighted sums over the maps on 2D surfaces (topological expansion)? The second question is whether there is a natural generalization of Gaussian beta ensembles and β-Dyson’s Brownian motion to electrical networks. For n 2, such a generalization was given in Section 6.2. “ Question 6.4. We are in the setting on an electrical network G V, E endowed with a killing measure K, as in Section 6.2. Given n 3 and β 2 , is there“ p a distributionq on the ě ą ´ n fields λ x λ1 x , λ2 x , . . . , λn x x V , with λ1 x λ2 x λn x , satisfying the followingp p properties?q “ p p q p q p qqq P p q ą p qą¨¨¨ą p q (1) For β 1, 2, 4 , λ is distributed as the fields of ordered eigenvalues in a GFF with values intoP t n un matrices, real symmetric β 1 , complex Hermitian β 2 , resp. quaternionic Hermitianˆ β 4 . p “ q p “ q (2) For β 0, λ is obtainedp by“ reorderingq n i.i.d. scalar GFFs (6.2). “ 21 (3) As β 2 , λ converges in law to Ñ´ n 1 1 1 φG,K, φG,K,..., φG,K , ´?n ?n ?n ¯

where φG,K is a scalar GFF (6.2). (4) For every x V , λ x GG,K x,x is distributed, up to reordering, as the GβE (3.1). P p q{ p q (5) For every x,y V , thea couple ?2λ x GG,K x,x , ?2ηλ y GG,K y,y , with η P p p q{ p q p q{ p qq given by (6.4), is distributed as the β-Dyson’sa Brownian motion (5.1)a at points 1 and η. 1 (6) The fields p1 λ and λ n p1 λ are independent. 1 p q ´ p q (7) The field ?n p1 λ is distributed as a scalar GFF (6.2). 1 p q 1 2 1 (8) The field 2 p2 λ n p1 λ is the α 2 -permanental field with kernel GG,K, where 1 p q´ p q ´ α 2 d β,n` , and in particular˘ is distributed as the occupation field of the continuous- “ p q α 1 ´ 2 time random walk loop-soup LG,K . 1 1 (9) The field 2 p2 λ is the α-permanental field with kernel GG,K, where α 2 d β,n , and in particularp isq distributed as the occupation field of the continuous-time“ randomp q walk α loop-soup LG,K (already implied by (6)+(7)+(8)). (10) The symmetric moments

m ν p q G,K pν λ xk k p p qq β,n A kź1 E “ are linear combination of products

a 1 GG,K xk,xk1 kk , p q 1 k źk1 m ν ď ď ď p q

with akk1 N and for every k J1,m ν K, P P p q

2akk akk1 νk, ` “ 1 k1ÿm ν ď k1ď k p q ‰ the coefficients of the linear combination being universal polynomials in β and n, not depending on the electrical network and its parameters; see also Question 6.3. V (11) Given K R , non uniformly zero, and λ˜ λ˜1, λ˜2,..., λ˜n the field associated to the P ` “ p q K K λ˜ killing measurer instead of , the law of has the following density with respect to that of λ: r 1 d β,n det ∆G K 2 p q 1 p´ ` q exp K x K x p2 λ x . ˆdet ∆G K ˙ ´ 2 p p q´ p qq p p qq r ´ xÿV ¯ p´ ` q P r (12) λ satisfies a BFS-Dynkin type isomorphism with continuous time random walks (already implied by (10)+(11)).

If the graph G is a tree, the answer for the properties (1),(2),(4),(5),(6),(7),(8),(9),(11),(12) is yes, at least for β 0. In absence of cycles, λ satisfies a Markov property, and along each branch of the tree oneě has the values of a β-Dyson’s Brownian motion at different positions. On the random walk loop-soup side, (8) and (9) is ensured by the covariance of the loop-soups under the rewiring of graphs; see Chapter 7 in [LJ11]. Constructing λ on a tree for β 2 , 0 P ´ n is a matter of constructing the corresponding β-Dyson’s Brownian motion. However,` if the˘ graph G contains cycles, constructing λ is not immediate, and we have not encountered such a construction in the literature. One does not expect a Markov property, since already for β 1, 2, 4 one has to take into account the angular part of the matrices. P t u 22 Appendix: A list of moments for GβE and the corresponding formal polynomials

2 p1 λ β,n n, x p q y “ P 1,1 nY11Y12, p q “ q β 2 β p2 λ β,n n 1 n d β,n , x p qy “ 2 ` ´ ´ 2 ¯ “ p q β 2 β P 2 n 1 n Y11 d β,n Y11, p q “ ´ 2 ` ´ ´ 2 ¯ ¯ “ p q 4 2 p1 λ β,n 3n , x p q y “ 2Y Y Y Y 2Y Y Y Y2 Y P 1,1,1,1 n 11 12 33 34 2n 11 12 22 23 34, p q “ ` q q q q q 2 β 3 β 2 p2 λ p1 λ β,n n 1 n 2n, x p q p q y “ 2 ` ´ ´ 2 ¯ ` β 3 β 2 Y Y Y Y2 Y2 Y P 2,1,1 n 1 n 11 22 23 2n 11 12 23, p q “ ´ 2 ` ´ ´ 2 ¯ ¯ ` β 3 β 2 q q q P 1,2,1 n 1 n 2n Y11Y12Y22Y23, p q “ ´ 2 ` ´ ´ 2 ¯ ` ¯ β 3 β 2 Y Y Y q Yq Y Y Y2 P 1,1,2 n 1 n 11 12 33 2n 11 12 22 23, p q “ ´ 2 ` ´ ´ 2 ¯ ¯ ` q q q 2 2 β 4 β β 3 p2 λ β,n n 2 1 n x p q y “ 4 ` 2 ´ ´ 2 ¯ β 2 β β 1 2 n2 2 1 n `´´ ´ 2 ¯ ` 2 ¯ ` ´ ´ 2 ¯ d β,n d β,n 2 , “ p qp p q` q 2 2 β 4 β β 3 β 2 Y Y P 2,2 n 2 1 n 1 n 11 22 p q “ ´ 4 ` 2 ´ ´ 2 ¯ ` ´ ´ 2 ¯ ¯ β 2 β Y2 Y2 2 n 2 1 n 11 12, `´ 2 ` ´ ´ 2 ¯ ¯ q β 2 β p3 λ p1 λ β,n 3 n 3 1 n, x p q p qy “ 2 ` ´ ´ 2 ¯ β 2 β Y2 Y P 3,1 3 n 3 1 n 11 12, p q “ ´ 2 ` ´ ´ 2 ¯ ¯ β 2 β q P 1,3 3 n 3 1 n Y11Y12Y22, p q “ ´ 2 ` ´ ´ 2 ¯ ¯ q 2 2 β 3 β β 2 β β p4 λ β,n 2 n 5 1 n 3 1 n, x p qy “ 4 ` 2 ´ ´ 2 ¯ ` ´ 2 ` ´ ´ 2 ¯ ¯ 2 2 β 3 β β 2 β β Y2 P 4 2 n 5 1 n 3 1 n 11, p q “ ´ 4 ` 2 ´ ´ 2 ¯ ` ´ 2 ` ´ ´ 2 ¯ ¯ ¯ 2 2 2 β 3 β β 2 β β p3 λ β,n 12 n 27 1 n 3 15 1 n, x p q y “ 4 ` 2 ´ ´ 2 ¯ ` ´ 2 ` ´ ´ 2 ¯ ¯ 2 2 β 3 β β 2 β Y2 Y Y P 3,3 9 n 2 1 n 1 n 11 12 22 p q “ ´ 4 ` 2 ´ ´ 2 ¯ ` ´ ´ 2 ¯ ¯ 2 2 β 3 β β 2 β βq Y3 Y3 3 n 3 1 n 2 1 n 11 12. ` ´ 4 ` 2 ´ ´ 2 ¯ ` ´ 2 ` ´ ´ 2 ¯ ¯ ¯ 23 q Acknowledgements The author thanks Guillaume Chapuy and J´er´emie Bouttier for discussions and references on the beta ensembles. The author thanks Yves Le Jan and Wendelin Werner for their feedback on the preliminary version of the article. This work was supported by the French National Research Agency (ANR) grant within the project MALIN (ANR-16-CE93-0003).

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CNRS and LPSM, UMR 8001, Sorbonne Universite,´ 4 place Jussieu, 75252 Paris cedex 05, France Email address: [email protected]

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