Inverting the coupling of the signed Gausssian free field with a loop soup

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by  Titus Lupu

INVERTING THE COUPLING OF THE SIGNED GAUSSSIAN FREE
FIELD WITH A LOOP SOUP
TITUS LUPU,CHRISTOPHESABOT, ANDPIERRE TARRE`S
Abstract. Lupu introduced a coupling between a random walk loop-soup and a Gaussian
free field, where the sign of the field is constant on each cluster of loops. This coupling is a
signedversionofisomorphismtheoremsrelatingthesquareoftheGFFtotheoccupationfield
7
ofMarkoviantrajectories. Hisconstructionstartswithaloop-soup,andbyaddingadditional
1
randomness samples a GFF out of it. In this article we provide the inverse construction:
0
2 startingfrom asigned freefieldandusingaself-interacting random walk related tothisfield,
weconstructarandomwalkloop-soup. OurconstructionreliesonthepreviousworkbySabot
n
and Tarr`es, which inverts the coupling from the square of the GFF rather than the signed
a
GFFitself. Asaconsequence,wealsodeduceaninversionofthecouplingbetweentherandom
J
current and the FK-Isingrandom cluster models introduced byLupu and Werner.
4

]
R
P
1. Introduction
.
h
t Let = (V,E) be a connected undirected graph, with V at most countable and each vertex
a G
m x V of finite degree. We do not allow self-loops, however the edges might be multiple. Given
∈
e E an edge, we will denote e and e its end-vertices, even though e is non-oriented and
[ on∈e can interchange e and e .+Each e−dge e E is endowed with a conductance W > 0.
+ e
1 − ∈
There may be a killing measure κ = (κ ) on vertices.
v x x V
∈
2 We consider (Xt)t 0 the Markov jump processes on V which being in x V, jumps along
≥ ∈
9 an adjacent edge e with rate W . Moreover if κ = 0, the process is killed at x with rate κ
e x x
0 6
(the process is not defined after that time). ζ will denote the time up to which X is defined.
1 t
If ζ < + , then either the process has been killed by the killing measure κ (and κ 0) or it
0
∞ 6≡
. has gone off to infinity in finite time (and V infinite). We will assume that the process X is
1
0 transient, which means, if V is finite, that κ 0. Px will denote the law of X started from x.
6≡
7 Let (G(x,y)) be the Green function of X :
x,y V t
1 ∈
: ζ
v G(x,y) = G(y,x) = E 1 dt .
Xi x(cid:20)Z0 {Xt=y} (cid:21)
r Let be the Dirichlet form defined on functions f on V with finite support:
a E
(1.1) (f,) = κ f(x)2+ W (f(e ) f(e ))2.
x e +
E − −
x V e E
X∈ X∈
P will be the law of (ϕ ) the centred Gaussian free field (GFF) on V with covariance
ϕ x x V
∈
E [ϕ ϕ ]= G(x,y). In case V is finite, the density of P is
ϕ x y ϕ
1 1
exp (f,f) df .
(2π)|V2|√detG (cid:18)−2E (cid:19)x V x
Y∈
2010 Mathematics Subject Classification. primary 60J27; 60J55; secondary 60K35; 82B20; 81T25; 81T60.
Key words and phrases. Gaussian free field;Ray-Knightidentity;self-interacting processes; loop-soups; ran-
dom currents;Ising model.
1
2 TITUSLUPU,CHRISTOPHESABOT,ANDPIERRETARRE`S
U,f
Given U a finite subset of V, and f a function on U, P will denote the law of the GFF ϕ
ϕ
conditioned to equal f on U. (ℓ (t)) will denote the family of local times of X:
x x V,t [0,ζ]
∈ ∈
t
ℓ (t) = 1 ds.
x Xs=x
Z0 { }
For all x V, u > 0, let
∈
τx = inf t 0;ℓ (t) > u .
u { ≥ x }
Recall the generalized second Ray-Knight theorem on discrete graphs by Eisenbaum, Kaspi,
Marcus, Rosen and Shi [2] (see also [8, 10]):
Generalized second Ray-Knight theorem. For any u >0 and x V,
0
∈
1
ℓx(τux0)+ 2ϕ2x under Px0(·|τux0 < ζ)⊗Pϕ{x0},0
(cid:18) (cid:19)x V
∈
has the same law as
12ϕ2x under Pϕ{x0},√2u.
(cid:18) (cid:19)x V
∈
Sabot and Tarr`es showed in [9] that the so-called “magnetized” reverse Vertex-Reinforced
JumpProcessprovidesaninversionofthegeneralized secondRay-Knighttheorem,inthesense
that it enables to retrieve the law of (ℓ (τx0),ϕ2) conditioned on ℓ (τx0)+ 1ϕ2 . The
jump rates of that latter process canxbeuinterxprxe∈tVed as the two-pointxfuunction2s oxfxt∈hVe Ising
(cid:0) (cid:1)
model associated to the time-evolving weights.
However in [9] the link with the Ising model is only implicit, and a natural question is
whether Ray-Knight inversion can be described in a simpler form if we enlarge the state space
of the dynamics, and in particular include the “hidden” spin variables.
The answer is positive, and goes through an extension of the Ray-Knight isomorphism
introduced by Lupu [6], which couples the sign of the GFF to the path of the Markov chain.
The Ray-Knight inversion will turn out to take a rather simple form in Theorem 3 of the
present paper, where it will be defined not only through the spin variables but also random
currents associated to the field though an extra Poisson Point Process.
The paper is organised as follows.
In Section 2 we recall some background on loop soup isomorphisms and on related couplings
and state and prove a signed version of generalized second Ray-Knight theorem. We begin in
Section 2.1 by a statement of Le Jan’s isomorphism which couples the square of the Gaussian
Free Field to the loop soups, and recall how the generalized second Ray-Knight theorem can
be seen as its Corollary: for more details see [4]. In Subsection 2.2 we state Lupu’s isomor-
phism which extends Le Jan’s isomorphism and couples the sign of the GFF to the loop soups,
using a cable graph extension of the GFF and Markov Chain. Lupu’s isomorphism yields
an interesting realisation of the well-known FK-Ising coupling, and provides as well a “Cur-
rent+Bernoulli=FK” coupling lemma [7], which occur in the relationship between the discrete
and cable graph versions. We briefly recall those couplings in Sections 2.3 and 2.4, as they are
implicit in this paper. In Section 2.5 we state and prove the generalized second Ray-Knight
“version” of Lupu’s isomorphism, which we aim to invert.
Section 3 is devoted to the statements of inversions of those isomorphisms. We state in
Section 3.1 a signed version of the inversion of the generalized second Ray-Knight theorem
through an extra Poisson Point Process, namely Theorem 3. In Section 3.2 we provide a
discrete-time description of the process, whereas in Section 3.3 we yield an alternative version
of that process through jump rates, which can be seen as an annealed version of the first one.
INVERTING THE COUPLING OF THE SIGNED GAUSSSIAN FREE FIELD WITH A LOOP SOUP 3
We deduce a signed inversion of Le Jan’s isomorphism for loop soups in Section 3.4, and an
inversion of the coupling of random current with FK-Ising in Section 3.5.
Finally Section 4 is devoted to the proof of Theorem 3: Section 4.1 deals with the case of a
finite graph without killing measure, and Section 4.2 deduces the proof in the general case.
2. Le Jan’s and Lupu’s isomorphisms
2.1. Loop soups and Le Jan’s isomorphism. The loop measure associated to the Markov
jumpprocess(X ) isdefinedasfollows. LetPt bethebridgeprobabilitymeasurefromx
t 0 t<ζ x,y
≤
to y in time t (conditionned on t < ζ). Let p (x,y) be the transition probabilities of (X ) .
t t 0 t<ζ
≤
Let µ be the measure on time-parametrised nearest-neighbour based loops (i.e. loops
loop
with a starting site)
dt
µ = Pt p (x,x) .
loop x,x t t
x V Zt>0
X∈
The loops will be considered here up to a rotation of parametrisation (with the corresponding
pushforward measure induced by µ ), that is to say a loop (γ(t)) will be the same as
loop 0 t tγ
≤ ≤
(γ(T +t)) (γ(T +t t )) , where denotes the concatenation of paths. A
0≤t≤tγ−T ◦ − γ tγ−T≤t≤tγ ◦
loop soup of intensity α > 0, denoted , is a Poisson random measure of intensity αµ . We
α loop
L
see it as a random collection of loops in . Observe that a.s. above each vertex x V,
α
G ∈ L
contains infinitely many trivial ”loops” reduced to the vertex x. There are also with positive
probability non-trivial loop that visit several vertices.
Let L( ) be the occupation field of on V i.e., for all x V,
. α α
L L ∈
tγ
L ( ) = 1 dt.
x α γ(t)=x
L (γ(t))0X≤t≤tγ∈LαZ0 { }
In [3] Le Jan shows that for transient Markov jump processes, L ( )< + for all x V a.s.
x α
L ∞ ∈
For α = 1 he identifies the law of L( ):
2 . Lα
1 1
Le Jan’s isomorphism. L( ) = L ( ) has the same law as ϕ2 = ϕ2
. L1/2 x L1/2 x V 2 2 x
∈ (cid:18) (cid:19)x V
under P . (cid:0) (cid:1) ∈
ϕ
Letusbrieflyrecall how LeJan’s isomorphismenables onetoretrieve thegeneralized second
Ray-Knight theorem stated in Section 1: for more details, see for instance [4]. We assume that
κ is supportedby x : thegeneral case can bedealt with by an argument similar to theproof of
0
Proposition 4.6. Let D = V x , and note that the isomorphism in particular implies that
0
\{ }
L( ) conditionally on L ( ) = u has the same law as ϕ2/2 conditionally on ϕ2 /2 = u.
. L1/2 x0 L1/2 x0
On the one hand, given the classical energy decomposition, we have ϕ = ϕD+ϕ , with ϕD
x0
theGFF associated totherestriction of to D, whereϕD andϕ areindependent. Now ϕ2/2
E x0
conditionally on ϕ2 /2 = u has the law of (ϕD+η√2u)2/2, where η is the sign of ϕ , which is
x0 x0
independentof ϕD. But ϕD is symmetric, sothat the latter also has thelaw of (ϕD+√2u)2/2.
On the other hand, the loop soup can be decomposed into the two independent loop
1/2
L
soups D contained in D and (x0) hitting x . Now L( D ) has the law of (ϕD)2/2 and
L1/2 L1/2 0 . L1/2
L( (x0)) conditionally on L ( (x0)) = u has the law of the occupation field of the Markov
. L1/2 x0 L1/2
chain ℓ(τx0) under P ( τx0 < ζ), which enables us to conclude.
u x0 ·| u
4 TITUSLUPU,CHRISTOPHESABOT,ANDPIERRETARRE`S
2.2. Lupu’s isomorphism. As in [6], we consider the metric graph ˜ associated to . Each
G G
edge e is replaced by a continuous line of length 1W 1.
2 e−
TheGFFϕon with law P canbeextended toaGFFϕ˜on ˜asfollows. Given e E, one
ϕ
G G ∈
considers inside e a conditionally independent Brownian bridge, actually a bridge of a √2
×
standard Brownian motion, of length 1W 1, with end-values ϕ and ϕ . This provides a
2 e− e− e+
continuous field on the metric graph which satisfies the spatial Markov property.
Similarly one can define a standard Brownian motion (B˜) on ˜, whose trace on
G 0 t ζ˜ G G
indexed by the local times at V has the same law as the Marko≤v≤process (X ) on V with
t t 0
≥
jump rate W to an adjacent edge e up to time ζ, as explained in Section 2 of [6]. One can
e
associate a measure on time-parametrized continuous loops µ˜, and let ˜ bethe Poisson Point
1
L2
Process of loops of intensity µ˜/2: the discrete-time loops can be obtained from ˜ by
1 1
L2 L2
taking the print of the latter on V.
Lupu introduced in [6] an isomorphism linking the GFF ϕ˜ and the loop soup ˜ on ˜.
1
L2 G
Theorem 1 (Lupu’s Isomorphism,[6]). There is a coupling between the Poisson ensemble of
loops ˜ and (ϕ˜ ) defined above, such that the two following constraints hold:
L21 y y∈G˜
For all y ˜, L (˜ ) = 1ϕ˜2
• ∈ G y L21 2 y
The clusters of loops of ˜ are exactly the sign clusters of (ϕ˜ ) .
• L21 y y∈G˜
Conditionally on (ϕ˜ ) , the sign of ϕ˜ on each of its connected components is distributed
| y| y ˜
independently and unifor∈mGly in 1,+1 .
{− }
Lupu’s isomorphism and the idea of using metric graphs were applied in [5] to show that on
the discrete half-plane Z N, the scaling limits of outermost boundaries of clusters of loops in
×
loop soups are the Conformal Loop Ensembles CLE.
Let (ϕ˜) (resp. (˜ )) be the set of edges e E such that ϕ˜ (resp. ˜ ) does not touch 0
1 1
O O L2 ∈ L2
on e, in other words such that all the edge e remains in the same sign cluster of ϕ˜ (resp. ˜ ).
1
L2
Let ( ) be the set of edges e E that are crossed (i.e. visited consecutively) by the trace
1
O L2 ∈
of the loops on V.
1
L2
InordertotranslateLupu’sisomorphismbackontotheinitialgraph ,oneneedstodescribe
G
ononehandthedistributionof (ϕ˜)conditionallyonthevaluesofϕ,andontheotherhandthe
O
distributionof (˜ )conditionally on andthecluster ofloops ( )onthediscretegraph
1 1 1
O L2 L2 O L2
G. These two distributions are described respectively in Subsections 2.3 and 2.4, and provide
realisations of the FK-Ising coupling and the “Current+Bernoulli=FK” coupling lemma [7].
2.3. The FK-Ising distribution of (ϕ˜) conditionally on ϕ.
O | |
Lemma 2.1. Conditionally on (ϕ ) , (1 ) is a family of independent random vari-
x x V e (ϕ˜) e E
ables and ∈ ∈O ∈
1 if ϕ ϕ < 0,
P(e (ϕ˜)ϕ) = e− e+
6∈ O | (cid:26) exp −2Weϕe−ϕe+ if ϕe−ϕe+ > 0.
Proof. Conditionally on (ϕx)x V, are cons(cid:0)tructed as ind(cid:1)ependent Brownian bridges on each
∈
edge, so that (1 ) are independentrandom variables, and it follows from the reflection
e (ϕ˜) e E
principle that, if∈ϕO ϕ ∈ > 0, then
e− e+
exp 1W (ϕ +ϕ )2
P(e (ϕ˜)ϕ) = −2 e e− e+ = exp 2W ϕ ϕ .
6∈ O | exp 1W (ϕ ϕ )2 − e e− e+
(cid:0)−2 e e− − e+ (cid:1)
(cid:0) (cid:1) (cid:3)
(cid:0) (cid:1)
INVERTING THE COUPLING OF THE SIGNED GAUSSSIAN FREE FIELD WITH A LOOP SOUP 5
Let us now recall how the conditional probability in Lemma 2.1 yields a realisation of the
FK-Ising coupling.
Assume V is finite. Let (J ) be a family of positive weights. An Ising model on V with
e e E
interaction constants (J ) is∈a probability on configuration of spins (σ ) +1, 1 V
e e E x x V
∈ ∈ ∈ { − }
such that
1
PIsg((σ ) ) = exp J σ σ .
J x x V Isg e e+ e−
∈ !
ZJ eX∈E
An FK-Ising random cluster model with weights (1 e 2Je) is a random configuration of
− e E
− ∈
open (value 1) and closed edges (value 0) such that
1
PFJK−Isg((ωe)e∈E)= FK−Isg2♯ clusters (1−e−2Je)ωe(e−2Je)1−ωe,
ZJ eY∈E
where ”♯ clusters” denotes the number of clusters created by open edges.
The well-known FK-Ising and Ising coupling reads as follows.
Proposition 2.2 (FK-Ising and Ising coupling). Given an FK-Ising model, sample on each
cluster an independent uniformly distributed spin. The spins are then distributed according
to the Ising model. Conversely, given a spins configuration σˆ following the Ising distribution,
consider each edge e, such that σˆ σˆ < 0, closed, and each edge e, such that σˆ σˆ > 0
e− e+ e− e+
open with probability 1 e 2Je. Then the open edges are distributed according to the FK-Ising
−
−
model. The two couplings between FK-Ising and Ising are the same.
ConsidertheGFFϕon distributedaccordingtoP . LetJ (ϕ)betherandominteraction
ϕ e
G | |
constants
J (ϕ) = W ϕ ϕ .
e | | e| e− e+|
Conditionally on ϕ, (sign(ϕ )) follows an Ising distribution with interaction constants
x x V
| | ∈
(J (ϕ)) : indeed, the Dirichlet form (1.1) can be written as
e e E
| | ∈
(ϕ,ϕ) = κ ϕ(x)2 + (ϕ(x))2( W ) 2 J (ϕ)sign(ϕ(e ))sign(ϕ(e )).
x x,y e +
E − | | −
x V x V y x e E
X∈ X∈ X∼ X∈
Similarly, when ϕ Pϕ{x0},√2u has boundary condition √2u 0 on x0, then (sign(ϕx))x V has
∼ ≥ ∈
an Ising distribution with interaction (J (ϕ)) and conditioned on σ = +1.
Now, conditionally on ϕ, (ϕ˜) has FeK|-Is|ineg∈Edistribution with weighxt0s (1 e 2Je(ϕ)) .
− | | e E
Indeed, the probability for eO (ϕ˜) conditionally on ϕ is 1 e 2Je(ϕ), by L−emma 2.1, as∈in
− | |
∈ O −
Proposition 2.2.
Note that, given that (ϕ˜) has FK-Ising distribution, the fact that the sign of on its con-
O
nected components is distributed independently and uniformly in 1,1 can be seen either
{− }
as a consequence of Proposition 2.2, or from Theorem 1.
Given ϕ = (ϕ ) on the discrete graph , we introduce in Definition 2.1 as the random
x x V
∈ G
set of edges which has the distribution of (ϕ˜) conditionally on ϕ = (ϕ ) .
x x V
O ∈
Definition 2.1. We let (ϕ) be a random set of edges which has the distribution of (ϕ˜)
O O
conditionally on ϕ = (ϕ ) given by Lemma 2.1.
x x V
∈
2.4. Distribution of (˜ ) conditionally on . The distribution of (˜ ) conditionally
1 1 1
O L2 L2 O L2
on can be retrieved by Corollary 3.6 in [6], which reads as follows.
1
L2
6 TITUSLUPU,CHRISTOPHESABOT,ANDPIERRETARRE`S
Lemma 2.3 (Corollary 3.6in[6]). Conditionally on , the events e (˜ ), e E ( ),
1 1 1
L2 6∈ O L2 ∈ \O L2
are independent and have probability
(2.1) exp 2W L ( )L ( ) .
− e e+ L21 e− L21
(cid:16) q (cid:17)
This result gives rise, together with Theorem 1, to the following discrete version of Lupu’s
isomorphism, which is stated without any recourse to the cable graph induced by .
G
Definition 2.2. Let (ω ) 0,1 E be a percolation defined as follows: conditionally on
e e E
∈ ∈ { }
, the random variables (ω ) are independent, and ω equals 0 with conditional probability
1 e e E e
L2 ∈
given by (2.1).
Let ( ) the set of edges:
+ 1
O L2
( ) = ( ) e E ω = 1 .
+ 1 1 e
O L2 O L2 ∪{ ∈ | }
Proposition 2.4 (Discrete version of Lupu’s isomorphism, Theorem 1 bis in [6]). Given a
loop soup , let ( ) be as in Definition 2.2. Let (σ ) 1,+1 V be random spins
1 + 1 x x V
L2 O L2 ∈ ∈ {− }
taking constant values on clusters induced by ( ) (σ = σ if e ( )) and such
O+ L12 e− e+ ∈ O+ L12
that the values on each cluster, conditional on and ( ), are independent and uniformly
1 + 1
L2 O L2
distributed. Then
σ 2L ( )
x x 1
L2 x V
(cid:16) q (cid:17) ∈
is a Gaussian free field distributed according to P .
ϕ
Proposition 2.4 induces the following coupling between FK-Ising and random currents.
If V is finite, a random current model on with weights (J ) is a random assignment to
e e E
G ∈
each edge e of a non-negative integer nˆ such that for all x V,
e
∈
nˆ
e
e adjacentto x
X
is even, which is called the parity condition. The probability of a configuration (n ) satis-
e e E
∈
fying the parity condition is
1 (J )ne
PRC( e E,nˆ = n )= e ,
J ∀ ∈ e e RC n !
ZJ e E e
Y∈
where actually RC = Isg. Let
ZJ ZJ
(nˆ)= e E nˆ > 0 .
e
O { ∈ | }
The open edges in (nˆ) induce clusters on the graph .
O G
Given a loop soup , we denote by N ( ) the number of times the loops in cross the
α e α α
L L L
nonoriented edge e E. The transience of the Markov jump process X implies that N ( )
e α
∈ L
is a.s. finite for all e E. If α = 1, we have the following identity (see for instance [11]):
∈ 2
Loop soup and random current. Assume V is finite and consider the loop soup . Con-
1
L2
ditionally on the occupation field (L ( )) , (N ( )) is distributed as a random current
x 1 x V e 1 e E
L2 ∈ L2 ∈
model with weights 2W L ( )L ( ) . If ϕ is the GFF on given by Le Jan’s or
(cid:18) e e− L21 e+ L12 (cid:19)e E G
Lupu’s isomorphism, thenqthese weights are (J∈(ϕ)).
e
| |
INVERTING THE COUPLING OF THE SIGNED GAUSSSIAN FREE FIELD WITH A LOOP SOUP 7
Conditionally on the occupation field (L ( )) , ( ) are the edges occupied by a
x 1 x V 1
L2 ∈ O L2
random current and ( ) the edges occupied by FK-Ising. Lemma 2.1 and Proposition 2.4
+ 1
O L2
imply the following coupling, as noted by Lupu and Werner in [7].
Proposition 2.5 (Random current and FK-Ising coupling, [7]). Assume V is finite. Let nˆ
be a random current on with weights (J ) . Let (ω ) 0,1 E be an independent
e e E e e E
percolation, each edge beinGg opened (value 1) in∈dependently w∈ith∈pro{babi}lity 1 e Je. Then
−
−
(nˆ) e E ω = 1
e
O ∪{ ∈ | }
is distributed like the open edges in an FK-Ising with weights (1 e 2Je) .
− e E
− ∈
2.5. Generalized second Ray-Knight “version” of Lupu’s isomorphism. We are now
in a position to state the coupled version of the second Ray-Knight theorem.
Theorem 2. Let x0 V. Let (ϕ(x0))x V with distribution Pϕ{x0},0, and define (ϕ(0)) as in
∈ ∈ O
Definition 2.1. Let X be an independent Markov jump process started from x .
0
Fix u > 0. If τux0 < ζ, we let Ou be the random subset of E which contains O(ϕ(x0)), the
edges used by the path (Xt)0 t τux0, and additional edges e opened conditionally independently
with probability ≤ ≤
1 eWe|ϕ(e0−)ϕ(e0+)|−Weq(ϕ(e0−)2+2ℓe−(τux0))(ϕ(e0+)2+2ℓe+(τux0)).
−
We let σ 1,+1 V be random spins sampled uniformly independently on each cluster
∈ {− }
induced by , pinned at x , i.e. σ = 1, and define
Ou 0 x0
ϕ(u) := σ ϕ(0)2 +2ℓ (τx0).
x x x x u
q
Then, conditionally on τux0 < ζ, ϕ(u) has distribution Pϕ{x0},√2u, and Ou has distribution
(ϕ(u)) conditionally on ϕ(u).
O
Remark 2.6. One consequence of that coupling is that the path (Xs)s τux0 stays in the positive
connected component of x for ϕ(u). This yields a coupling between ≤the range of the Markov
0
chain and the sign component of x0 inside a GFF Pϕ{x0},√2u.
Proof of Theorem 2: The proof is based on [6]. Let D = V x , and let ˜ be the loop
0 1
\{ } L2
soup of intensity 1/2 on the cable graph ˜, which we decompose into ˜(x0) (resp. ˜D) the loop
G L1 L1
2 2
soup hitting (resp. not hitting) x , which are independent. We let and (x0) (resp. D) be
0 L12 L21 L12
the prints of these loop soups on V (resp. on D = V x ). We condition on L ( ) = u.
\{ 0} x0 L21
Theorem 1 implies (recall also Definition 2.1) that we can couple ˜D with ϕ(0) so that
L1
2
L ( D)= ϕ(0)2/2 for all x V, and (˜ ) = (ϕ(0)).
x L12 x ∈ O L12 O
Define ϕ(u) = (ϕ(u)) from ˜ by, for all x V,
x x V 1
∈ L2 ∈
ϕ(u) = 2L ( )
| x | x L21
q
and ϕ(u) = σ ϕ(u) , where σ 1,+1 V are random spins sampled uniformly independently
x x x
| | ∈ {− }
on each cluster induced by (˜ ), pinned at x , i.e. σ = 1. Then, by Theorem 1, ϕ(u) has
O L12 0 x0
distribution Pϕ{x0},√2u.
8 TITUSLUPU,CHRISTOPHESABOT,ANDPIERRETARRE`S
For all x V, we have
∈
(0)2
ϕ
L (˜ ) = x +L ( (x0)).
x L21 2 x L21
On the other hand, conditionally on L( ),
. 1
L2
P(e (˜ )) P(e (˜ ) e ( ))
1 1 1
P(e (˜ ) e (˜D) ( )) = 6∈ O L2 = 6∈O L2 | 6∈ O L2
6∈ O L21 | 6∈ O L12 ∪O L21 P(e 6∈ O(L˜D12)∪O(L21)) P(e 6∈ O(L˜D21)|e 6∈ O(L12))
P(e (˜ ) e ( ))
1 1
= 6∈ O L2 | 6∈ O L2 = exp W L ( )L ( )+W L ( D)L ( D) ,
P(e6∈ O(L˜D21)|e 6∈ O(LD12)) (cid:18)− eq e− L21 e+ L21 eq e− L21 e+ L21 (cid:19)
where we use in the third equality that the event e (˜D) is measurable with respect to the
6∈ O L1
2
σ-field generated by ˜D, which is independent of ˜(x0), and where we use Lemma 2.3 in the
L1 L1
2 2
fourth equality, for ˜ and for ˜D.
L12 L21
We conclude the proof by observing that (x0) conditionally on L ( (x0)) = u has the law
L1 x0 L1
of the occupation field of the Markov chain ℓ(τ2x0) under P ( τx0 < ζ). 2 (cid:3)
u x0 ·| u
3. Inversion of the signed isomorphism
In [9], Sabot and Tarr`es give a new proof of the generalized second Ray-Knight theorem
togetherwithaconstructionthatinvertsthecouplingbetweenthesquareofaGFFconditioned
by its value at a vertex x and the excursions of the jump process X from and to x . In this
0 0
paper we are interested in inverting the coupling of Theorem 2 with the signed GFF : more
precisely, we want to describe the law of (Xt)0 t τux0 conditionally on ϕ(u).
We present in section 3.1 an inversion inv≤olv≤ing an extra Poisson process. We provide
in Section 3.2 a discrete-time description of the process and in Section 3.3 an alternative
descriptionviajumprates. Sections3.4and3.5arerespectively dedicated toasignedinversion
of Le Jan’s isomorphism for loop soups, and to an inversion of the coupling of random current
with FK-Ising.
3.1. A description via an extra Poisson point process. Let (ϕˇ ) be a real function
x x V
on V such that ϕˇ = +√2u for some u >0. Set ∈
x0
Φˇ = ϕˇ , σ = sign(ϕˇ ).
x x x x
| |
We define a self-interacting process (Xˇ ,(nˇ (t)) ) living on V NE as follows. The process
t e e E
Xˇ starts at Xˇ(0) = x . For t 0, we set ∈ ×
0
≥
Φˇ (t) = (Φˇ )2 2ℓˇ (t), x V, and J (Φˇ(t)) = W Φˇ (t)Φˇ (t), e E.
x x x e e e e+
− ∀ ∈ − ∀ ∈
q
where ℓˇ (t) = t ds is the local time of the process Xˇ up to time t. Let (N (u)) be
an indepxendent0P1o{iXsˇsso=nx}Point Processes on R with intensity 1, for each edge e eE. Wu≥e0set
R + ∈
N (2J (t)), if σ σ = +1,
e e e e+
nˇe(t)= −
(0, if σe σe+ = 1.
− −
INVERTING THE COUPLING OF THE SIGNED GAUSSSIAN FREE FIELD WITH A LOOP SOUP 9
We also denote by ˇ(t) E the configuration of edges such that nˇ (t) > 0. As time increases,
e
the interaction paraCmet⊂ers J (Φˇ(t)) decreases for the edges neighboring Xˇ , and at some ran-
e t
dom times nˇ (t) may drop by 1. The process (Xˇ ) is defined as the process that jumps only
e t t 0
≥
at the times when one of the nˇ (t) drops by 1, as follows:
e
if nˇ (t)decreases by 1at timet, butdoes notcreate anew cluster in ˇ, thenXˇ crosses
e t t
• C
the edge e with probability 1/2 or does not move with probability 1/2,
if nˇ (t) decreases by 1 at time t, and does create a new cluster in ˇ, then Xˇ moves/or
e t t
• C
stays with probability 1 on the unique extremity of e which is in the cluster of the
origin x in the new configuration.
0
We set
Tˇ := inf t 0, x V, s. t. Φˇ (t) = 0 ,
x
{ ≥ ∃ ∈ }
clearly, the process is well-defined up to time Tˇ.
Proposition 3.1. For all 0 t Tˇ, Xˇ is in the connected component of x of the configu-
t 0
ration ˇ(t). If V is finite, the≤pro≤cess ends at x , i.e. Xˇ = x .
C 0 Tˇ 0
Theorem 3. Assume that V is finite. With the notation of Theorem 2, conditioned on ϕ(u) =
ϕˇ, (Xt)t τux0 has the law of (XˇTˇ t)0 t Tˇ.
Moreo≤ver, conditioned on ϕ(u−) =≤ϕˇ≤, (ϕ(0), (ϕ(0))) has the law of (σ Φˇ (Tˇ), ˇ(Tˇ)) where
O x′ x C
(σ ) 1,+1 V are random spins sampled uniformly independently on each cluster in-
dux′cexd∈bVy∈ˇ({Tˇ−), with}the condition that σ = +1.
C x′0
If V is infinite, then Pϕ{x0},√2u-a.s., Xˇt (with the initial condition ϕˇ = ϕ(u)) ends at x0, i.e.
Tˇ < + and Xˇ = x . All previous conclusions for the finite case still hold.
∞ Tˇ 0
3.2. Discrete time description of the process. We give a discrete time description of the
process (Xˇ ,(nˇ (t)) ) that appears in the previous section. Let t = 0 and 0< t < < t
t e e E 0 1 j
∈ ···
be the stopping times when one of the stacks n (t) decreases by 1, where t is the time when
e j
one of the stacks is completely depleted. It is elementary to check the following:
Proposition 3.2. The discrete time process (Xˇ ,(nˇ (t )) ) is a stopped Markov pro-
ti e i e E 0 i j
∈ ≤≤
cess. The transition from time i 1 to i is the following:
−
first chose e an edge adjacent to the vertex Xˇ according to a probability proportional
• ti−1
to nˇ (t ),
e i 1
−
decrease the stack nˇ (t ) by 1,
e i 1
• if decreasing nˇ (t ) by−1 does not create a new cluster in ˇ , then Xˇ crosses the
• e i−1 Cti−1 ti−1
edge e with probability 1/2 or does not move with probability 1/2,
if decreasing nˇ (t ) by 1 does create a new cluster in ˇ , then Xˇ moves/or stays
• e i−1 Cti−1 ti−1
with probability 1 on the unique extremity of e which is in the cluster of the origin x
0
in the new configuration.
3.3. An alternative description via jump rates. We provide an alternative description of
the process (Xˇ , ˇ(t)) that appears in Section 3.1.
t
C
Proposition 3.3. The process (Xˇ , ˇ(t)) defined in section 3.1 can be alternatively described
t
by its jump rates : conditionally on iCts past at time t, if Xˇ = x, y x and x,y ˇ(t), then
t
∼ { } ∈ C
(1) Xˇ jumps to y without modification of ˇ(t) at rate
C
Φˇ (t)
y
W
x,yΦˇ (t)
x
10 TITUSLUPU,CHRISTOPHESABOT,ANDPIERRETARRE`S
(2) the edge x,y is closed in ˇ(t) at rate
{ } C
2W Φˇy(t) e2Wx,yΦˇx(t)Φˇy(t) 1 −1
x,yΦˇ (t) −
x
(cid:16) (cid:17)
and, conditionally on that last event:
- if y is connected to x in the configuration ˇ(t) x,y , then Xˇ simultaneously jumps
C \{ }
to y with probability 1/2 and stays at x with probability 1/2
- otherwise Xˇ moves/or stays with probability 1 on the unique extremity of x,y
t
{ }
which is in the cluster of the origin x in the new configuration.
0
Remark 3.4. It is clear from this description that the joint process (Xˇ , ˇ(t),Φˇ(t)) is Markov
t
C
process, and well defined up to the time
Tˇ := inf t 0: x V, s.t. Φˇ (t) = 0 .
x
{ ≥ ∃ ∈ }
Remark 3.5. One can also retrieve the process in Section 3.1 from the representation in
Proposition 3.3 as follows. Consider the representation of Proposition 3.3 on the graph where
each edge e is replaced by a large number N of parallel edges with conductance W /N. Consider
e
now nˇ(N)(t) the number of parallel edges that are open in the configuration ˇ(t) between x and
x,y
C
y. Then, when N , (nˇ(N)(t)) , converges in law to (nˇ(t)) , defined in section 3.1.
t 0 t 0
→ ∞ ≥ ≥
Proof ofProposition 3.3: AssumeXˇ = x,fixy xandlete = x,y . Recallthat x,y ˇ(t)
t
∼ { } { } ∈ C
iff nˇ (t) 1.
e
≥
Let us first prove (1):
P Xˇ jumps to y on time interval [t,t+∆t] without modification of ˇ(t) x,y ˇ(t)
C |{ } ∈C
1
=(cid:0) P(nˇ (t) nˇ (t+∆t) =1, nˇ (t+∆t) 1 nˇ (t) 1) (cid:1)
e e e e
2 − ≥ | ≥
1 Φˇ (t)
y
= (2J (t) 2J (t+∆t))+o(∆t)= W ∆t+o(∆t).
2 e − e xyΦˇ (t)
x
Similarly, (2) follows from the following computation:
P x,y closed in ˇ(t+∆t) x,y ˇ(t) =P(n (t+∆t)= 0 nˇ (t) 1)
e e
{ } C |{ } ∈ C | ≥
(cid:0)P(nˇe(t) = 1, nˇe(t+∆t)= 0) e−2Je(t)(cid:1)2Je(t)
= = (J (t) J (t+∆t))+o(∆t)
P(nˇe(t) ≥ 1) 1−e−2Je(t) e − e
(cid:3)
Weeasily deducefromtheProposition 3.3andTheorem6thefollowing alternative inversion
of the coupling in Theorem 2.
Theorem 4. With the notation of Theorem 2, conditionally on (ϕ(u),Ou), (Xt)t τux0 has the
law of self-interacting process (Xˇ ) defined by jump rates of Proposition≤3.3 starting
Tˇ t 0 t Tˇ
with − ≤ ≤
Φˇ = (ϕ(0))2+2ℓ (τx0) and ˇ(0) = .
x x x u u
C O
q
Moreover (ϕ(0), (ϕ(0))) has the same law as (σ Φˇ(T), ˇ(Tˇ)) where (σ ) is a configuration
O ′ C x′ x∈V
of signs obtained by picking a sign at random independently on each connected component of
ˇ(T), with the condition that the component of x has a + sign.
0
C