Weighted graphs and complex Gaussian free fields

We prove a combinatorial lemma about the distribution of directed currents in a complex"loop soup"and use it to give a new proof of the isomorphism relating loop measures and complex Gaussian fields.


Introduction
Loops and related measures are useful tools in the analysis of random walks. They have recently come under study in [7] as a discrete analogue of the Brownian loop soup introduced in [8], which itself was motivated by the study of the Schramm-Loewner evolution. Such measures were also explored in a continuous setting by Yves Le Jan [3]. One of his findings was the connection between the Gaussian free field and the occupation field of a Poissonian ensemble of Markov loops. This connection can be viewed as a version of the Dynkin's isomorphism theorem [1]. In [2] this isomorphism was extended to connect certain non symmetric Markov processes and complex Gaussian fields.
A version of the isomorphism theorem using the discrete time loop soup was proved in [4] and [6]. Random walk on a finite graph can be fully described by a substochastic transition matrix Q. Any event in this setting is essentially a union of chain trajectories, and its probability is an additive function on sets of trajectories, which is related to Q. It is not uncommon in statistical physics to interpret events and their probabilities as configuration collections and weights, respectively. Even if Q takes complex values, in some cases we can still build meaningful objects that have probabilistic analogues, such as loop soups, by putting potentially complex weights on paths. That proof involves the loop soup at intensity 1/2 and uses undirected currents.
This generalization is one of the main points of the discussion in [6]. The complex Gaussian free field is introduced there as a pair of real Gaussian free fields with potentially negative correlations between fields and within each field. A version of the isomorphism theorem is formulated and proved there by comparing the Laplace transforms of a complex Gaussian field squared and a continuous occupation field of a complex loop soup. A combinatorial proof of the isomorphism can be found in [4], which is one the one hand discussed under the assumption that weights correspond to a certain probability space, and on the other hand does not use that assumption in a significant way. This note heavily relies on, and serves as a continuation those two papers. Here we adapt the arguments from [4] to the complex setting and extend some of the results of [6] to a wider range of weights. The key new results here are the exact distribution on directed currents, induced by the random walk loop soup at intensity 1 (which is presented in our Proposition), and the isomorphism theorem, which connects the continuous occupation field of the loop soup and the absolute value of a complex Gaussian free field squared. To prove the latter, we do not utilize the Laplace transform, which is commonly used to show the isomorphism in the literature. Another advantage of the theorem presented here is that it involves the random walk loop soup measure at intensity 1, which is easier to analyze than the loop soup at intensity 1/2. Hopefully, our proof sheds some light on a seemingly accidental connection between the loop measures and the Gaussian free field. The isomorphism theorems proved in [4] and [6] become a special case of our Theorem. This paper is structured as follows. We first introduce the setup and basic notations. Then we state the main results of the paper, including the isomorphism theorem. All the proofs are contained in the final section.

Basic definitions
Consider a finite complete digraph G = (V, E) with N = |V | vertices. Directed edges E ∼ = V × V are identified with ordered pairs of vertices; note that we allow self-edges. The set of vertices is ordered: V = (v j ) N j=1 := (v 1 , v 2 , . . . , v N ) and whenever we take an ordered subset of it, we preserve the order between the vertices. We will often use the following ordered subsets: . For future reference, we use (·) and {·} to denote ordered and unordered sets, respectively.
We call functions q : E → C weights on directed edges. For any u, v ∈ V , we write q uv instead of q(u, v) for brevity. Let Q = (q uv : u, v ∈ V ) ∈ C N ×N , and call q a Hermitian weight, if Q is Hermitian. We say that q is integrable, if ρ(|Q|) < 1, where ρ denotes the spectral radius and |Q| = |q uv | : u, v ∈ V . The notation |Q| always refers to the matrix of absolute values; to denote the matrix determinant, we use det. If U ⊆ V , then Q U denotes the restriction of Q to rows and columns that correspond to vertices in U , that is, A path ω of length |ω| = k in U ⊆ V is a sequence of k + 1 vertices in U : Paths of length 0 are called trivial. Equivalently, a path ω is a sequence of k = |ω| directed edges, where the second vertex of each edge is the same as the first vertex of the next edge in the sequence: Concatenation of edges is represented by ⊕ here. It can be applied to paths similarly, as long as each next path in the sequence that we concatenate starts where the previous path has ended.
If {u, v} ⊆ U , let P U (u, v) denote the set of paths in U starting at u and ending at v. Paths P U (v) := P U (v, v) are called loops rooted at v in U and contain the trivial loop consisting of a single vertex. We use P U to denote ∪ u,v∈U P U (u, v). We have previously defined q as a function on directed edges, and we will also use it to denote the following function on P V : and we let q be equal to 1 on the trivial loops in P V . Note that if q is integrable, then it defines a complex measure on P V .
Note that we are not counting the visit at time 0. In particular, n = 0 on trivial loops. We We can extend the definition of q to currents. If C ∈ C V , we set An (oriented) unrooted loop is an equivalence class of nontrivial rooted loops under cyclic permutations: The set of unrooted loops is denoted by L. If a rooted loop ω represents l ∈ L, we will write ω ∈ l. The set of unrooted loops whose representatives stay in U ⊆ V and visit v ∈ U at least once is denoted by L U (v). The definitions of q, n and c are extended from P V to L by taking any rooted representatives: Such an extension does not depend on the choice of the representative.
If X is any countable set, we let N X fin stand for finite multisets of elements from X , that is, the set of functions X → N, which are supported on a finite set. Local times n and currents c can be viewed as functions on N L fin : Main results

Loop measures and occupation fields
If q is an integrable weight on V , we define the unrooted loop measure m by where d(l) is the largest integer d such that every representative of l consists of the concatenation of d identical loops. If q is integrable, m is a complex measure on L.
The (random walk) loop soup (at intensity 1) is a collection of independent Poisson random variables indexed by L with intensity e −m(l) . If q is complex, we interpret this as the measure on finite multisets of unrooted loops: We write ν c and ν * for the pushforwards of this measure as measures on C V and N V : where the sums are over all s ∈ N L fin that produce the current C and the local time n ′ , respectively. We call ν c the (directed) current field and ν * the discrete occupation field.
Our first result, gives the distribution for the current field of any integrable weight (not necessarily Hermitian).
Proposition. If q is an integrable weight and C ∈ C V , then (3. 3) The proof of this fact is combinatorial in nature and revolves around the identity (4.9), which can be viewed as a useful result on its own.
Given a discrete occupation field, the continuous occupation field is obtained by independently at each vertex u replacing n u with the sum of n u +1 independent exponential random variables with mean one. We write this distribution as ν n . We can give its density with respect to Lebesgue measure λ N on R N + : Due to the proposition above, this can be written as

Bubble soup
In order to prove the Proposition, we define and analyze certain auxiliary measures. For v ∈ U ⊆ V , a growing loop in U at v induced by q (at time t = 1) is a "random" rooted loop in P U (v) sampled as follows. If ν g denotes the measure on the growing loop on P U (v), then where G U (v, v) denotes the Green's function. Note that Indeed, the first equation follows from a standard renewal argument and uses the fact that q is a complex measure on paths with finite total variation. For the final expression, see Lemma 3.1 in [6].
The bubble measure ν b is the measure on N -tuples ω = (ω j : ω j ∈ P V j (v j )) N j=1 given by the product measure Here we have used the following well-known formula (for example, see the Proposition 3.5 in [4]): Note that the definition depends on the ordering of V , but if we forget the order in ω, then it is immediate from (3.6), that the resulting measure will not depend on the order in V .
The following statement allows us to work with bubble soup instead of the unrooted loop soup to derive the current distribution (3.3).
Lemma. For any ordering of V , the measure induced on currents by ν b is ν c .
This follows immediately from the Proposition 5.8 of [4] for general intensities when Q is a substochastic matrix. In the case of intensity one and positive weights a similar result was established in the Proposition 9.4.1 of [5]. Unfortunately, there was a misstatement in the latter proof of Problem 9.1 which was part of the proof. Because of this unfortunate misprint, we will redo the proof here. We will also show that the argument applies to general integrable weights.

Isomorphism theorem
We now assume that q is Hermitian, and thus G = (I −Q) −1 is a positive definite Hermitian matrix. The (discrete centered) complex Gaussian free field Z = (Z v : v ∈ V ) on V with covariance G is a random complex vector in C N with density with respect to the Lebesgue measure on C N ; here ·, · denotes the dot product of complex vectors. Z satisfies the following covariance relations: We can decompose the Green's function into the real and imaginary parts: G = G R + iG I . Since G is Hermitian, G R is symmetric and G I is antisymmetric. A complex Gaussian free field on a set of N elements can be viewed as a real field on 2N elements. Indeed, let where ⊕ denotes the concatenation of sequences. According to the Proposition 4.5 in [6], that is, the probability distributions of these complex random vectors are the same.
Let f |Z| 2 denote the density of |Z| 2 = (Z u Z u ) u∈V with respect to Lebesgue measure λ N on R N + . According to the Theorem 2 in [6], we should expect that the continuous occupation field at intensity 1 has the same density as the square of the absolute value of a complex Gaussian free field. The following generalizes the isomorphism theorems as stated in [4] and [6].
Theorem. If q is an integrable, Hermitian weight, then the continuous occupation field ν n has the same distribution as |Z| 2 /2 where Z is a complex Gaussian free field with covariance matrix G = (I − Q) −1 .
In view of (3.9), this result can be interpreted differently. If Z ′ and Z ′′ are two real Gaussian free fields with correlation structure as in (3.8), then ν n has the same distribution as |Z ′ | 2 + |Z ′′ | 2 /2.
If Q is a nonnegative integrable weight, then the distribution of |Z| 2 is the same as that of |X| 2 + |Y | 2 where X, Y are independent real Gaussian fields with covariance matrix (I − Q) −1 . In this case, the result above reduces to the usual isomorphism theorem, which states that |X| 2 /2 has the same distribution as the continuous occupation field at time 1/2.

Proof of the Lemma
We start by combining (3.5) and (3.7) to see that det G = e m(L) . In view of that, and also (3.2) and (3.6), we see that the goal is to prove that where the first sum is over all such tuples ω = (ω j ) N j=1 , that N j=1 c(ω j ) = C.
Fix any j ∈ [N ]. Let u = v j , L = L j and P = P V j (u) for brevity. We shall now construct a mapping from N L fin to P . Take any s ∈ N L fin and order the unrooted loops in it arbitrarily.
where, as before, d(l) is the largest integer such that any rooted representative of l is a concatenation of d(l) identical rooted loops, and S s = l∈L s l .
Note that l∈L q(l) s l = q(ω) whenever ψ(s, o) = ω for some o ∈ O(s). We can now see from (4.2) and (4.3), that it is sufficient to prove that for any ω ∈ P with n 0 = n u (ω) ≥ 1, 1 S s ! l∈L n u (l) s l = 1, There is a natural bijection between ψ and finite sequences of positive integers (n j ) k j=1 with k j=1 n j = n 0 , which we call seq(k, n 0 ). Multiplying both sides of (4.4) by n 0 !, we see that it is equivalent to the identity ∞ k=1 seq(k,n 0 ) To establish this we need to show that the left-hand side equals the number of permutations of n 0 elements. To see this, suppose (n j ) k j=1 are given and (a j ) n 0 j=1 = (a 1 , a 2 , . . . , a n 0 ) is a permutation of (1, 2, . . . , n 0 ). Then we get another permutation by putting parentheses down: (a 1 , . . . , a n 1 ), (a n 1 +1 , . . . , a n 1 +n 2 ), . . . , (a n 1 +n 2 +...+n k−1 +1 , . . . , a n 0 ), and viewing this as a representation of a permutation by its cycle structure. However, there are many ways to represent the same permutation. There are k! ways to permute the elements of (n j ) k j=1 , and for the cycle corresponding to n j there are n j choices for which element to call a n 1 +n 2 +...+n j−1 +1 . This establishes our claim.

Proof of the Proposition
We will prove this by induction on the number of vertices N = |V |, viewing the current measure as the pushforward of a bubble soup under the mapping c. If C uv = 0 for some u, v with q uv = 0, then both sides of (3.3) equal zero. Hence we will assume that C is a current such that q uv = 0 if C uv = 0.
If V = {x} is a singleton with q = q xx , then the bubble soup consists only of self-loops ω k = (x, x, . . . , x) at x in V with |ω k | = k and

Therefore (3.3) holds in this case.
Now suppose that V has N = |V | ≥ 2 vertices, x ∈ V , and let U = V \ {x}. The induction assumption is that (3.3) holds for the currents C ∈ C U , where Q U denotes Q restricted to U . We call this measure ν 0 . Using (3.7), we see that If we order the vertices in V in such a way that x is the first vertex, then the construction of a bubble soup in V starts by growing loops at x in V , and the growing loops that follow are fully contained in U . Let ν + be the measure on currents induced by a growing loop at x in V : where L(C + ) is the set of loops in P V (x, x), that induce a current C + : Note that for any ω ∈ P V we have q(ω) = q c(ω) , therefore all the summands in (4.6) are equal, since they correspond to the same current. We can now rewrite (4.6) in a simplified form: where W (C + ) = |L(C + )|. To get the distribution on currents induced by a bubble soup in V , we calculate the measure ν 0 induced by a bubble soup in U , the measure ν + induced by a loop growing at x in V , and take their convolution: where C ∈ C V and Since q(C + +C 0 ) = q(C + ) q(C 0 ), we can combine (3.3), (4.7) and (4.8) to see that it suffices to prove the following combinatorial statement: Note that the products can be written as multinomial coefficients: Let us fix an ordering of V starting with x and we use the same ordering on U (ignoring x). For every u ∈ V , let n u = n u (C) and let S u = S u (C) be the set of n u -tuples a u = (a u 1 , . . . , a u n u ) in V n u that contain C uv elements v for every v ∈ V . Let S(C) = S u u∈V be the collection of such sequences. Note that the left-hand side of (4.9) is equal to |S| = u∈V |S u |. Now take any pair (C + , C 0 ) ∈ P C and let n u + , S + u , n u 0 and S u 0 be the corresponding quantities. Define The right-hand side of (4.9) is |S ′ |, thus it suffices to give a bijection between S and S ′ .
Suppose (a u ) u∈V ∈ S(C) are given. To map S to S ′ , we define ω ∈ P V (x, x) by means of an algorithm.
• Set ω = (x). If n x = 0, stop and output the trivial loop.
• Otherwise, let ω = (x, a x 1 ), remove a x 1 from a x and reset n x → n x − 1.
• Otherwise, if ω j = u, let ω j+1 equal a u 1 , remove a u 1 from a u , and reset n u → n u − 1.
If the algorithm is correct, then clearly ω ∈ L(C + ) for C + = c(ω), also C 0 := C − C + ∈ C U and (a u ) u∈V ∈ S(C 0 ). The correctness follows from the current property of C ∈ C V . We cannot encounter a situation where ω j = u = x and n u = 0, because that would imply that To get the inverse mapping, run the algorithm in reverse, that is, map an ω ∈ L(C + ) to an element in S(C + ) and concatenate it with a vector from S(C 0 ).

Proof of the Theorem
Fix an ordering V . To avoid cumbersome notation, identify vertices with integers: V = (1, 2, . . . , N ). For z ∈ C N , let x = Re[z] and y = Im[z]. If we do the change of variables x j + iy j = √ t j e iθ j , t j ∈ R + and θ j ∈ [0, 2π), we get To get the marginal density f |Z| 2 (t) for t ∈ R N + , we integrate over θ ∈ T := [0, 2π) N : t j t k q jk e i(θ k −θ j )    dθ (4.10) Next we find the density of the occupation field using its current representation. Suppose that we have a matrix C ∈ N N ×N . Then Since the right-hand side is always an integer for C ∈ N N ×N , we see that To find the density dν n /dλ N , we use (3.4) and (2.1): To see that this is equal to (4.10) and finish the proof, we use (4.11): exp t j t k q jk e i(θ k −θ j ) .