aa r X i v : . [ m a t h . P R ] S e p Loop-erased walks and random matrices
Jonas Arista and Neil O’Connell
Abstract.
It is well known that there are close connections between non-intersecting processes in one dimension and random matrices, based on thereflection principle. There is a generalisation of the reflection principle formore general (e.g. planar) processes, due to S. Fomin, in which the non-intersection condition is replaced by a condition involving loop-erased paths.In the context of independent Brownian motions in suitable planar domains,this also has close connections to random matrices. An example of this wasfirst observed by Sato and Katori (Phys. Rev. E, 83, 2011). We presentfurther examples which give rise to various Cauchy-type ensembles. We alsoextend Fomin’s identity to the affine setting and show that in this case, byconsidering independent Brownian motions in an annulus, one obtains a novelinterpretation of the circular orthogonal ensemble.
1. Introduction
It is well known that there are close connections between non-intersecting pro-cesses in one dimension and random matrices, based on the reflection principle.There is a generalisation of the reflection principle for more general processes, dueto S. Fomin [ ], in which the non-intersection condition is replaced by one involv-ing loop-erased paths. In the context of independent Brownian motions in suitableplanar domains, this also has close connections to random matrices, specificallyCauchy-type ensembles. An example of this was first observed by Sato and Ka-tori [ ]. We will present further examples, in particular, based on some domainswhich were discussed in Fomin’s original paper. We will also consider the circularsetting, with periodic boundary conditions, for this we extend Fomin’s identity tothe affine setting; we show that in this case, by considering independent Brownianmotions in an annulus, we obtain a novel interpretation of the Circular OrthogonalEnsemble of random matrix theory. Determinant formulas for the total weight of one-dimensional non-intersecting processes have many variations, both in continuous and discrete set-tings. They are also known as the Karlin-McGregor formula for Markov pro-cesses [ , , , ], or the Lindstr¨om-Gessel-Viennot lemma in enumerativecombinatorics [ , , , ]. Roughly speaking, the argument behind all these Mathematics Subject Classification. determinant formulas is the classical reflection principle, which allows the construc-tion of a particular one-to-one ‘path-switching’ map from a set of intersecting pathsonto itself, such that the map is its own inverse (see Section 2.1).For two-dimensional state space processes, it is not clear how to perform theclassical reflection principle, since the paths under consideration are allowed to haveself-intersections (or loops). However, there is a generalisation of the reflectionprinciple for more general (e.g. planar) paths, due to S. Fomin [ ], in which thenon-intersecting condition is replaced by one involving loop-erased paths . Thenit is possible to obtain a determinant formula (Theorem 2.2) for the total weightof discrete planar processes which satisfy Fomin’s non-intersection condition, herestated in the context of Markov chains: Fomin’s identity.
Consider a time-homogeneous Markov chain whose statespace is a discrete subset V of a simply connected domain Ω. Assume that thetransitions of the chain are determined by a (weighted) planar directed graph (withvertex set V ). Multiple loops are allowed. Distinguish a subset ∂ Γ ⊂ V of boundaryvertices and assume they all lie on the topological boundary ∂ Ω. Assume thatvertices a n , ..., a ⊂ V and b , ..., b n ⊂ ∂ Γ lie on the boundary ∂ Ω and are orderedcounterclockwise (along ∂ Ω), as in Figure 4. Therefore, if h ( a i , b j ) 1 ≤ i, j ≤ n, denotes the probability (or hitting probability ) that the Markov chain, starting at a i , will first hit the boundary ∂ Γ at vertex b j (if a i ∈ ∂ Γ, the chain is supposed towalk into V \ ∂ Γ before reaching b j ), then the n × n determinantdet( h ( a i , b j )) ni,j =1 , (1.1)is equal to the probability that n independent trajectories of the Markov chain X , ..., X n , starting at a , ..., a n , respectively, will first hit the boundary ∂ Γ at loca-tions b , ..., b n , respectively, and furthermore the trajectory X j will never intersectthe loop-erasure LE ( X i ) of X i , for all i < j , that is, X j ∩ LE ( X i ) = ∅ , for all 1 ≤ i < j ≤ n. (1.2)The above identity is the non-acyclic analogue of the determinant formula fornon-intersecting one-dimensional processes of Karlin-McGregor/Gessel-Viennot. Inthis respect, the following details are worth to remark: because of the nature ofthe underlying graph, trajectories of the Markov chain are allowed to have loopsand therefore, for a given trajectory, we can properly define its loop-erasure asthe self-avoiding path resulting from erasing its loops chronologically. Moreover,the determinant (1.1) gives the locations of the hitting points b , ..., b n along theboundary ∂ Γ, and the condition on the trajectories is given by (1.2), which forcesthe loop-erased paths to repel each other (see Section 2.2). The counterclockwisearrangement of paths is just a particular case in the more general combinatorialidentity given by S. Fomin in [ ], which can be applied to a wide range of configura-tions of n distinct paths, depending on the location of the initial and final verticesand the topology of the planar domain Ω.Section 3 is a first step towards the extension of the previous framework tonon-simply connected domains of the complex plane. There, we state and prove anaffine (circular) version of Fomin’s identity (Proposition 3.3), which can be seen asan extension of Fomin’s identity to the setting of the affine symmetric group ˜ A n . In Section 5 we relate this affine version with the Circular Orthogonal Ensemble(COE) of random matrix theory. In the context of Markov chains, our affine versionof Fomin’s identity can be stated as follows:
An affine version of Fomin’s identity.
Consider a time-homogeneousMarkov chain whose transitions are determined by the (directed) lattice strip G = Z × { , , ..., N } . Assume that the transition probabilities are space-invariant withrespect to a fixed horizontal translation S : G → G . If vertices a n , ..., a , b , ..., b n are ordered counterclockwise along the boundary (as in Figure 5), then the n × n determinant det X k ∈ Z ζ k h ( a i , S k b j ) ! ni,j =1 , (1.3)of hitting probabilities h ( a i , S k b j ), where S k = S ◦ S k − , k ∈ Z , and ζ = (cid:26) n is odd − n is even , is equal to the probability that n independent trajectories of the Markov chain X , ..., X n , starting at a , ..., a n , respectively, will first hit the upper boundary ∂ Γ = Z × { N } at any of the n cyclic permutations of the vertices b , ..., b n , shifted also byall possible horizontal translations by S k , k ∈ Z , and furthermore the trajectoriesare constrained to satisfy P j ∩ LE ( P j − ) = ∅ , < j ≤ n, and P ∩ LE ( S P n ) = ∅ . It is important to note that the non-intersection condition above is related tothe one between trajectories in a cylindrical lattice (or annulus on the complexplane), see Figure 1 and the introduction of Section 3. In an acyclic graph, theabove affine case agrees with the Gessel-Zeilberger formula for counting paths inalcoves [ ] (see Section 3.4). Our interest in the above determinant formulas reliesupon their applicability in the context of suitable scaling limits of Fomin’s identityand its affine version. It is well known that the two-dimensional Brownian motion B is the scaling limit of simple random walks on different planar graphs [ ]. Moreover,the loop-erasure of those random walks converges (in a certain sense) to a randomself-avoiding continuous path in the complex plane called SLE(2), which belongsto the family of Schramm-Loewner evolutions, or SLE( k ), k ≥
0, for short [ , , ]). As we might expect from our intuition, the latter SLE(2) path is, in fact,a loop-erasure LE ( B ) of the Brownian motion B in a sense which can be madeprecise [ ]. The previous considerations offer the possibility of interpreting, at leastinformally, the scaling limit of Fomin’s identity and its affine version in terms of two-dimensional Brownian motions, in suitable complex domains. For example, sincethe determinants (1.1) and (1.3) involve hitting probabilities for a single Markovchain, it continues to make sense when h ( a, b ) is the Poisson kernel (or hittingdensity) of two-dimensional Brownian motion in suitable simply connected domainsΩ with smooth boundaries. One might expect that determinants of hitting densitiesare the scaling limits of the corresponding determinants of hitting probabilities forsimple random walks, in square grid approximations of Ω and, moreover, that theformer determinants express non-crossing probabilities between Brownian pathsand SLE(2) paths. This scaling limit has been rigorously achieved in the case of b b n − : : :: : : a a a n − a n b b b n − b n : : : a n + mb n + m Figure 1.
Affine setting. n = 2 paths [ – ], while ongoing works related to the general case n > , , , , ].Our contribution in the previous context is the connection with random matrixtheory that emerges from the following setting: assume that Ω is a suitable complex(connected) domain with smooth boundary and h ( z , y ) is the (hitting) density ofthe harmonic measure µ z , Ω ( A ) = P z ( B T ∈ A ) , A ⊂ ∂ Ω , with respect to one-dimensional Lebesgue measure (lenght), where B under P z denotes a two-dimensional Brownian motion starting at z ∈ Ω, and T = inf { t > B t / ∈ Ω } is the first exit time of Ω (see Section 4.1). More generally, h ( z , y ) canbe the hitting density of a diffusion in a suitable complex domain, with absorbingand normal reflecting boundary conditions (this idea is originally discussed in [ ]).Therefore, for m ∈ R and appropriately chosen (parametrized) positions x , ..., x n and y , ..., y n along the boundary ∂ Ω, the determinants of hitting densities H ( x, y ) = det ( h ( x i , y j )) ni,j =1 dy · · · dy n , (1.4)and H ( x, y ) = det X k ∈ Z ζ k h ( x i , y j + mk ) ! ni,j =1 dy · · · dy n , (1.5)where ζ = (cid:26) n is odd − n is even , can be interpreted, informally, as the probability that n independent ‘Brownianmotions’ B i , i = 1 , ..., n , starting at positions x , x , ..., x n , respectively, will firsthit an absorbing boundary ∂ Γ ⊂ ∂ Ω at (parametrized) positions in the intervals( y i + dy i ), i = 1 , ..., n , and whose trajectories are constrained to satisfy the condition B j ∩ LE ( B i ) = ∅ , for all 1 ≤ i < j ≤ n, in (1.4), or B j ∩ LE ( B j − ) = ∅ , < j ≤ n, and B ∩ LE ( m + B n ) = ∅ , in the affine case (1.5). We remark that in the affine case, we assume Ω to beinvariant under a fixed (horizontal) translation by m ∈ R , and therefore m + B n is the horizontal translation by m of the Brownian path B n , see Figure 1. We remark that some hitting densities h ( x, y ) can be calculated explicitly for a numberof important domains, like disks and half-planes, and many others can be deducedfrom these by reflection and conformal invariance of the two-dimensional Brownianmotion. We consider examples of determinants of hitting densities in Sections 4and 5.Finally, if ∂ Γ = ∂ Ω and then the whole boundary ∂ Ω is absorbing, we requirea different notion of hitting density h ( x, y ) (since the paths need to ‘walk’ into theinterior Ω ◦ = Ω \ ∂ Ω before reaching their destination). Therefore, in order to studydeterminants of the form (1.4) and (1.5), we consider the so-called excursion Poissonkernel . In this context, an example of a similar interpretation of determinants ofhitting densities of the form (1.4) was first observed by Sato and Katori [ ] (seeSection 4.6). Non-intersecting processesin one dimension have long been an integral part of random matrix theory, at leastsince the pioneering work of Dyson [ ] in the 1960s. For example, it is well knownthat, if one considers n independent one-dimensional Brownian particles, started atthe origin and conditioned not to intersect up to a fixed time T (see Section 4.2 fordetails), then the locations of the particles at time T have the same distribution asthe eigenvalues of a random real symmetric n × n matrix with independent centeredGaussian entries, with variance T on the diagonal and T / ] or Proposition 5.4 below.In two dimensions , we can consider appropriate limits of the formlim ( x ,..., x n ) ∈ Cx i → x ∈ ∂ Ω ˜ H ( x, y ) , ( y , ..., y n ) ∈ C, (1.6)where ˜ H ( x, y ) is an appropriate normalisation of the determinants H ( x, y ) in (1.4)and (1.5), and the positions x , ..., x n , y , ..., y n are determined by chambers (al-coves) C of R n . These limits give the locations of the n hitting points y , ..., y n along the absorbing boundary ∂ Γ, when the processes start at a single commonpoint x ∈ ∂ Ω. In a way, this is the two-dimensional analogue of the model describedin the preceding paragraph. In Section 4 we show that the limits (1.6) agree witheigenvalue densities of Cauchy type random matrix ensembles, for determinants ofthe form (1.4) (see [ ] and Section 4.6 for similar asymptotic considerations re-garding excursion Poisson kernels). For determinants of the form (1.5), in Section 5we show that, by considering the hitting density of the two-dimensional Brownianmotion in an annulus on the complex plane, certain limit of the form 1.6 agrees withthe Circular Orthogonal Ensemble (COE) of random matrix theory (Proposition5.5). The paper is structured into two parts thatcan be read (essentially) independently. The first part (Sections 2 and 3) is mainlyconcerned to the combinatorial results of Section 1.1. In Section 2 we give somebackground on the reflection principle and Fomin’s generalisation for loop-erasedwalks in discrete lattice models. In Section 3, we present the affine version ofFomin’s identity. The second part (Sections 4 and 5) shows calculations and limitsfor determinants of hitting densities of the form (1.4) and (1.5). In Section 4, weshow that for suitable simply connected domains, the determinants associated with
Fomin’s identity converge, in a certain sense, to some known ensembles of randommatrix theory. In Section 5 we consider the affine setting and, after revisiting themodel of non-intersecting one-dimensional Brownian motion on the circle [ ], weshow that a determinant of the form (1.5), in the context of independent Brownianmotions in an annulus, converges in a suitable limit to the Circular OrthogonalEnsemble. Acknowledgements.
We gratefully acknowledge the support of the European Re-search Council (Grant number 669306) and CONACYT (PhD scholarship number411059). We would also like to thank the anonymous referees for their careful read-ing and suggestions, in particular for drawing our attention to the paper [ ], whichhave led to a much improved version of the paper.
2. The reflection principle and Fomin’s generalisation
In this section, we consider the discrete versions of some of the determinantformulas considered in the Introduction. This combinatorial approach has someadvantages and will be particularly convenient in Section 2.2, where some of themain concepts are defined for discrete paths. Let G = ( V, E, ω ) be a directed graphwith no multiple edges, countable vertex set V and edge set E ⊂ V × V . Thegraph G need not be acyclic, so multiple loops are allowed. The set ω is a family ofpairwise distinct formal indeterminates { ω ( e ) } e ∈ E that we will call the weights ofthe edges. The imposed restriction on edge multiplicity is not essential, but mostof the applications we have in mind share this condition.Let us introduce the notation and terminology we will use through all thefollowing sections. A directed edge e from vertex a ∈ V to vertex b ∈ V will bedenoted as a e → b , and a path or walk P will mean a finite sequence of (directed)edges and vertices P : a e → a e → a e → · · · e n → a n . In this case, we say that P is a path from a to a n of length n . For any pair ofvertices a, b ∈ V , we denote the set of all paths in G from a to b by H ( a, b ), and,if a = ( a , ..., a n ) and b = ( b , ..., b n ) are two n -tuples of vertices, then H ( a , b ) willdenote the set of n -tuples of paths H ( a , b ) = { P = ( P , ..., P n ) : P i ∈ H ( a i , b i ) , for 1 ≤ i ≤ n } . The weight ω ( P ) of a path P is defined as the product of its edge weights ω ( P ) = n Y i =1 ω ( e i ) , if P is given as above. Analogously, the weight of an n -tuple P = ( P , ..., P n ) isthe product of the corresponding path weights ω ( P ) = Q ni =1 ω ( P i ). A quantity ofinterest will be the generating function h ( a, b ) = X P ∈H ( a,b ) ω ( P ) , a, b ∈ V, which encodes all paths P ∈ H ( a, b ) according to their weight. This expres-sion should be understood as a formal power series in the independent variables { ω ( e ) } e ∈ E . x x y y P P x x y y ~ P ~ P Figure 2.
The association ( P , P ) ϕ ( ˜ P , ˜ P ) is an involution, ϕ = id .Finally, two paths P and P in G intersect if they share at least one vertex (intheir vertex-sequence definitions) and we will write this as P ∩ P = ∅ . A family ofpaths P ∈ H ( a , b ) is intersecting if any two of them intersect. We will say that P is self-avoiding or has no loops if it does not visit the same vertex more than once,that is, if a i = a j in the vertex sequence definition of P , for all 0 ≤ i < j ≤ n . If the graph G = ( V, E, ω ) is acyclic(loops are not allowed), the reflection principle relies upon the following property.Consider two paths P and P in G , and assume that they intersect (see Figure2). Fix a total order for the set of vertices V and let A = { v α : α ∈ I } be the setof intersection vertices between P and P , which is finite. Among all intersectionvertices, let v α be the minimal with respect to the given order, and split the paths P and P at the vertex v α , into the corresponding subpaths: P : a P ′ −→ v α P ′′ −→ a n P : a ′ P ′ −→ v α P ′′ −→ a ′ m . Now interchange the parts P ′′ and P ′′ above. This procedure creates two new paths˜ P and ˜ P given by ˜ P : a P ′ −→ v α P ′′ −→ a ′ m ˜ P : a ′ P ′ −→ v α P ′′ −→ a n . The paths ˜ P and ˜ P also intersect (in particular, v α is an intersection vertex)and, more importantly, their set of intersection vertices is also A = { v α : α ∈ I } .This means that the intersection vertices are invariant under the map ( P , P ) ( ˜ P , ˜ P ) and hence so is the minimum vertex v α . Therefore, if we perform the sameprocedure to the paths ˜ P and ˜ P , we recover the original paths P and P . In otherwords, the map ( P , P ) ( ˜ P , ˜ P ) is an involution. Moreover, the weights are alsoinvariant under this operation: ω ( P ) ω ( P ) = ω ( ˜ P ) ω ( ˜ P ).A careful application of the above argument leads to the following enumerationformula for non-intersecting paths by Karlin and McGregor [ ] (in the context ofMarkov chains) and Lindstr¨om [ ] (further developed by Gessel-Viennot [ ]): Theorem . In an acyclic graph, let ∂ Γ ⊂ V be the distinguished set ofvertices: ∂ Γ = { a ∈ V : ∄ a e → b } . For arbitrary sets A = { a , ..., a n } ⊂ V and B = { b , ..., b n } ⊂ ∂ Γ , it holds X σ ∈ S n sgn( σ ) X P ∈H ( a , b σ ) P i ∩ P j = ∅ , i = j ω ( P ) = det ( h ( a i , b j )) ni,j =1 , where b σ = ( b σ (1) , ..., b σ ( n ) ) . If the graph G = ( V, E, ω )is not acyclic, and the paths P and P intersect, then the invariance of the inter-section vertices described in the previous section is no longer guaranteed, since anintersection vertex can be part of a loop. However, there is a modification of thereflection principle for general graphs, due to Fomin [ ], which we describe below.We briefly present the key concept of loop-erased walks introduced by G. Lawler [ ]. Definition . For each path P in G = ( V, E, ω ) of the form a e → a e → a e → · · · e n → a n , the loop-erasure of P , denoted LE ( P ) , is the self-avoiding path obtained by chrono-logical loop-erasure of P , as follows: • Let j = max { j : a j = a } ; • recursively, if j k < n , then j k +1 = max { j : a j = a j k +1 } ; • if j k = n , then LE ( P ) is the path a j e j −→ a j e j −→ a j e j −→ · · · e jk − −→ a j k . This procedure erases loops in P in the order they appear, and the operationis iterated until no loop remains. In particular, note that LE ( P ) is a subpath ofthe original path P , with the same starting and end points a and a n , respectively.Using the above procedure, Fomin [ ] introduced the so-called loop-erasedswitching for paths that are allowed to self-intersect. The loop-erased switchingis as follows: consider two paths P : a = x → ... → a n = y and P : a ′ = x → ... → a ′ m = y in the graph G , starting from different vertices a = a ′ , and assume P and LE ( P ) intersect at least at one common vertex, that is, P ∩ LE ( P ) = ∅ (see Figure 3). Among all such intersection vertices, let v = a j i be the one withminimal index along the vertex sequence of LE ( P ), (see Definition 1), and splitthe path P at the end of the edge a j i − e ji − −→ v into two subpaths: P : a P ′ ( v ) −→ v P ′′ ( v ) −→ a n . This partition ensures that all possible loops of P ‘rooted’ at v are part of P ′′ ( v ),so that P ′′ ( v ) does not intersect the path LE ( P ′ ( v )) at any vertex different from v . Now, if we split the path P at its first visit to v , we have P : a ′ P ′ ( v ) −→ v P ′′ ( v ) −→ a ′ m . Then, by construction of v , P ′′ ( v ) does not visit any other vertex of LE ( P ′ ( v )),except for v , so it shares the same property as P ′′ ( v ). The latter common condition x x y y v x x y y vP P ~ P ~ P Ω Ω
Figure 3.
The loop-erased switching. The path P intersects theloop-erased part of P and v is the ‘first intersection’ (left). Inter-changing the paths at v , the new paths ˜ P (black) and ˜ P (gray)satisfy the same property, that is, ˜ P intersects the loop-erasedpart of ˜ P and v is the ‘first intersection’ (right).allows us to interchange the parts P ′′ ( v ) and P ′′ ( v ) at the vertex v , and create newpaths ˜ P : a P ′ ( v ) −→ v P ′′ ( v ) −→ a ′ m ˜ P : a ′ P ′ ( v ) −→ v P ′′ ( v ) −→ a n . Note that the new paths ˜ P and LE ( ˜ P ) also intersect ( v is an intersection ver-tex), and therefore ˜ P ∩ LE ( ˜ P ) = ∅ . These conditions ensure that the map( P , P ) ( ˜ P , ˜ P ) is an involution, and the ‘minimality’ of the intersection vertex v is preserved, exactly as in Section 2.1. We also have ω ( P ) ω ( P ) = ω ( ˜ P ) ω ( ˜ P ).The following theorem (Theorem 7.1 in [ ]) is an application of the above loop-erased switching procedure. Fix a distinguished subset of vertices ∂ Γ ⊂ V and callit the absorbing boundary . For a ∈ V and b ∈ ∂ Γ, denote by H + ( a, b ) ⊂ H ( a, b ) theset of all paths of positive length a e → a e → a e → · · · e n → b, such that all the internal vertices a , ..., a n − lie in V \ ∂ Γ. If a ∈ ∂ Γ, we assume n ≥
2, so that the path walks into V \ ∂ Γ before reaching the vertex b . Analogously,define H + ( a , b ) for n -tuples of paths P = ( P , ..., P n ) as at the beginning of Section2, that is H + ( a , b ) = { P = ( P , ..., P n ) : P i ∈ H + ( a i , b i ) , for 1 ≤ i ≤ n } . Theorem . Let G = ( V, E, ω ) be a graph satisfying theabove assumptions and ∂ Γ ⊂ V . Let A = { a , ..., a n } ⊂ V and B = { b , ..., b n } ⊂ ∂ Γ be two labelled sets of different vertices. Therefore X σ ∈ S n sgn( σ ) X P ∈H + ( a , b σ ) P j ∩ LE ( P i )= ∅ , i 2, so that the path walks into V \ ∂ Γ before reaching the vertex b . Analogously,define H + ( a , b ) for n -tuples of paths P = ( P , ..., P n ) as at the beginning of Section2, that is H + ( a , b ) = { P = ( P , ..., P n ) : P i ∈ H + ( a i , b i ) , for 1 ≤ i ≤ n } . Theorem . Let G = ( V, E, ω ) be a graph satisfying theabove assumptions and ∂ Γ ⊂ V . Let A = { a , ..., a n } ⊂ V and B = { b , ..., b n } ⊂ ∂ Γ be two labelled sets of different vertices. Therefore X σ ∈ S n sgn( σ ) X P ∈H + ( a , b σ ) P j ∩ LE ( P i )= ∅ , i The graph G is embedded into Ω and the vertices a n , ..., a , b , ..., b n are ordered counterclockwise along ∂ Ω. Remark. Note that the above theorem agrees with Theorem 2.1 if the graphunder consideration is acyclic. Also, note that Theorem 2.2 does not give the totalweight of families of non-intersecting paths in G connecting A and B (in the strictsense of non-intersection). However, the paths are constrained to satisfy P j ∩ LE ( P i ) = ∅ , for all i < j, which forces the corresponding loop-erased parts to repeal each other. Corollary . Assume that G is planar and it is also embedded into a con-nected planar domain Ω in such a way that the vertices in the absorbing boundary ∂ Γ lie on the topological boundary ∂ Ω . Let A ⊂ V and B ⊂ ∂ Γ be as in Theorem2.2, and, whenever i > i ′ and j < j ′ , assume that every path P ∈ H + ( a i , b j ) inter-sects every path P ′ ∈ H + ( a i ′ , b j ′ ) at a vertex in V \ ∂ Γ (see Figure 4). In this case,the only allowable permutation in (2.1) is the identity permutation, and therefore X P ∈H + ( a , b ) P j ∩ LE ( P j − )= ∅ , Assume that the vertex set V is the state space of a time-homogeneousMarkov chain X and the possible transitions between states are determined by theplanar graph G . That is, the transition probabilities p ( a, b ) are positive if and onlyif there is an edge a e → b , in which case ω ( e ) = p ( a, b ). Then the assertion ofCorollary 2.3 has the following probabilistic interpretation: the generating function h ( a, b ) = X P ∈H + ( a,b ) ω ( P ) , a ∈ V, b ∈ ∂ Γ , is the hitting probability P a ( X T = b, T < ∞ ), where T is the first time the chain X hits the boundary ∂V (if a ∈ ∂ Γ, the Markov chain is supposed to walk into V \ ∂ Γbefore reaching ∂ Γ). Then the left hand side of (2.2) is equal to the probabilitythat n independent trajectories X , ..., X n of the Markov process X , starting atlocations a , a , ..., a n , respectively, will hit the boundary ∂ Γ for the first time atthe points b , b , ..., b n , respectively, and furthermore the trajectory X j will notintersect the loop-erased path LE ( X i ) at any vertex in V \ ∂ Γ, for all i < j , that is, X j ∩ LE ( X i ) = ∅ , for all 1 ≤ i < j ≤ n. 3. Affine version of Fomin’s identity In this section, we extend Fomin’s identity to the setting of the affine symmetricgroup (Theorem 3.1) and consider its natural projection onto the cylindrical lattice(Proposition 3.3). Our main motivation is to present a preliminary extension ofthe framework considered in Section 4 to non-simply connected domains, and show,in Section 5, an interesting connection with circular ensembles of random matrixtheory.As we discussed in Section 2.2, the interaction between n paths P , P , ..., P n imposed in Fomin’s identity (Theorem 2.2) is given by the condition P j ∩ LE ( P i ) = ∅ , for all i < j. (3.1)In particular, restricted to the lattice strip G of Figure 5, the above conditionensures a type of ‘repulsion’ between consecutive paths from left to right, that is,every path P j will not intersect the loop-erased part LE ( P j − ) of the path to itsright. Theorem 3.1 below is an extension of Fomin’s identity in the sense that weconsider families of paths P , P , ..., P n subject to (3.1) and also subject to an extranon-intersection condition between the path P and the translation to the right of P n , given by a fixed translation S of the graph G (see Figure 6), that is P ∩ LE ( S P n ) = ∅ . (3.2)This type of interaction is helpful when the lattice strip G is projected onto the cylindrical lattice ˜ G , modulo the translation S (or affine setting, see Sections 3.2 and3.3). In this case, the conditions (3.1) and (3.2) jointly ensure that the projectedpaths ˜ P , ˜ P , ..., ˜ P n , in ˜ G , also satisfy the analogous ‘left to right’ non-intersectioncondition ˜ P j ∩ LE ( ˜ P j − ) = ∅ , < j ≤ n, and ˜ P ∩ LE ( ˜ P n ) = ∅ . Consider the lattice strip G =( V, E, ω ), given by the vertex set V = Z × { , , ..., N } and connected by directedhorizontal and vertical edges, in both positive and negative directions. We alsoassume that the weights { ω ( e ) } e ∈ E are invariant under horizontal translations by v = ( M, M ∈ Z . Let ∂ Γ = { ( i, N ) : i ∈ Z } denote the upperboundary of the lattice strip and consider the set H + ( a , b ) for a = ( a , ..., a n ), b = ( b , ..., b n ) vectors of vertices, as defined in Section 2.2. We have the following. Theorem . Consider integers i n < i n − < ... < i < i n + M and j n The lattice strip G with vertex set V = Z × { , , ..., N } . If S : G → G is the horizontal translation by ( M, , the n +1) vertices a n , ..., a , S a n , S b n , b , ..., b n are ordered counterclockwise along the topological boundary of the lat-tice strip G (see Figure 5). Therefore X P ∈H + ( a , b ) P j ∩ LE ( P j − )= ∅ , Unlike Fomin’s identity, the extra condition P ∩ LE ( S P n ) = ∅ in(3.3) forces us to consider families of paths where the n end vertices are permutations and translations of the originals ( b , ..., b n ), see the proof below. In particular,the end vertices should vary among n -tuples S k b σ = ( S k b σ (1) , ..., S k n b σ ( n ) ), with σ ∈ S n and k + ... + k n = 0, k i ∈ Z . This can be thought of as the action of the(infinite) affine symmetric group ˜ A n on the vertices ( b , ..., b n ). Remark. In the acyclic case, the above theorem agrees with the Gessel-Zeilbergerformula for counting paths in alcoves [ ]. Proof of Theorem 3.1. We will follow the strategy of proof of Fomin’s iden-tity (Theorem 6.1 in [ ]), that is, we will give a sign-reversing involution on the setof summands on the right hand side of (3.3) which violate the condition P j ∩ LE ( P i ) = ∅ , for all 1 ≤ i < j ≤ n, and(3.4) P j ∩ LE ( S P n ) = ∅ , for all 1 ≤ j ≤ n. As a consequence, the sum of all of the latter terms will vanish and, the sum ofthe remaining terms, the ones which satisfy (3.4), will be simplified to the desiredexpression in the left-hand side of (3.3).The sign-reversing involution is as follows. For n ≥ 2, let σ ∈ S n and k i ,1 ≤ i ≤ n , integers such that k + k + ... + k n = 0. Consider a family of paths P ∈ H ( a , S k b σ ) which violates the condition (3.4). We will construct a new familyof paths ˜ P ∈ H ( a , S ˜k b ˜ σ ), with ˜ σ ∈ S n and ˜ k + ˜ k + ... + ˜ k n = 0, that alsoviolates the condition (3.4), and satisfes ω ( ˜ P ) = ω ( P ) and sgn(˜ σ ) = − sgn( σ ). The construction of the new family ˜ P is essentially an application of the Fomin’s loop-erased switching (Section 2.2) over the paths P n , P n − , ..., P , S P n . This construction will also ensure that the correspondence P ˜ P is one-to-one, asdesired.To make the notation simpler, let us denote a := S a n and the correspondingpath starting at S a n by P := S P n . Choose indexes i ′ and j ′ as follows. Since thefamily P ∈ H ( a , S k b σ ) violates (3.4), the set of indexes 0 ≤ i < j ≤ n such that P j ∩ LE ( P i ) = ∅ is not empty. Therefore, we can choose 0 ≤ i ′ < n the minimumamong those indexes and consider the path LE( P i ′ ). Along the latter path, choosea vertex v ′ and index j ′ as follows: • Along the vertex sequence of the path LE( P i ′ ), choose v ′ as the ‘closest’(that is, with minimal index) intersection vertex to the starting vertex a i ′ . • Now consider the set of indexes { j : 1 ≤ i ′ < j ≤ n } such that P j intersectsLE( P i ′ ) at v ′ (in other words, v ′ ∈ P j ∩ LE( P i ′ )), and let j ′ the minimumof this set.We have two different scenarios, depending on weather P i ′ is the path S P n or not. If i ′ = 0 (and P i ′ is not the path S P n ), we perform the usual loop-erasedswitching (Section 2.2) over the paths P i ′ and P j ′ at the vertex v ′ , that is, we definenew paths ˜ P i ′ : a i ′ P ′ i ′ −−→ v ′ P ′′ j ′ −−→ S k j ′ b σ ( j ′ ) ˜ P j ′ : a j ′ P ′ j ′ −−→ v ′ P ′′ i ′ −−→ S k i ′ b σ ( i ′ ) . For the remaining paths, i / ∈ { i ′ , j ′ } , we define ˜ P i := P i . The original family P ∈H ( a , S k b σ ) is then mapped to a new family of paths ˜ P ∈ H ( a , S ˜k b ˜ σ ), where ˜ k =( k , ..., k j ′ , ..., k i ′ , ..., k n ) and ˜ σ = σ ◦ ( i ′ , j ′ ) ∈ S n are the vector k and permutation σ , with the entries i ′ and j ′ interchanged. Note that the sum of the entries of ˜ k is zero, as desired, and sgn(˜ σ ) = − sgn( σ ). Moreover, the family ˜ P also violatesthe condition (3.4) since the paths ˜ P i ′ and ˜ P j ′ share the vertex v ′ . Note that theweights are also preserved: ω ( ˜ P ) = ω ( P ).In the second case, when i ′ = 0, a more careful selection of paths is needed: weperform the loop-erased switching over the paths S P n and P j ′ : S P n : S a n ( S P n ) ′ −−−−→ v ′ ( S P n ) ′′ −−−−−→ SS k n b σ ( n ) P j ′ : a j ′ P ′ j ′ −−→ v ′ P ′′ j ′ −−→ S k j ′ b σ ( j ′ ) , and create the two new paths˜ P n : a n P ′ n −−→ S − v ′ S − P ′′ j ′ −−−−−→ S − S k j ′ b σ ( j ′ ) ˜ P j ′ : a j ′ P ′ j ′ −−→ v ′ ( S P n ) ′′ −−−−−→ SS k n b σ ( n ) , (see Figure 6). The rest of the paths remain invariant, ˜ P i := P i for i / ∈ { n, j ′ } . Thus,the new family ˜ P = ( ˜ P , ..., ˜ P n ) satisfies ˜ P ∈ H ( a , S ˜k b ˜ σ ), with ˜ k = ( k , ..., k n +1 , ..., k n − , k j ′ − 1) and ˜ σ = σ ◦ ( j ′ , n ) ∈ S n . Note again that the sum of the entriesof ˜ k is zero and sgn(˜ σ ) = − sgn( σ ). Moreover, since the weight ω is invariant under a a a n − a n b b b n − b n : : : : : : S b n S a n b b b n − b n a a a n − a n S b n S a n S − b : : :: : : Figure 6. Loop-erased switching over the paths P and S P n .horizontal translations, we have ω ( ˜ P ) = ω ( P ). We only need to show that thefamily ˜ P violates the condition (3.4) as well, but this is clearly the case since thepaths ˜ P j ′ and LE ( S ˜ P n ) intersect at the vertex v ′ , that is˜ P j ′ ∩ LE ( S ˜ P n ) = ∅ . Therefore, in both cases, applying the loop-erased switching to the family ˜ P , werecover the original family P , so the corresponding map from P to ˜ P is an involutionon the set of paths that violate (3.4). Moreover, since sgn( σ ) ω ( ˜ P ) = − sgn(˜ σ ) ω ( ˜ P ),the sum of all these terms vanishes on the right hand side of (3.3), and thereforethe total sum is X σ ∈ S n X k i ∈ Z k + k + ... + k n =0 sgn( σ ) X P ∈H + ( a , S k b σ ) P satisfies ( . ) ω ( P ) . (3.5)Finally, in the expression above, if a family P ∈ H + ( a , S k b σ ) satisfies (3.4), theloop-erased parts LE ( P j ), 1 ≤ j ≤ n , are pairwise disjoint and then σ must be theidentity permutation and k = k = ... = k n = 0, as required. In this case, thecondition (3.4) on paths can be simplified to the one in the left hand side of (3.3). (cid:3) As described in the introduction ofSection 3, a useful application of Theorem 3.1 is when we consider the projection ofthe lattice strip G onto the cylindrical lattice , modulo a translation S (see Figure 7).Intuitively, a family of n (loop-erased) paths can wind around the cylinder severaltimes (equivalently, translations of the end vertex by S m = S ◦ S m − , m ∈ Z , inthe strip) before reaching its destination. Moreover, there are exactly n differentways in which the n paths can reach their destination without intersecting, givenby the n ‘cyclic permutations’ of the end vertices. ~ a ~ b Figure 7. A path in the cylindrical lattice ˜ G with winding num-ber k = 1.Corollary 3.2 and Proposition 3.3 make the above considerations precise. Theseconsiderations give a more tractable form of Theorem 3.1, first as a sum of n determinants in Corollary 3.2 and then as a single determinant in Proposition 3.3. Corollary . In the context of Theorem 3.1, by summing up in (3.3) overall the weights of all families of paths P = ( P , ..., P n ) starting at a = ( a , ..., a n ) ,and ending at all possible translations of b = ( b , ..., b n ) by S m , m ∈ Z , we obtain X P ∈ S m ∈ Z H + ( a , S m b ) P j ∩ LE ( P j − )= ∅ , Using the identity (3.3) and summing up over all the weights as indi-cated in the statement of the corollary, the left hand side of (3.6) takes the form X m ∈ Z X σ ∈ S n X k i ∈ Z k + k + ... + k n =0 sgn( σ ) X P ∈H + ( a , S m + k b σ ) ω ( P ) , which, in turn, can be easily simplified to the desired expression on the right-handside. For the second part, note that, if η = e i πn is a complex root of unity, then wecan eliminate the condition P ni =1 k i = 0 mod n by using the identity1 n n − X u =0 η u P ni =1 k i = (cid:26) P ni =1 k i = 0 , mod n . Then, the right-hand side of (3.6) can be written as1 n n − X u =0 X σ ∈ S n sgn( σ ) X k ,...,k n ∈ Z η u P ni =1 k i X P ∈H + ( a , S k b σ ) ω ( P ) , and the latter as1 n n − X u =0 X σ ∈ S n sgn( σ ) n Y i =1 X k ∈ Z η uk X P ∈H + ( a i , S k b σ ( i ) ) ω ( P ) . The above expression is (3.7). (cid:3) Proposition . Denote by [ ℓ ] ∈ S n the cyclic permutation shifted by ℓ =0 , , ..., n − : [ ℓ ]( k ) = k − ℓ, mod n, in { , ..., n } . Let a = ( a , ..., a n ) and b = ( b , b , ..., b n ) be the vectors of vertices of Theorem 3.1.For each [ ℓ ] ∈ S n , ℓ = 0 , ..., n − , define the n -tuple: k ℓ = (1 , ..., , | {z } ℓ times , ..., | {z } n − ℓ times ) . (3.8) We have the following X [ ℓ ] ∈ S n ℓ =0 ,...,n − X P ∈ S m ∈ Z H + ( a , S m + k ℓ b [ ℓ ] ) P j ∩ LE ( P j − )= ∅ , Let G ( a , b ) denote the left-hand side of (3.6). Using Corollary 3.2, asimple calculation shows that for each ℓ = 0 , ..., n − G ( a , S k ℓ b [ ℓ ] ) = 1 n n − X u =0 η − ℓu sgn([ ℓ ]) det X k ∈ Z η uk h ( a i , S k b j ) ! . Therefore, the left hand side of (3.9) can be expressed as n − X ℓ =0 G ( a , S k ℓ b [ ℓ ] ) = n − X u =0 n n − X ℓ =0 η − ℓu sgn([ ℓ ]) ! det X k ∈ Z η uk h ( a i , S k b j ) ! , which is a sum of n determinants.Case 1. If n is odd, sgn([ ℓ ]) = 1 for all ℓ = 0 , ..., n − n n − X ℓ =0 η − ℓu = (cid:26) − u = 0 , mod n , therefore, the only remaining determinant is the one corresponding to u = 0, andso ζ = η u = 1. Case 2. If n is even, sgn([ ℓ ]) = ( − ℓ for all ℓ = 0 , ..., n − n n − X ℓ =0 ( − ℓ η − ℓu = 1 n n − X ℓ =0 η ℓ ( n − u ) . The above sum is 1 if and only if n − u = 0 mod n , and zero otherwise. The onlyremaining determinant is then u = n , and therefore ζ = η u = − 1, which concludesthe proof. (cid:3) Remark. Assume that the vertex set V = Z × { , , ..., N } is the state spaceof a time-homogeneous Markov chain X and the possible transitions between statesare determined by the lattice strip G introduced at the beginning of the section. Inother words, the transition probabilities p ( u, v ) are positive if and only if there is anedge u e → v , in which case ω ( e ) = p ( u, v ). Assume that the transition probabilitiesare space-invariant with respect to a fixed horizontal translation S : G → G . Then,the assertion of Proposition 3.3 has the following probabilistic interpretation: ifvertices a n , ..., a , b , ..., b n are ordered counterclockwise along the boundary (as inFigure 6), then the n × n determinantdet X k ∈ Z ζ k h ( a i , S k b j ) ! ni,j =1 , of hitting probabilities h ( a i , S k b j ), where S k = S ◦ S k − , k ∈ Z , and ζ = (cid:26) n is odd − n is even , is equal to the probability that n independent trajectories of the Markov chain X , ..., X n , starting at a , ..., a n , respectively, will first hit the upper boundary ∂ Γ = Z × { N } at any of the n cyclic permutations of the vertices b , ..., b n , shifted also byall possible horizontal translations by S k , k ∈ Z , and furthermore the trajectoriesare constrain to satisfy X j ∩ LE ( X j − ) = ∅ , < j ≤ n, and X ∩ LE ( S X n ) = ∅ . In this section, weconsider families of paths defined in the directed cylindrical lattice ˜ G of Figure 7 andreview some properties regarding their loop-erasures. As in the previous sections,the cylindrical lattice need not be acyclic (loops are allowed) and, if the numberof paths is odd, we can obtain a variant of Proposition 3.3 by applying Fomin’sidentity directly (see Proposition 3.4 below). However, there is a slight differencebetween these two approaches, since the loop-erasure of a path in ˜ G may differ fromthe projection of the loop-erasure of the corresponding path in the lattice strip G (see the remark just after Proposition 3.4).Define the (directed) cylindrical lattice (or, just cylinder ) ˜ G = ( ˜ V , ˜ E ) as the di-rected graph with vertex set ˜ V = Z M × { , , ..., N } and connected by edges in bothpositive and negative directions. Here, we consider the canonical representation of Z M as Z /M Z = { [0] , ..., [ M − } . Let’s distinguish the set of (boundary) vertices ∂ ˜ G = Z M × { N } . If we consider the lattice strip G = ( V, E, ω ) of Theorem 3.1and the notation thereof, there is a natural correspondence between paths in thecylinder ˜ G and paths in the strip G . In particular, every path ˜ P in ˜ G starting at ~ a ~ b ~ b ~ a a a b b S a S b P P a b P Figure 8. Three paths, and their projections onto the cylinder.˜ a = ([ i ] , 0) and ending at ˜ b = ([ j ] , N ), for i, j ∈ { , ..., M − } , and with all internalvertices lying in ˜ G \ ∂ ˜ G , can be seen as the image of any path P ℓ ∈ G of the form P ℓ ∈ H + ( S ℓ a, S ℓ + k b ) , ℓ ∈ Z , with a = ( i, ∈ V , b = ( j, N ) ∈ V , and a unique k ∈ Z . The integer k is usuallycalled the winding number of the path ˜ P (see Figure 7). Since the weight function ω defined on the strip G is invariant under the translation S by ( M, G inherits canonically a weight function ˜ ω on ˜ E and the path ˜ P inherits the weight˜ ω ( ˜ P ) := ω ( P ), whenever P ∈ H + ( a, S k b ) is the projection of ˜ P .Let C + (˜ a, ˜ b ) be the set of all paths in the cylinder ˜ G of positive length, startingat ˜ a ∈ ˜ G and ending at ˜ b ∈ ∂ ˜ G , with all internal vertices in ˜ G \ ∂ ˜ G . Similarly, define C + (˜ a , ˜ b ) for families of paths ˜ P , ..., ˜ P n , starting at ˜ a = (˜ a , ..., ˜ a n ) and ending at˜ b = (˜ b , ..., ˜ b n ). We have the following: Proposition . Consider integers ≤ i n < i n − < ... < i < M and ≤ j n < j n − < ... < j < M . Define two n -tuples of vertices in the cylinder ˜ G as ˜ a k := ([ i k ] , , ˜ b k := ([ j k ] , N ) , ≤ k ≤ n. If n is odd and we consider the n cyclic permutations defined in Proposition 3.3,we obtain X σ cyclic X ˜ P ∈C + (˜ a , ˜ b σ )˜ P j ∩ LE ( ˜ P i )= ∅ , i Note that the right hand side of (3.10) can be written as the determi-nant det X ˜ P ∈C + (˜ a i , ˜ b j ) ˜ ω ( ˜ P ) ni,j =1 , Figure 9. Hexagonal latticeand, since sgn( σ )=1 for all σ cyclic if n is odd, the equality (3.10) is a directapplication of Fomin’s identity (Theorem 2.2), according to the weight function ˜ ω on ˜ G . (cid:3) Remark. The right hand side of (3.10) agrees with the right hand side ofidentity (3.9), for n odd. This implies that the left hand sides of (3.10) and (3.9)are equal, which is not immediately obvious from the definitions. For example, wecan consider the paths P , P and P in the lattice strip of Figure 8. There, we havethat the corresponding projections onto the cylinder satisfy ˜ P j ∩ LE ( ˜ P i ) = ∅ , for i < j , and, in particular ˜ P ∩ LE ( ˜ P ) = ∅ . However, P ∩ LE ( S P ) = ∅ , and then( P , P , P ) is not considered in the left hand side of (3.9). It would be interestingto have a direct combinatorial proof of this identity. Remark. In the acyclic case, one can obtain determinant formulas for an even number of non-intersecting walks on a cylindrical lattice by introducing modifiedweights which keep track of windings [ , ]. However, in the general case, wedo not see how to adopt this approach and the only way we know how to studythe case of an even number of particles is via the affine version of Fomin’s identityintroduced in Theorem 3.1. The results ofthis section were formulated for the square lattice but are equally valid for moregeneral periodic planar graphs, for example, the hexagonal lattice shown in Figure9. In the acyclic case, Theorem 3.1 agrees with the Gessel-Zeilberger formula [ ](see also [ ]). We note that in this context, the identity (3.9) gives a directconnection between the Gessel-Zeilberger formula, for counting paths in alcoves,and the Karlin-McGregor formula [ ] (Lindstr¨om-Gessel-Viennot lemma [ ]) forcounting non-intersecting paths on a cylinder; this answers positively a question ofFulmek [ ], where the problem of finding such a direct connection was posed asan open question. Moreover, in the continuous case, it also shows that the Karlin-McGregor (for n odd) and Liechty-Wang (for n even) formulas [ , ] for thetransition probability density of n (indistinguishable) non-intersecting Brownianmotions on the circle can be obtained directly from the (labelled) model of Hobson-Werner [ ], which is a continuous version of the Gessel-Zeilberger formula in thecase of the affine symmetric group ˜ A n (we review this in Section 5.1 below).0 In the acyclic case, one can obtain determinant formulas for an even number of non-intersecting walks on a cylindrical lattice by introducing modifiedweights which keep track of windings [ , ]. However, in the general case, wedo not see how to adopt this approach and the only way we know how to studythe case of an even number of particles is via the affine version of Fomin’s identityintroduced in Theorem 3.1. The results ofthis section were formulated for the square lattice but are equally valid for moregeneral periodic planar graphs, for example, the hexagonal lattice shown in Figure9. In the acyclic case, Theorem 3.1 agrees with the Gessel-Zeilberger formula [ ](see also [ ]). We note that in this context, the identity (3.9) gives a directconnection between the Gessel-Zeilberger formula, for counting paths in alcoves,and the Karlin-McGregor formula [ ] (Lindstr¨om-Gessel-Viennot lemma [ ]) forcounting non-intersecting paths on a cylinder; this answers positively a question ofFulmek [ ], where the problem of finding such a direct connection was posed asan open question. Moreover, in the continuous case, it also shows that the Karlin-McGregor (for n odd) and Liechty-Wang (for n even) formulas [ , ] for thetransition probability density of n (indistinguishable) non-intersecting Brownianmotions on the circle can be obtained directly from the (labelled) model of Hobson-Werner [ ], which is a continuous version of the Gessel-Zeilberger formula in thecase of the affine symmetric group ˜ A n (we review this in Section 5.1 below).0 4. Connections to random matrix theory As explained in Section 1.2 of the introduction, there is a natural way to con-sider diffusion scaling limits of both Fomin’s identity (Corollary 2.3) and its affineversion (Proposition 3.3). Regarding Fomin’s identity, this idea is originally dis-cussed in [ ], where some examples for two-dimensional Brownian motion are de-scribed in detail. For our purposes, the connection with random matrix theoryemerges from the following considerations: assume that Ω is a suitable complex(connected) domain with smooth boundary and h ( z , y ) is the (hitting) density ofthe harmonic measure µ z , Ω ( A ) = P z ( B T ∈ A ) , A ⊂ ∂ Ω , with respect to one-dimensional Lebesgue measure (lenght), where B under P z denotes a two-dimensional Brownian motion starting at z ∈ Ω, and T = inf { t > B t / ∈ Ω } is the first exit time of Ω (see Section 4.1). Therefore, for m ∈ R and appropriately chosen (parametrized) positions x , ..., x n and y , ..., y n alongthe boundary ∂ Ω, the determinants of hitting densities H ( x, y ) = det ( h ( x i , y j )) ni,j =1 dy · · · dy n , (4.1)and H ( x, y ) = det X k ∈ Z ζ k h ( x i , y j + mk ) ! ni,j =1 dy · · · dy n , (4.2)where ζ = (cid:26) n is odd − n is even , can be interpreted, informally, as the probability that n independent ‘Brownianmotions’ B i , i = 1 , ..., n , starting at positions x , x , ..., x n , respectively, will firsthit an absorbing boundary ∂ Γ ⊂ ∂ Ω at (parametrized) positions in the intervals( y i + dy i ), i = 1 , ..., n , and whose trajectories are constrained to satisfy B j ∩ LE ( B i ) = ∅ , for all 1 ≤ i < j ≤ n, in (4.1), or B j ∩ LE ( B j − ) = ∅ , < j ≤ n, and B ∩ LE ( m + B n ) = ∅ , in the affine case (4.2). We remark again that, in the affine case, we assume Ω tobe invariant under a fixed (horizontal) translation by m ∈ R , and therefore m + B n is the horizontal translation by m of the Brownian path B n .Our main interest is the determination of the behaviour of the n hitting points y , ..., y n along the boundary, when the starting points x , ..., x n merge into a singlecommon point in ∂ Ω. In other words, for determinants of the form (4.1), thissection considers certain limitslim ( x ,..., x n ) ∈ Cx i → x ∈ ∂ Ω ˜ H ( x, y ) , ( y , ..., y n ) ∈ C, where ˜ H ( x, y ) is an appropriate normalisation of H ( x, y ) and the positions x , ..., x n , y , ..., y n are determined by chambers C of R n . Determinants of the affine form (4.2)are considered in Section 5.In Sections 4.3, 4.4 and 4.5, we revisit the examples considered in [ ] (seeFigure 10). We will see that the consideration of the above limits reveals some x n x x : : : x x x n : : :x x x n : : :y n y y ::: θ θ y y y n : : : Positive quadrant a ) Half unit disk c )Strip b ) Figure 10. Simply connected domains in the complex plane C .natural connections to random matrices, particularly Cauchy type ensembles [ ].An example of this connection was first observed by Sato and Katori [ ], in thecontext of excursion Poisson kernel determinants , and we discuss this in Section4.6. Section 5 considers the affine (circular) case and shows that it is also relatedin a natural way to circular ensembles of random matrix theory.As a warm-up, in Section 4.2 we recall a well-known connection between non-intersecting one-dimensional Brownian motions and the Gaussian Orthogonal En-semble (GOE) of random matrix theory. TheRiemann mapping theorem asserts that any two proper simply connected domainsof C can be conformally mapped into each other. More precisely, if Ω ⊂ C andΩ ′ ⊂ C are two proper simply connected domains with z ∈ Ω and z ′ ∈ Ω ′ ,then there exists a unique conformal (analytic with non-vanishing derivative) map f : Ω → Ω ′ such that f ( z ) = z ′ and f ′ ( z ) > 0. In addition, it is well known thatthe two-dimensional Brownian motion is invariant under conformal transformations[ ]: Proposition . If B is a two-dimensional Brownian motion starting at z ∈ Ω and T = inf { t > B t / ∈ Ω } is the exit time of the domain Ω , then there existsa (random) time change σ : [0 , T ′ ] → [0 , T ] such that the process ( f ( B σ ( t ) ) , ≤ t < T ′ ) is again a two-dimensional Brownian motion, starting at f ( z ) ∈ Ω ′ and stopped atits first exist T ′ of Ω ′ . These properties ensure that, under mild conditions on ∂ Ω (for example, if ∂ Ω is determined by a Jordan curve; see also Section 2.3 of [ ] for more generalconditions), we have that for all A ⊂ ∂ Ω P z ( B T ∈ A ) = P f ( z ) ( f ( B T ) ∈ f ( A )) = P z ′ ( B ′ T ′ ∈ f ( A )) , (4.3)where B ′ is another two-dimensional Brownian motion. If we set µ z , Ω ( A ) = P z ( B T ∈ A ), A ⊂ ∂ Ω , then µ z , Ω defines a measure on ∂ Ω, which is called the harmonic measure or hitting measure on ∂ Ω. Therefore, identity (4.3) becomes µ z , Ω ( A ) = µ z ′ , Ω ′ ( f ( A )) , for all A ⊂ ∂ Ω . (4.4) If both measures are absolutely continuous with respect to one-dimensional Lebesguemeasure, or lenght (which is the case in all the examples considered in this paper),then, from (4.4) we obtain (see also [ , ]): Proposition . Let Ω and Ω ′ be two simply connected domains with z ∈ Ω .Let f : Ω → Ω ′ be a conformal map and set z ′ = f ( z ) . Assume that we can defineharmonic measures µ z , Ω and µ z ′ , Ω ′ and both are absolutely continuous with respectto one-dimensional Lebesgue measure (lenght), therefore h Ω ( z , y ) = | f ′ ( y ) | h Ω ′ ( f ( z ) , f ( y )) , (4.5) where h Ω ( z , · ) and h Ω ′ ( z ′ , · ) are the corresponding densities of µ z , Ω and µ z ′ , Ω ′ ,respectively. Definition . When the harmonic measure µ z , Ω has a density h Ω ( z , y ) withrespect to one-dimensional Lebesgue measure (lenght), we call this density the hittingdensity or Poisson kernel of Ω . We often drop the suffix Ω in the definition above and simply write h = h Ω .In practice, the explicit computation of the harmonic measure (or its density) foran arbitrary simply connected domain Ω is not an easy task, but there are someexamples where this computation can be easily performed. In Sections 4.3, 4.4and 4.5 we consider the positive quadrant Ω , the infinite strip Ω and the upperhalf-circle Ω , respectively (see Figure 10):Ω = { z ∈ C : Re ( z ) > , Im ( z ) > } , Ω = { z ∈ C : 0 < Im ( z ) < t } , t > , Ω = { z ∈ C : | z | < , Im ( z ) > } . Consider a sys-tem of n independent one-dimensional Brownian motions conditioned not to inter-sect up to a fixed time t > 0, starting at positions x n < x n − < ... < x , respectively.This is the n -dimensional Brownian motion starting at x = ( x , ..., x n ) ∈ R n andconditioned to stay in the chamber C = { y ∈ R n : y n < y n − < ... < y } up totime t > 0. Since the n -dimensional Brownian motion is a strong Markov processwith continuous paths, the Karlin-McGregor formula [ ] gives the (unnormalised)density of the positions of the process at time t :ˆ p t ( x, y ) = det [ p t ( x i , y j )] ni,j =1 , x, y ∈ C, (4.6)where p t ( x, y ) = 1 √ πt e − ( x − y )22 t . Let M t,x be the normalisation constant for (4.6), that is, M t,x = ˆ C ˆ p t ( x, y ) dy. We have the following: Proposition . The positions at time t > of n independent one-dimensionalBrownian motions, started at the origin, and conditioned not to intersect up to time t > , are given by lim x ∈ Cx → M t,x ˆ p t ( x, y ) = 1 M ′ t e − t P ni =1 y i Y ≤ i The above expression agrees with the joint density of the eigenvaluesof an n × n GOE random matrix with variance parameter t [ , ]. Proof. A simple calculation shows that1 M t,x ˆ p t ( x, y ) = 1 M t,x πt ) n/ e − t P ni =1 ( x i + y i ) det (cid:16) e t x i y j (cid:17) ni,j =1 , and, dividing both numerator and denominator by the Vandermonde determinant∆( x ) = Y ≤ i 0, with density h ( x, y ) = h ′ ( x, y ) + h ′ ( x, − y ) = 2 π xx + y , y > . (4.7)Consider the determinant of hitting densities H ( x, y ) = det ( h ( x i , y j )) ni,j =1 , x, y ∈ D, where D = { x ∈ R n : 0 < x n < x n − < ... < x } . Proposition . For any y ∈ D , and t > , lim x ∈ Dx → t ˜ H ( x, y ) = 1 M t Y ≤ j ≤ n ( t + y j ) − n Y ≤ i In particular, when t = 1, the above density takes the form1 M Y ≤ j ≤ n (1 + y j ) − n Y ≤ i For all x, y ∈ D , the function H ( x, y ) is positive and a Cauchy deter-minant (see [ ]). Therefore H ( x, y ) = (cid:18) π (cid:19) n n Y i =1 x i Y ≤ i,j ≤ n ( x i + y j ) − Y ≤ i For all x, y ∈ C , the function H t ( x, y ) is positive (see [ ]), and thefollowing explicit expression for H t ( x, y ) can be obtained H t ( x, y ) = 1(2 t ) n Y ≤ i,j ≤ n sech (cid:16) π t ( y j − x i ) (cid:17) Y ≤ i 0. We have that for all x ∈ C such that | x i | < ε , 1 ≤ i ≤ n , and for all y ∈ C | F x ( t, y ) | ≤ n Y j =1 (cid:18) ce π t y j + c − e − π t y j (cid:19) n Y ≤ i 1, and stopped until it hits the point z = e iθ , 0 < θ < π , is given by thewell-known formula (see [ ], section 1.10): h ( x, θ ) = 1 π − x − x cos θ + x , < θ < π. As before, consider the determinant of hitting densities H ( x, θ ) = det( h ( x i , y j )) ni,j =1 , x ∈ N, θ ∈ Θ , where N = { x ∈ R n : − < x n < x n − < ... < x < } andΘ = { θ ∈ R n : 0 < θ < θ < ... < θ n < π } . Proposition . For any θ ∈ Θlim x ∈ Nx → ˜ H ( x, θ ) = 1 M Y ≤ i The above density can be thought of as the β = 1 version of theeigenvalue density of a random matrix in SO (2 n ), which is the subgroup of unitarymatrices consisting of 2 n × n orthogonal matrices with determinant one (see [ ]). Proof. The function H ( x, θ ) can be expressed explicitly as H ( x, θ ) = H ′ ( x ) Y ≤ i,j ≤ n (1 − x i cos θ j + x i ) − Y ≤ i 1, and assume that | x i | < ε , for all 1 ≤ i ≤ n . We have that | − x i cos θ j + x i | > (1 − ε ) and therefore | F x ( θ ) | ≤ n ( n − (1 − ε ) − n , for all θ ∈ Θ . Since Θ is a bounded set, by the bounded convergence theorem it follows that forany θ ∈ Θ lim x ∈ Nx → ˜ H ( x, θ ) = 1 M Y ≤ i 1. The next proposition is the excursion Poisson kernel analogueof Proposition 4.6. Proposition . As in Proposition 4.6, let Θ be the set Θ = { θ ∈ R n : 0 < θ < θ < ... < θ n < π } . Then lim x ,...,x n → det( h ∂ Ω ( x i , θ j )) ni,j =1 ´ Θ det( h ∂ Ω ( x i , θ j )) ni,j =1 dθ = 1 M n Y j =1 sin θ j Y ≤ i Note thatdet( h ∂ Ω ( x i , θ j )) ni,j =1 = (cid:18) π (cid:19) n n Y i =1 (1 − x i ) n Y j =1 sin θ j det ( B ) , where B = ( b i,j ) is the n × n matrix with positive entries b i,j = 1(1 − x i cos θ j + x i ) . The determinant det( B ) can be expressed as the product (see [ ]):det( B ) = det (cid:18) − x i cos θ j + x i (cid:19) ni,j =1 per (cid:18) − x i cos θ j + x i (cid:19) ni,j , (4.14)where the permanent of a square matrix is defined asper( a i,j ) ni,j =1 = X σ ∈ S n n Y i =1 a i,σ ( i ) . The determinant in the right hand side of (4.14) was considered in Section 4.5.Therefore, we can conclude thatdet( h ∂ Ω ( x i , θ j )) ni,j =1 = G ( x ) P ( x, θ ) n Y j =1 sin θ j Y ≤ i 0, 1 ≤ i ≤ n , and | Q x ( θ ) | ≤ n ! 2 n ( n − (1 − ε ) − n + n ) , for all θ ∈ Θ , whenever | x i | < ε , 0 < ε < 1, for all 1 ≤ i ≤ n . Since Θ is bounded, the desiredresult follows from the bounded convergence theorem. (cid:3) Proposition 4.7 agrees with certain asymptotics of an excursion Poisson kerneldeterminant in [ ], in the context of rectangular domains of the complex plane. 5. Circular ensembles In this section we consider limits of determinants of hitting densities of the(affine) form (4.2) H ( x, y ) = det X k ∈ Z ζ k h ( x i , y j + mk ) ! ni,j =1 dy · · · dy n , (5.1)where ζ = (cid:26) n is odd − n is even , and reveal some natural connections with circular ensembles of random matrix the-ory, similar to the connections described in Section 4 with Cauchy type ensembles.In particular, by considering the hitting density of the two-dimensional Brownianmotion in an annulus on the complex plane, we obtain a novel interpretation ofthe Circular Orthogonal Ensemble (COE) (see Section 5.2). Another example isgiven in Section 5.1, where we review the well-known model of n non-intersecting(one-dimensional) Brownian motions on the circle [ ] and detail its connectionwith the Circular Orthogonal Ensemble. An interesting consequence is Proposi-tion 5.3, which recovers the Karlin-McGregor (for n odd) and Liechty-Wang (for n even) determinant formulas [ , ], for the transition density of n indistinguishable non-intersecting Brownian motions on the circle, from the one in [ ].0 In this section we consider limits of determinants of hitting densities of the(affine) form (4.2) H ( x, y ) = det X k ∈ Z ζ k h ( x i , y j + mk ) ! ni,j =1 dy · · · dy n , (5.1)where ζ = (cid:26) n is odd − n is even , and reveal some natural connections with circular ensembles of random matrix the-ory, similar to the connections described in Section 4 with Cauchy type ensembles.In particular, by considering the hitting density of the two-dimensional Brownianmotion in an annulus on the complex plane, we obtain a novel interpretation ofthe Circular Orthogonal Ensemble (COE) (see Section 5.2). Another example isgiven in Section 5.1, where we review the well-known model of n non-intersecting(one-dimensional) Brownian motions on the circle [ ] and detail its connectionwith the Circular Orthogonal Ensemble. An interesting consequence is Proposi-tion 5.3, which recovers the Karlin-McGregor (for n odd) and Liechty-Wang (for n even) determinant formulas [ , ], for the transition density of n indistinguishable non-intersecting Brownian motions on the circle, from the one in [ ].0 As a warm up before Section5.2, we describe the model of n non-intersecting Brownian motions on the unit circle,originally studied by Hobson and Werner in [ ]. Here, the Brownian motions on T = { e iθ : − π ≤ θ < π } are given by β k := e iB k , ≤ k ≤ n, where B , B , ..., B n are n independent one-dimensional Brownian motions and weassume n ≥ 2. The following proposition shows that the above model can bestudied by considering the exit time of the n -dimensional Brownian motion B =( B , B , ..., B n ) of the domain˜ A n := { ν ∈ R n : ν n < ν n − < ... < ν < ν < ν n + 2 π } . Proposition . Let B and ˜ A n as above. The transitiondensity of the Brownian motion B killed at its first exit from ˜ A n is given by q t ( θ, ν ) = X σ ∈ S n X k + k + ... + k n =0 sgn( σ ) n Y i =1 p t ( θ i , ν σ ( i ) + 2 πk i ) , t > , (5.2) where θ = ( θ , ..., θ n ) ∈ ˜ A n , ν = ( ν , ..., ν n ) ∈ ˜ A n , and p t ( x, y ) = 1 √ πt e − ( x − y )22 t is the normal density with mean x and variance t . The method of proof of the last proposition is by a path-switching argument,similar to the one of Theorem 3.1. The following corollary is a restatement of part( i ) of the main theorem in [ ] and describes the transition density for n labelled particles in Brownian motion on the circle, constrained not to intersect until a fixedpositive time. Corollary . The (unnormalised) transition density of n non-intersectingBrownian motions ( β , ..., β n ) on the circle is q ∗ t ( e iθ , e iν ) = X σ ∈ S n X k + k + ... + k n =0mod n sgn( σ ) n Y i =1 p t ( θ i , ν σ ( i ) + 2 πk i ) , t > where e iθ = ( e iθ , ..., e iθ n ) ∈ T n , e iν = ( e iν , ..., e iν n ) ∈ T n , and θ, ν ∈ C = ˜ A n ∩ { ν ∈ R n : − π ≤ ν n < π } . Moreover, q ∗ t ( e iθ , e iν ) can be expressed as the sum of n determinants: q ∗ t ( e iθ , e iν ) = 1 n n − X u =0 det X k ∈ Z η uk p t ( θ i , ν j + 2 πk ) ! ni,j =1 . (5.3) Proof. Since any point in the circle is the projection of an infinite set of pointsin the real line modulo 2 π , the first part follows immediately by summing up in(5.2) over all the images of ν = ( ν , ..., ν n ) ∈ ˜ A n under translations of 2 π , that is q ∗ t ( e iθ , e iν ) = X ℓ ∈ Z q t ( θ, ν + 2 πℓ (1 , ..., . For the second part, if η = e i πn is a complex root of unity, we can eliminate thecondition k + k + ... + k n = 0, mod n , by using the identity1 n n − X u =0 η u P ni =1 k i = (cid:26) P ni =1 k i = 0 , mod n , and (5.3) follows. (cid:3) Interestingly, if we do not label the n Brownian particles in Corollary 5.2 (andtherefore the locations at time t > n cyclic permutations ofthe vector ( e iν , ..., e iν n ) along the circle), then the corresponding transition densitybecomes a single determinant: Proposition . The (unnormalised) transition density of n ‘indistinguish-able’ non-intersecting Brownian motions on the circle is given by H t ( e iθ , e iν ) = det X k ∈ Z e i πxk p t ( θ i , ν j + 2 πk ) ! ni,j =1 , θ, ν ∈ C, (5.4) where x = (cid:26) if n is odd , if n is even . Remark. In particular, Proposition 5.3 recovers the Karlin-McGregor (for n odd) and Liechty-Wang (for n even) determinant formulas [ , ], for the transitiondensity of n indistinguishable non-intersecting Brownian motions on the circle. Remark. Using modular transformations for Jacobi theta functions, the en-tries of the matrix in (5.4) can be written in terms of theta functions as follows: π θ (cid:16) − ( ν j − θ i )2 , e − t/ (cid:17) if n is odd , π θ (cid:16) − ( ν j − θ i )2 , e − t/ (cid:17) if n is even , where θ k ( z, q ) is the k -Jacobi theta function, z ∈ C , | q | < ]). Proof of Proposition 5.3. The method of proof is by summing-up, in (5.3),the n different destinations of the labelled process of Corollary 5.2. Fix θ ∈ C and ν ∈ C . If [ ℓ ] ∈ S n is the shift by ℓ = 0 , , ..., n − 1, let ν [ ℓ ] be the unique representativeof ( ν [ ℓ ](1) , ..., ν [ ℓ ]( n ) ) in C . Then, the n different ‘cyclic permutations’ of the vector e iν = ( e iν , ..., e iν n ) ∈ T n along the unit circle are given by e iν [ ℓ ] , ℓ = 0 , , ..., n − . With the notation of Corollary 5.2, it holds that q ∗ t ( e iθ , e iν [ ℓ ] ) = 1 n n − X u =0 η − ℓu sgn( σ ) det X k ∈ Z η uk p t ( θ i , ν j + 2 πk ) ! ni,j =1 . Finally, following the same argument as in the proof of Proposition 3.3, we obtain n − X ℓ =0 q ∗ t ( e iθ , e iν [ ℓ ] ) = det X k ∈ Z e i πxk p t ( θ i , ν j + 2 πk ) ! ni,j =1 , where x = (cid:26) n is odd , if n is even . (cid:3) Following the notation of Proposition 5.3, consider now the normalised density˜ H t ( e iθ , e iν ) = 1 M t,θ H t ( e iθ , e iν ) , θ, ν ∈ C, where M t,θ = ˆ C H t ( e iθ , e iν ) dν. The following proposition is essentially a reformulation of ( ii ) and ( iii ) of the maintheorem in [ ], here stated in the case of n indistinguishable non-intersectingBrownian motions on the circle. We also take into consideration the correspondingnormalisation constants. Proposition . For any θ, ν ∈ C , lim t →∞ ˜ H t ( e iθ , e iν ) = 1 M Y ≤ i The above limit agrees with the eigenvalue density of the CircularOrthogonal Ensemble (COE), defined on C = ˜ A n ∩ { ν ∈ R n : − π ≤ ν n < π } . Proof of Proposition 5.4. Using the Poisson summation formula for eachentry of the matrix array in (5.4), we have X k ∈ Z e i πxk p t ( θ i , ν j + 2 πk ) = 12 π X k ∈ Z e − i ( ν j − θ i )( x + k ) e − t ( x + k ) . Alternatively, the above can be seen as a direct consequence of the definitions byinfinite series of the Jacobi’s theta functions θ and θ (see second remark afterProposition 5.3). Now, by standard properties of determinants we obtain H t ( e iθ , e iν ) = 1(2 π ) n X k Mapping the strip onto the annuluswith normal reflection on the real axis and absorbing boundary Im ( z ) = | log r | , seeFigure 11. From Section 4.4, we know that if the process β starts at a point θ ∈ R ,then the distribution of its first hitting point at Im ( z ) = | log r | has the density h ( θ, ν ) = 12 | log r | sech (cid:18) π | log r | ( ν − θ ) (cid:19) , ν ∈ R . Consider the bounded set C = ˜ A n ∩ { ν ∈ R n : − π ≤ ν n < π } , where ˜ A n := { ν ∈ R n : ν n < ν n − < ... < ν < ν < ν n + 2 π } . Definition . Let η = e i πn be the n -th root of unity. Define, for θ, ν ∈ C , H ar ( e iθ , e iν ) = det X k ∈ Z e i πxk h ( θ i , ν j + 2 πk ) ! ni,j =1 , (5.6) where x = (cid:26) if n is odd , if n is even . Remark. The strip Ω ′ ⊂ C is clearly invariant under horizontal translations by2 πk , k ∈ Z , and therefore the determinant (5.6) is a determinant of hitting densitiesof the affine form (4.2), described at the beginning of Section 4. Since (5.6) isdefined as a natural continuous analogue of the determinant in Proposition 3.3, weexpect the determinant H ar ( e iθ , e iν ) to be positive and be interpreted (informally)as the probability that n independent trajectories B , ..., B n of the process B in theannulus Ω, starting at positions re iθ j , j = 1 , ..., n, will hit the unit circle by first time at points e iν j , j = 1 , ..., n, with an angle in each of the intervals ( ν j , ν j + dν j ), j = 1 , ..., n , and whose trajec-tories are constrained to satisfy B j ∩ LE ( B j − ) = ∅ , < j ≤ n, and B ∩ LE ( B n ) = ∅ . Note that we do not require that the trajectory which started at point re iθ j hitsthe unit circle at the corresponding point e iν j . Remark. If n is odd, the entries of the matrix in (5.6) can be written as12 π θ (0 , r ) θ (0 , r ) θ (cid:16) iπ | log r | ( ν j − θ i ) , r (cid:17) θ (cid:16) iπ | log r | ( ν j − θ i ) , r (cid:17) , where θ k ( z, q ) is the k -Jacobi theta function, z ∈ C , | q | < ]).Consider the normalised density˜ H ar ( e iθ , e iν ) = 1 M r,θ H ar ( e iθ , e iν ) , where M r,θ = ˆ C H ar ( e iθ , e iν ) dν. The following proposition gives the limit of ˜ H ar ( e iθ , e iν ) as the inner radius r goesto zero. This models the situation where the n Brownian motions start at the originof the complex plane. Proposition . For any θ, ν ∈ C , lim r → ˜ H ar ( e iθ , e iν ) = 1 M Y ≤ i The above limit agrees with the eigenvalue density of a randommatrix belonging to the Circular Orthogonal Ensemble (COE), defined on C [ , ]. Proof of Proposition 5.5. By Lemma A.2 and standard properties of de-terminants, we can express (5.6) as the sum H ar ( e iθ , e iν ) = 1(2 π ) n X k 0. As in the proof ofProposition 5.4, the bounded convergence theorem implies that, for each θ, ν ∈ C lim r → ˜ H ar ( e iθ , e iν ) = 1 ´ C det (cid:0) e − iν j ( x + k ′ i ) (cid:1) dν det (cid:16) e − iν j ( x + k ′ i ) (cid:17) ni,j =1 = 1 M Y ≤ i 6. Conclusions We have developed connections between loop-erased walks in two dimensionsand random matrices, based on an identity of S. Fomin [ ]. This complementsearlier work of Sato and Katori [ ], where an example of this type of connection wasexhibited in a slightly different context, as explained in Sections 1.3 and 4.6. Theseconnections resemble the well-known relations between non-intersecting processesin one dimension and random matrices. For two-dimensional Brownian motionsin suitable simply connected domains, conditioned (in an appropriate sense) tosatisfy a certain non-intersection condition, we obtain, in particular scaling limits,eigenvalue densities of Cauchy type.As a first step towards the consideration of non-simply connected domains, wehave formulated and proved an affine (circular) version of Fomin’s identity. Apply-ing this in the context of independent Brownian motions in an annulus, conditionedto satisfy a circular version of Fomin’s non-intersection condition, we obtain, in aparticular scaling limit, the circular orthogonal ensemble of random matrix theory. Exploring relations between random matrices, SLE and related combinatorialmodels, seems to be an interesting direction for future research. We hope that ourpreliminary findings will motivate further developments in this direction. Appendix A. For any n -tuple of complex numbers x = ( x , ..., x n ) let∆( x ) = det (cid:0) x i − j (cid:1) ni,j =1 = Y ≤ i A.1 . Let h i , ≤ i ≤ n , be functions which are (complex) analytic at x .Let x = ( x, ..., x ) and y = ( y , . . . , y n ) . Then lim y → x y − x ) det( h j ( y i )) ni,j =1 = C · det (cid:0) ∂ i − x h j ( x ) (cid:1) ni,j =1 , where C = Q n − j =1 1 j ! . Proof. By the Weierstrass preparation theorem, it suffices to prove the state-ment for y = x ε with ε → 0, where x ε = x + εδ, ε > , δ = ( n − , n − , ..., . The statement now follows from arguments presented, for example, in [ ]. This isgiven as follows. Define the difference operator D with increment ε by D h ( x ) = h ( x ) , Dh ( x ) = h ( x + ε ) − h ( x ) D i +1 h ( x ) = D ( D i h ( x )) , n ≥ . Then D i h ( x ) = i X k =0 ( − i + k (cid:18) ik (cid:19) h ( x + kε ) , i ≥ , (A.1)and the operators ∂ x and D are related through the identitylim ε → D i h ( x ) ε i = ∂ ix h ( x ) . Using the relation (A.1), we have the matrix decomposition: (cid:2) D i − h j ( x ) (cid:3) ni,j =1 = " i − X k =0 ( − i − k (cid:18) i − k (cid:19) h j ( x + kε ) ni,j =1 = (cid:20) ( − i − k − (cid:18) i − k − (cid:19)(cid:21) ni,k =1 [ h j ( x + ( k − ε )] nk,j =1 . Note that the matrix in the middle is lower triangular, so its determinant is theproduct of its diagonal entries and thereforedet (cid:0) D i − h j ( x ) (cid:1) ni,j =1 = det ( h j ( x + ( k − ε )) nk,j =1 . As a consequence, we obtain the following identity:det (cid:0) ∂ i − x h j ( x ) (cid:1) ni,j =1 = lim ε → ε − ( n ) det ( h j ( x + ( i − ε )) ni,j =1 . Finally, note that∆( x ε − x ) = ∆( εδ ) = ε ( n )∆( δ ) = ( − ε )( n ) n − Y j =1 j ! , and thereforedet (cid:0) ∂ i − x h j ( x ) (cid:1) ni,j =1 = n − Y j =1 j ! lim ε → x ε − x ) det ( h j ( x + ( n − i ) ε )) ni,j =1 , as required. (cid:3) Lemma A.2 . Let η = e i πn be the n -th root of unity. Therefore X k ∈ Z e i πxk h ( θ, ν + 2 πk ) = 12 π X k ∈ Z sech ( | log r | ( x + k )) e − i ( ν − θ )( x + k ) . Proof. If η = e i πn is the n -th root of unity, the left-hand side above can beexpressed as X k ∈ Z e i πxk h ( θ, ν + 2 πk ) = 12 | log r | X k ∈ Z e i πxk ˆ f ( k ) , (A.2)where ˆ f ( k ) is the Fourier transform of the function f ( ξ ) = | log r | π sech( | log r | ξ ) e − i ( ν − θ ) ξ . Therefore, applying the Poisson summation formula to the right hand side of (A.2),we obtain 12 | log r | X k ∈ Z e i πxk ˆ f ( k ) = 12 | log r | X k ∈ Z f ( x + k )= 12 π X k ∈ Z sech( | log r | ( x + k )) e − i ( ν − θ )( x + k ) . (cid:3) References [1] L. V. Ahlfors, Complex Analysis , 3rd edition, McGraw-Hill, New York (1979)[2] L. Carlitz and J. Levine, An identity of Cayley, American Mathematical Montly , 571–573(1960)[3] B. Conrey, Notes on eigenvalue distributions for the classical compact groups, Recent per-spectives in random matrix theory and number theory, LMS Lecture Note Series, vol. 322.Cambridge University Press (2005)[4] J. Dub´edat, Euler integrals for commuting SLEs, J. Stat. Phys. , 1183–1218 (2006)[5] R. Durrett, Brownian Motion and Martingales in Analysis , Wadsworth Inc., Belmont, CA,(1984)[6] R. Durrett, Probability: theory and Examples , 4th edition, Cambridge University Press, Cam-bridge (2010)[7] F. J. Dyson, A Brownian motion model for the eigenvalues of a random matrix, J. Math.Phys. (1962).[8] S. Fomin, Loop-erased walks and total positivity, Trans. Am. Math. Soc. , 3563–3583(2000)[9] P. Forrester, Log-Gases and Random Matrices , London Mathematical Society Monographs,vol. 34, (2010) [10] M. Fulmek, Nonintersecting lattice paths on the cylinder. S´em. Lothar. Combin. , 16(2004/07)[11] I. Gessel and C. Krattenthaler, Cylindric partitions, Trans. Am. Math. Soc. , 429-479(1997)[12] I. Gessel and D. Zeilberger, Random walk in a Weyl chamber, Proc. Am. Math. Soc. , 27–31(1992)[13] I. Gessel and X. Viennot, Determinants, paths and plane partitions, preprint (1989)[14] D. Grabiner, Brownian motion in a Weyl chamber, non-colliding particles, and random ma-trices, Ann. Inst. H. Poincar´e Probab. Stat. , 177-204 (1999)[15] D. Grabiner, Random walk in an alcove of an affine Weyl group, and non-colliding randomwalks on an interval, J. Combin. Theory Ser. A , 285–306 (2002).[16] D. Hobson and W. Werner, Non-colliding Brownian motions on the circle, Bull. London Math.Soc. , 643–650 (1996)[17] L. Jones and N. O’Connell, Weyl chambers, symmetric spaces and number variance saturation,ALEA Lat. Am. J. Probab. Math. Stat. , 91–118 (2006)[18] S. Karlin, Total positivity , Stanford University Press, Stanford (1968)[19] S. Karlin and J. McGregor, Coincidence probabilities, Pacific J. Math. , 1141–1164 (1959)[20] A. Karrila, Multiple SLE type scaling limits: from local to global. Available on Arxiv:1903.10354 (2019)[21] A. Karrila, K. Kyt¨ol¨a and E. Peltola, Boundary correlations in planar LERW and UST,Available on Arxiv: 1702.03261 (2017)[22] W. K¨oning and N. O’Connell, Eigenvalues of the Laguerre process as non-colliding squaredBessel processes, Elect. Comm. in Probab. , 107–114 (2001)[23] M. Kozdron. The scaling limit of Fomins identity for two paths in the plane, C. R. Math.Acad. Sci. Soc. R. Can. (3), 6580 (2007)[24] M. Kozdron and G. Lawler, The configurational measure on mutually avoiding SLE paths,Fields Institute Communications , 199-224 (2007)[25] M. Kozdron and G. Lawler, Estimates of random walk exit probabilities and application toloop-erased walk, Electron. J. of Probab. , 1442-1467 (2005)[26] G. Lawler, Conformally Invariant Processes in the Plane , American Mathematical Society,Ithaca (2005)[27] G. Lawler, Topics in loop measures and the loop-erased walk, Probability Surveys , 28–101(2018)[28] G. Lawler and V. Limic, Random Walk: A Modern Introduction , Cambridge University Press,Cambridge (2010)[29] Y. Le Jan, Markov Paths, Loops, and Fields , Lecture Notes in Mathematics, vol. 2026.Springer, New York (2011)[30] K. Liechty and D. Wang, Nonintersecting Brownian motions on the unit circle, Ann. Probab. , 1134–1211 (2016)[31] B. Lindstr¨om, On the vector representations of induced matroids, Bull. London Math. Soc. , 85-90 (1973)[32] M. Mehta, Random Matrices , 3rd edition, Academic Press, Amsterdam (2004)[33] NIST Digital Library of Mathematical Functions. http://dlmf.nist.gov/, Release 1.0.23 of2019-06-15. F. W. J. Olver, A. B. Olde Daalhuis, D. W. Lozier, B. I. Schneider, R. F. Boisvert,C. W. Clark, B. R. Miller, and B. V. Saunders, eds.[34] M. Sato and M. Katori, Determinantal correlations of Brownian paths in the plane withnonintersection condition on their loop-erased parts, Phys. Rev. E , (2011)[35] O. Schramm, Scaling limits of loop-erased random walk and uniform spanning trees, Israel J.Math. , 221–288 (2000)[36] J. Stembridge, Nonintersecting paths, Pfaffians, and plane partitions, Adv. Math. , 96-131(1990)[37] C. Wenchang, Finite differences and determinant identities, Linear Algebra and its Applica-tions , 215–228 (2009)[38] N. Witte and P. Forrester, Gap probabilities in the finite and scaled Cauchy random matrixensembles, Nonlinearity , 1965–1986 (2000)[39] A. Yadin and A. Yehudayoff, Loop-erased random walks and Poisson kernel on planar graphs,Ann. Probab. , (2011)[40] D. Zhan, Loop-erasure of planar Brownian motion, Commun. Math. Phys. , 709 (2011)0