Loop cluster on the discrete circle

The loop clusters of a Poissonian ensemble of Markov loops on a finite or countable graph have been studied in \cite{Markovian-loop-clusters-on-graphs}. In the present article, we study the loop clusters associated with a rotation invariant nearest neighbor walk on the discrete circle $G^{(n)}$ with $n$ vertices. We prove a convergence result of the loop clusters on $G^{(n)}$, as $n\rightarrow\infty$, under suitable condition of the parameters. These parameters are chosen in such a way that the rotation invariant nearest neighbor walk on $G^{(n)}$, as $n\rightarrow\infty$, converges to a Brownian motion on circle $\mathbb{S}^{1}=\mathbb{R}/\mathbb{Z}$ with certain drift and killing rate. In the final section, we show that several limit results are predicted by Brownian loop-soup on $\mathbb{S}^{1}$.

Define the clockwise edges set E µ n (l = (x 1 , . . . , x k )) = 1 k (Q (n) ) x 1 x 2 · · · (Q (n) ) x k−1 x k (Q (n) ) x k x 1 . (1) A loop is an equivalence 2 class of pointed loops. The loop measure is the corresponding push-forward measure on the space of loops. For simplicity of notations, we use the same notation l for a loop and µ for a loop measure.
Denote by L (n) α the Poisson ensemble (or "loop soup") of non-trivial loops with intensity measure αµ n where α > 0 is a fixed parameter. As in [LJL12], we define the loop cluster model as follows: two vertices are in the same cluster iff. they are linked through a sequence of connecting non-trivial loops. Another equivalent way is to define the closed edges: an undirected edge {x, y} is closed iff. there is no loop in the loop soup L (n) α which covers (x, y) or (y, x). Otherwise, we say the undirected edge {x, y} is open. Then the loop cluster is just the cluster connected by open edges. In [LJL12], as one of the examples, it is shown that the closed edges in loop cluster model on N form a renewal process. Moreover, the scaling limit of that process is a subordinator with potential den- , see Section 3 of [LJL12] in which this is proved in the finite marginal sense.
In this article, we consider the loop cluster C (n) α in the discrete circle G (n) . To be more precise, C (n) α is the collection of discrete arc separated by the closed edges. In Section 2, we calculate the probability that an edge is closed. Conditionally on {{1, n} is closed}, we show that the closed edges in this model form a conditioned renewal process.
It has the same distribution as closed edges in the model of a discrete interval. Thus, similar conditional result of the closed edges holds given that the loops avoid a particular vertex.
In Section 3, we identify the scaling limit of the conditioned renewal process as a subordinator conditioned to approach 1 continuously.
In Section 4, we strengthen the above convergence result to a convergence in the sense of Skorokhod, see Theorem 4.5.
In Section 5, we calculate the limit distribution of the cluster containing a particular vertex. By combining with the previous result of the limit distribution of the closed edges in 1 A pointed loop l = (x 1 , . . . , x k ) is a bridge on the graph from x 1 back to itself: x 1 → x 2 → · · · → x k → x 1 . For a pointed loop l = (x 1 , . . . , x k ), k is called the length of this pointed loop. A loop is non-trivial iff. k ≥ 2.
2 Two pointed loop are equivalent iff. they are the same under a circular permutation.
a discrete interval, we give a description of the limit conditional distribution of clusters given that the number of clusters is strictly larger than 1. Under the assumption that lim n→∞ n 2 c n = ǫ, we calculate the probability that there is only one cluster in the scaling limit. We summarize these results in Theorem 5.10.
In the final section, we provide an informal explanation for the convergence results from the point of view of the convergence of Poissonian loop ensembles.
Finally, we briefly present the difficulties and the techniques in the following. We would like to make use of the convergence result of the unconditioned processes proved in [LJL12]. Thus, we firstly calculate the Radon-Nikodym derivatives of the conditioned renewal process (resp. subordinator) with respect to the unconditioned renewal process (resp. subordinator) on a family of sub-σ-fields. 3 Then, we show the convergence of the Radon-Nikodym derivatives on these sub-σ-fields. Together with the convergence result in [LJL12], we conclude the convergence of the corresponding conditioned processes. In order to strengthen the convergence result to a convergence in the sense of Skorokhod, we need the tightness of the family of conditioned renewal process which is equivalent 4 to a uniform control of the càdlàg modulus of continuity. Thanks to the stationary and independent increments, we get the tightness of the unconditioned process as an application of Aldous' criteria, see Lemma 3.1. The difficulty of passing through the unconditioned processes to the conditioned processes is due to the absence of the Radon-Nikodym derivatives between the conditioned processes and the corresponding unconditioned processes on the whole path. Then, we have to cut the whole path into two parts and then prove the tightness for each part. For the first half part of the paths, the Radon-Nikodym derivatives exist and hence the tightness is established. 5 For the second half part of the paths, we use time reversal to get back to the first half part. For that purpose, we study in the end of Section 3 the left limit at the lifetime and the time reversal distribution of the conditioned subordinator. Finally, in order to get the limit distribution of the clusters in the multi-clusters case, we have to calculate the limit distribution of the cluster containing a particular vertex, e.g. x 0 . More precisely, recall that the edges uncovered by those loops avoiding the particular vertex x 0 is described by the closure of the range of a conditioned subordinator in the scaling limit. Now, we consider the loops passing through the vertex x 0 which are not too large to cover the whole space. This cluster might cover some edges which are not covered by the loops avoiding x 0 . Accordingly, we erase a part of the range of the conditioned subordinator. Then, the remaining part of the range of the subordinator represents the closed edges in the scaling limit. For this part, the key is the independence between the loops avoiding x 0 and those loops passing through x 0 which is guaranteed by the Poisson loop soup construction. In order to express the results explicitly, we calculate the Lévy measure of the subordinator in Lemma 5.6 by inverse Laplace transform.
2 Closed edges as a conditioned renewal process In order to calculate the probability that a particular edge is closed, we need the following classical result on the determinant of Toeplitz matrices. Please refer to Proposition 2.2 and Example 2.8 in [BG05].
Lemma 2.1 ( [BG05]). Let T 3,n be a n × n tri-diagonal Toeplitz matrix of the following Let S n be the following circulant n × n matrix: Let x 1 , x 2 be the roots of x 2 − ax + bc = 0. Then, One can check the following result from the definition of pointed loop measure.
Lemma 2.2. Define a modified generatorL (n) from L (n) by replacing (L (n) ) 1 n and (L (n) ) n 1 by 0. Letμ n be the pointed loop measure associated withL (n) . Then, As a corollary, conditionally on that the edge {1, n} is closed, the loop soup L (n) α is the Poisson loop ensemble of intensity measure αμ n .
Next, we present some useful properties which are frequently used throughout the paper.
Definition 2.1. Let F be a subset of the state space S and l is a loop on S. We say l is inside F if l does not visit any vertex in S \ F , which is denoted by l ⊂ F . Lemma 2.3. Let µ be the Markovian loop measure associated with a generator L on a state space S. Let F be a finite subset of the state space S. Then, µ(l is non-trivial , l ⊂ F, dl) is the Markovian loop measure associated with the generator L| F ×F . Moreover, Proof. One can check from definition that µ(l ⊂ F, dl) is the Markovian loop measure associated with the generator L| F ×F . It remains to prove that for a Markovian loop measure µ associated with generator L on a finite state space S, We see that Then, we give in the following proposition the probability that a particular edge is closed.
Proposition 2.4. Set Then, Proof. For a loop l, denote by N i j (l) the number of jumps from i to j. Then, for two adjacent vertices i and j, a loop l visits both i and j iff. N i j (l) + N j i (l) > 0. Recall that an undirected edge {x, y} is closed iff. there is no loop jumping between x and y among the Poissonian loop soup. Then, we have P[{1, n} is closed] = exp(−αµ n (N 1 n (l) + N n 1 (l) > 0)) = exp{−α(µ n (1) − µ n (N 1 n (l) + N n 1 (l) = 0))}.
By Lemma 2.2, whereμ (n) is the same as in Lemma 2.2. Then, by Lemma 2.3 and Lemma 2.1, In the following context, we consider the loop cluster conditionally on {1, n} being closed.
for all m ∈ Z, and L is null elsewhere.
According to Proposition 3.1 in [LJL12], in the case of loop cluster model on Z, conditionally on {{0, 1} is closed}, the left points of the closed edges form a renewal process.
Finally, in our situation, conditionally on {1, n} being closed, we can identify the left end points of closed edges as a renewal process conditioned to jump at n.
Remark 2.1. It is not hard to find the correspondence between the killing parameter κ in [LJL12] and our parameters c and p: 3 Finite-marginal convergence towards a conditioned subordinator In the following context, we always assume the following relation between the parameters c, p and κ: As mentioned in the proof of Proposition 2. with generator L such that L m m+1 = L m+1 m = 1/2, L m m = −(1 + κ/2) for m ∈ N + and L is null elsewhere. Then, associated with this L, we have a loop measure µ (κ) and a Poisson point process of loops of intensity αµ (κ) . The corresponding loop probability depends on κ and we will denote it by P (κ) .
In the following context, we always assume the following whenever the domain of α is not specified.
Hypothesis 3.1. Assume α ∈]0, 1[. For a càdlàg process X and a subset A of the state space, denote by T A the entrance time of A: T A def = {t ≥ 0 : X t ∈ A}. We denote by X t− the left hand limit lim s↑t X s .
Let T be a (F (ǫ) t , t ≥ 0)-stopping time. Then, ⌊ǫ α−1 T ⌋ is a (F n , n ∈ N) stopping time. In order to show the tightness, it is enough to verify the following Aldous' criteria (see [JS03]): for each strictly positive M and δ, M is the collection of (F (ǫ) t ) t -stopping times bounded by M. Since we already know the finite marginals convergence, condition (2) reduces to P[X (κ) ⌈ǫ α−1 θ⌉ |. By the finite marginals convergence, and the proof is complete.
Then, we get the convergence in the sense of Skorokhod. Using this result, we will show the convergence of the corresponding conditioned processes in the following proposition.
The law of the truncated process Z (κ/n 2 ) under P (κ/n 2 ) depends on n, κ and is denoted by Q n,κ . Conditionally on {n, n + 1} being closed, the left points of the closed edges together with the left point of {0, 1} form a renewal process conditioned to jump at n. Define a conditioned loop probability as follows:P n,κ [·] = P (κ/n 2 ) [·|{n, n + 1} is closed]. LetQ n,κ be the law of Z (κ/n 2 ) underP n,κ . As n tends to infinity, underP n,κ , (Z (κ/n 2 ) Before proving this, let us describe the law of (X 2. The conditioned process 6 Y (κ) is a h-transform of the original subordinator with respect to the excessive function x → u(1 − x). To be more precise, for y ∈ [x, 1[, its semi-group is given by t (x, dy).
Let Q x stand for the law of the Markov process with sub-Markovian semi-group t (x, dy) and initial state x. (We choose the càdlàg version of Proof.

The subordinator (X
for y > x. When y tends to x, U(x, y) tends to ∞. As a consequence, the drift coefficient d = 0, see Proposition 1.7 in [Ber99]. It is proved by H. Kesten [Kes69] that for a fixed x > 0, x does not belong to the range of the subordinator with probability 1, see Proposition 1.9 in [Ber99]. By applying the strong Markov property to any stopping time S, Then, we use Lemma 1.10 in [Ber99]: 6 More precisely, the process defined by the probability ]. 7 Here,Π represents the tail of the characteristic measure of the subordinator.
Then, we see that In particular, for fixed time t: Then, the rest will follow from the classical results on the h-transform, see Chapter 11 of [CW05].
Take a positive function g, we have P t Ug = ∞ t P s g ds and Ug = ∞ 0 P s g ds. Then, for all positive function g, we have P t Ug ≤ Ug and P t Ug increases to Ug as t decreases to 0. As a consequence, except for a set N of z of zero Lebesgue measure, y → u(y, z) is an excessive function, i.e.
Take a decreasing sequence (z n ) n with limit 1 which is outside of the negligible set N. As the increasing limit of a sequence of excessive functions y → u(y, z n ), y → u(y, 1) is excessive.
3. Before providing the proof, we would like to give a short explanation. From the symmetry of the loop model on the discrete segment, the graphs of the conditional renewal processes are centrosymmetric. Then, as the scaling limit, the conditional subordinator has a centrosymmetric graph. Thus, Y In the following, we will not use the discrete approximation described above. In- Let's begin to prove Y (κ) ζ− = 1. In order to prove this, it is enough to show that By applying Theorem 11.9 of [CW05] to the stopping time If X follows the law P 0 , then X + x has the law P x . Therefore, the above quantity Consequently, Performing the change of variable Then, we have that By the right-continuity of the path, Then, Then we see that the semi-group (Q t , t ≥ 0) is Feller. 5. By a classical result about time reversal, the reversed process is a moderate Markov process, its semi-groupQ t (x, dy) is given by the following formula: Then we use the duality between P t andP t : This implies that the semi-group (Q t , t ≥ 0) associated with the reversed process of Y is given byQ Then, by a change of variable, we find that it equals to the semi-group of 1 − Y (κ) .
By result 3 in this lemma, the reversed process starts from 1. Then, it is exactly the left-continuous modification of 1 − Y (κ) for Y (κ) starting from 0.
The above proposition gives the Radon-Nikodym derivative between the subordinator and its bridge on a sub-σ-field. We will prove Proposition 3.2 by showing the convergence of the Radon-Nikodym derivatives from the discrete case to the continuous case.
From Lemma 3.1, we know that the sequence of renewal processes S (κ/n 2 ) converges towards the subordinator X (κ) in the sense of Skorokhod. By the coupling theorem of Skorokhod and Dudley, we can suppose that S (κ/n 2 ) converges to X (κ) almost surely as long as our result only depends on the law. The Proposition 3.1 in [LJL12] gives the density of the renewal measure of the subordinator (X Since U(x, x+) = ∞, the drift of the subordinator is zero, see Theorem 5 in Chapter 3 of [Ber96]. Then, by Theorem 4 in Chapter 3 of [Ber96], for any x > 0, X T ]x,∞[ − holds with probability 1. Thus, almost surely.
Let Q x stand for the law of the Markov process with sub-Markovian semi-group Q t (x, dy) = u(1−y) u(1−x) P t (x, dy) and initial state x. By Lemma 3.3, where P x is the law of the process X (κ) starting from x. We fix any δ > 0, lim n→∞Q n,κ 1 n Z (κ/n 2 ) We have thatQ n,κ [ 1 n Z (κ/n 2 ) ≤ 1]. Therefore, for any fixed t, (under the lawQ n,κ and Q 0 respectively) as n tends to infinity. In particular, we have Taking any bounded continuous function f , by the coupling assumption and dominated Therefore, we have the finite marginals convergence.

Convergence in the sense of Skorokhod
We will strengthen 9 the finite marginals convergence to the convergence in the sense of Skorokhod.
8 The dominating sequence is where the sub-Markovian process Y (κ) is the conditioned subordinator defined in Lemma 3.3.
Remark 4.1. The purpose of taking the value "−1" is to ensure that the last jump is larger than a strictly positive constant.
By Theorem 3.21 of [JS03], (Z (κ/n 2 ) ⌊n 1−α t⌋ /n, t ≥ 0) is tight iff. we have tightness for (H (n) , n ≥ 1) and (R (n) , n ≥ 1). Roughly speaking, the tightness is equivalent to a uniform control of the càdlàg modulus of continuity. The tightness of (H (n) , n ≥ 1) gives the uniform control of the càdlàg modulus of continuity on the first half parts of the paths. Whereas the tightness of (R (n) , n ≥ 1) gives the uniform control of the càdlàg modulus of continuity on the second half parts. In this way, we can find a uniform control over the whole path and then we can conclude the tightness of ((Z (κ/n 2 ) ⌊n 1−α t⌋ /n, t ≥ 0), n ≥ 1). The details are presented in the proof of Lemma 4.4 in the following context.
For the purpose of self-containedness, we will explain several notations and state a criterion of tightness for stochastic process. They are taken from [JS03]. See also [EK86].
10 Notice that there is no restriction on the last interval. 2. for all N ∈ N + , η > 0, ǫ > 0, there exists n 0 ∈ N and θ ∈]0, 1[ such that Then we turn to prove tightness for (H (n) , n ≥ 1) and (R (n) , n ≥ 1). ⌊n 1−α s⌋ , s ≥ 0) exceeds 1 2 . It is enough 11 to prove that for any bounded continuous f , The proof is very similar to that of Proposition 3.2. Thus, we merely provide a stretch of proof here.
By the coupling theorem of Skorokhod and Dudley, we can suppose that the renewal process S (κ/n 2 ) converges to the subordinator X (κ) almost surely as long as our result only depends on the law. Since for any x > 0, X T ]1/2,∞[ . Consequently, the quantity ( * * ) converges to almost surely. For fixed δ > 0, the quantity ( * * ) is uniformly bounded from above as long as 1 n Z (κ/n 2 ) ⌊n 1−α T Then, by dominated convergence, for all δ > 0, we have lim n→∞Q n,κ 1 n Z (κ/n 2 ) It implies that Taking any bounded continuous f , by the coupling assumption and dominated conver-  Consequently, there exists n 0 ∈ N + and θ > 0 such that for n ≥ n 0 , there exists a partition The dominating sequence is ( * * ).

The limit distribution of the loop clusters.
In this section, we would like to have a description of the loop clusters together with its scaling limit. We will always assume the following assumption when we talk about its scaling limit: We have seen in the above sections that the conditioned renewal processes (formed by the closed edges conditionally on the closeness of a particular edge) converges towards a 13 T  Proposition 5.1. The closed edges associated with the ensemble of loops avoiding the vertex 1 can be viewed 14 as a conditioned renewal process. The limit of these conditioned renewal processes is a conditioned subordinator described in Lemma 3.3.
Thus, it remains to study the ensemble of loops passing the vertex 1. For this part, we need to introduce several notations: We know that Z is a covering space of the discrete circle G (n) under the following mapping π (n) : π (n) (i + kn) = i + 1 for k ∈ Z and i = 0, . . . , n − 1. iff. the following conditions are all fulfilled: • Rot(l) = 0, • l passes through the vertex 1 in the discrete circle G (n) , • suppose l pt,1 and l pt,2 are both in the equivalence class l and they both start from the vertex 1 in G (n) . Denote byl pt,i (i = 1, 2) the unique pointed loop on Z starting from 0 as the lift of l pt,i (i = 1, 2). Then,l pt,1 andl pt,2 are equivalent pointed loops. Lemma 5.2. Let µ n,Z be the non-trivial loop measure on Z associated with the following generator L: Remark 1.1 in [LJL12]. Thus, the Markovian loop measure remains the same if we replace the above generator by the following one By the definition of the Poisson random measure, (L (n) α,i , i = 1, 2, 3, 4) are independent. We see that L α,4 . It is sufficient to study L  α,4 do not change if we replace (p n , 1−p n , 1+c n ) by ( p n (1 − p n ), p n (1 − p n ), 1 + c n ) (or ( 1 2 , 1 2 , 1 + κ (n) 2 ) equivalently). The non-symmetry only affects the distribution of L (n) α,2 . Therefore, we will use the symmetrized parameter ( 1 2 , 1 2 , 1 + κ (n) 2 ) when we study the distribution of L where x 1 , x 2 are the roots of the polynomial x 2 −(1+ κ (n) 2 )x+ 1 4 . As n tends to infinity, under the assumption that lim n→∞ n 2 κ (n) = κ, the pair of random variables ( An n , Bn We calculate the above determinants by using Lemma 2.1:

Proof. We fix a sub-interval of
where x 1 and x 2 are the roots of the polynomial x 2 − 1 + κ (n) 2 x + 1 4 . Therefore, .
In particular, µ n,Z (l is a non-trivial loop passing through 0, l is contained in [−n + 1, n − 1]) . Therefore, µ n,Z (l passes through 0, l ⊂ [−n + 1, n − 1], but l is not contained in [−m n , M n ]) . Then, . We see that Finally, we get the convergence result for (− An n , Bn n ) under the assumption that lim n→∞ n 2 κ (n) = κ.
We are mostly interested in the scaling limit 15 of P[(random clusters given by L α,3 ) ∈ ·|there exists at least two clusters]. 15 We normalize the discrete intervals to have length 1.
(As we have explained in the paragraph below Remark 5.1, it is as same as the distribution of random clusters given that there exists at least 2 clusters.) We have described in Proposition 5.3 the distribution of the random interval covered by the loops in Lift(L (n) α,3 ). Moreover, we also identify the clusters of L (n) α,1 as the complement of the closure of the range of a conditioned subordinator in the scaling limit, see Proposition 5.1. By combining these two results together, we get the distribution of the closed edges in the scaling limit. Let (A, B) be a pair of real valued variables independent of Y (κ) and with the following distribution: Then, in the sense of Skorokhod.
There is another way to express the scaling limit. Firstly, we could calculate the limit distribution of the cluster containing the vertex x 0 for a fixed point x 0 in the discrete circle. (By rotation invariance of the model, we previously take x 0 to be 1.) In the discrete model, this cluster is a discrete arc containing x 0 . We map it to Z such that x 0 is mapped to 0. When there exists a closed edge, we identify this cluster as a random interval [−G n , D n ] such that G n , D n ≥ 0 and G n + D n ≤ n − 1. We will give the limit distribution of (G n /n, D n /n) given that the cluster at 0 does not contain all the vertices. Next, we calculate the limit distribution of other clusters given the cluster containing the vertex x 0 . From the construction of the discrete model, given the cluster containing the vertex x 0 , the closed edges have the same conditional law as the closed edges in the random discrete interval J x 0 where J x 0 is the complement of the cluster at x 0 . As a result 16 , we can identify the limit of closed edges. The details will be presented in Theorem 5.10. For this alternative way to express the scaling limit, it is enough for us to state the limit joint distribution of P[G n /n ∈ dx, D n /n ∈ dy].
In fact, we have the following proposition: 16 See Remark 4.4 Proposition 5.5.
Conditionally on the existence of closed edges, (G n /n, D n /n) converges in distribution towards G, D where the density q(x, y) of (G, D) is given by Remark 5.2. Proposition 5.5 implies that the following probability goes to 0 as n tends to infinity: The reason is as follows: Since P[(G n /n, D n /n) ∈ ·|∃ closed edges] converges towards (G, D) and the distribution of G + D has no atom, we must have lim n→∞ P[G n + D n = n − 1|∃ closed edges] = 0.
To prove Proposition 5.5, we need the following lemmas.
Lemma 5.6. The Lévy measure Π of the subordinator of the renewal density u(x) = ( 2 √ κ 1−e −2 √ κx ) α is given by the following expression: Proof. The Lévy measure Π and the renewal density u are related through the Laplace exponent of the subordinator as follows: whereΠ is the tail mass of Π. Then, we compute Φ(λ) from u: We change the variable x by log(1−s) Then, we use the following equality 17 that Beta(x, y) · Beta(x + y, 1 − y) = π x sin(πy) .
We see that Now, we change the variable y by e −2 √ κu : Thus,Π Finally, we find Π(dt) by calculating the derivative ofΠ: Lemma 5.7. For the subordinator X (κ) of potential density u( Proof. According to Lemma 1.10 in [Ber99], By Lemma 5.6, Thus, By performing the change of variable t = 1−e −2 √ κz 1−e −2 √ κx , we see that Lemma 5.8. Consider the càdlàg subordinator bridge Y (κ) introduced 18 in the above sections, see Lemma 3.3. Let Q 0 stand for the law of this Markov process starting from 0.
Fix a positive measurable function f : [0, 1] 2 → R + and 0 < a, b < 1 such that a + b < 1. Then, T [a,∞[ <1−b} 18 We get this process by applying the Doob's harmonic transform to a subordinator with potential density U (x, y) = 1 {y>x} α with respect to the excessive function x → U (x, 1). This is the process conditioned to approach 1 continuously and killed when it exceeds 1.
Proof. Let X (κ) be the subordinator with the potential density U(x, y) = 1 {y>x} Let P 0 stand for its law with starting state 0. We have seen that this subordinator has zero drift. Consequently, for any fixed a > 0, P 0 [a belongs to the closure of the range of X (κ) ] = 0 which is frequently used to replace "≤" or "≥" by strict inequalities "<" or ">". Set u(x) = U(0, x) and denote by Π its Lévy measure. According to Lemma 1.10 in [Ber99], By using the strong Markov property at time T [a,∞[ for the subordinator X (κ) , we see that where φ is a positive measurable function. Therefore, for a positive measurable function φ, we have As a result, T [a,∞[ <1−b} = 0<z 1 <a<z 1 +z 2 <z 1 +z 2 +z 3 <1−b<z 1 +z 2 +z 3 +z 4 <1 By performing the change of variables x = z 1 + z 2 , y = 1 − z 1 − z 2 − z 3 : Finally, the calculation is finished by using Lemma 5.7.
Proof of Proposition 5.5. Firstly, we deduce from Proposition 5.3 the density ρ(a, b) of Take an independent subordinator bridge Y (κ) defined in Lemma 3.3. Set Moreover, conditionally on the existence of closed edges, (G n /n, D n /n) converges in distribution towards (G, D) whose density equals By Lemma 5.8, for x > 0, y > 0, x + y < 1, Therefore, We make a change of variable as follows: Then, For the simplicity of notation, set δ = pq dp dq We take w = 1 1+δp : pq dp dq By Euler's reflection formula, Beta(1 − α, α) = π sin(πα) . Thus, for x > 0, y > 0, x + y < 1, Then we have that We have

By integration by parts,
Then, Finally, one can deduce the distribution of (G, D).
Next, we will calculate the probability that there is no closed edge. For that part, we need the following lemma.
• By Proposition 5.5, Conditionally on (G ′ n , D ′ n ), Z ′(n) is the process of left end points of closed edges 19 on discrete interval {1, . . . , n−1−G ′ n −D ′ n } with killing measure κ (n) . By assumptions, lim n→∞ κ (n) n 2 = κ. Therefore, lim n→∞ (n−1−G ′ n −D ′ n ) 2 κ (n) = (1−G ′ −D ′ ) 2 κ. By Theorem 4.5 and its following remarks, 6 Informal relation with convergence of loop soups Finally, we would like to give informal remarks of the previous results from the point of view of the scaling limit of the loop soup. Please refer to [Lup13] for the Markovian loop soup of one dimensional diffusions.
Firstly, let us give an informal explanation of the convergence result for the closed edges in the loop cluster model on N which is proved in [LJL12].
It is known that the Brownian loop soup is the scaling limit of simple random walk loop soup. Intuitively, the scaling limit of the closed edges probably 20 has some relationship with the zero set of the occupation field of the Brownian loop. In fact, as an application of Proposition 3.4 of [Lup13], the occupation field of Brownian loop soup with killing rate κ 2 within ]0, ∞[ is a homogeneous branching process with immigration. It is the solution of the following SDE: where B is a Brownian motion and X 0 = 0. (It belongs to the Cox-Ingersoll-Ross (CIR) family of diffusions which could be viewed as a generalization of squared Bessel process. More precisely, it is a radial Ornstein-Uhlenbeck process of dimension 2α with parameter − √ κ, see [GJY03].) To be more precise, when we apply Proposition 3.4 in [Lup13], we take the non-increasing positive harmonic function to be u ↓ (x) = e − √ κx and take the non-decreasing positive harmonic function to be u ↑ (x) = 2 √ κ sinh( √ κx) such that the Green function density with respect to the Lebesgue measure G(x, y) is given by G(x, y) = u ↑ (x)u ↓ (y) for x ≤ y. (This normalization is required in order to apply Proposition 3.4 in [Lup13]). We see that w(x) = Wronskian(u ↓ , u ↑ ) = 2. One can check that the zero set is given by the range of the subordinator with potential density Next, we consider the loop cluster over a discrete interval which is considered in this article.
If we suppose the approximation by Brownian loop soup within ]0, 1[ works, then we expect that the limit distribution of the closed edges is the zero set of the occupation field of this Brownian loop soup. By Proposition 3.4 of [Lup13], we know that the occupation field over the interval ]0, 1[ indexed by the position t ∈]0, 1[ is the solution of the following SDE: sinh( √ κ(1 − t)) Y t dt, t ∈ [0, 1].
In fact, it is the bridge of a square radial OU process of dimension 2α of parameter −λ from 0 to 0 of fixed time duration 1. Please refer to [FPY93] for Markovian bridge and 20 There is not an immediate consequence of convergence of loop soup. That's why our explanation stays informal.
refer to [GJY03] for the transition density of square radial OU process and its relationship with squared Bessel process. Let D t be the first time of hitting 0 after time t. Then, the Radon-Nikodym derivative of the bridge process over the square radial OU process is 1 {Dt<1} 1 − e − √ κ 1 − e −2 √ κ(1−Dt) α restricted on the sub-σ-field up to time D t . This is exactly the same as U (Dt,1) U (0,1) which is used to construct our subordinator bridge. Then, one can check that the zero set of the bridge of the square radial OU process agrees with the range of the conditioned subordinator defined in Lemma 3.3.
Finally, we would like to point out the way to get the limit distribution of (A, B) in Ray-Knight theorem for diffusions, conditioned on the total local time at 0, U and V are two independent copies of square radial OU processes of dimension zero and parameter − √ κ, see e.g. Proposition 3.1 [Lup13]. Thus, it is enough to compute the first hitting time of 0, and then integrate them with respect to the total local time. The density of the first hitting time of 0 for our square radial OU process is given by see e.g. [ELY99] (Corollary 3.19). Finally, we get the joint density of the first hitting times of 0 for U and V . We see that it is exactly the same density as the limit distribution of (A n /n, B n /n) as n → ∞ up to a normalization constant, see Lemma 5.3.