Interlaced processes on the circle
aa r X i v : . [ m a t h . P R ] N ov Interlaced processes on the circle
Anthony P. Metcalfe, Neil O’Connell, and Jon Warren
Abstract.
When two Markov operators commute, it suggests that we cancouple two copies of one of the corresponding processes. We explicitly con-struct a number of couplings of this type for a commuting family of Markovprocesses on the set of conjugacy classes of the unitary group, using a dynam-ical rule inspired by the RSK algorithm. Our motivation for doing this is todevelop a parallel programme, on the circle, to some recently discovered con-nections in random matrix theory between reflected and conditioned systemsof particles on the line. One of the Markov chains we consider gives rise toa family of Gibbs measures on ‘bead configurations’ on the infinite cylinder.We show that these measures have determinantal structure and compute thecorresponding space-time correlation kernel.
1. Introduction
When two Markov operators commute, it suggests that we can couple two copiesof one of the corresponding processes. Such couplings have been described in [ ]in a general context. In this paper, we explicitly construct a number of couplings ofthis type for a commuting family of Markov processes on the set of conjugacy classesof the unitary group, using a dynamical rule inspired by the Robinson-Schensted-Knuth (RSK) algorithm. Our motivation for doing this is to develop a parallelprogramme, on the circle, to some recently discovered connections in random matrixtheory between reflected and conditioned systems of particles on the line (see, forexample, [
2, 4, 19, 32, 40 ]). The RSK algorithm is a combinatorial device whichplays an important role in the representation theory of GL ( n, C ) and lies at theheart of these developments. We refer the reader to [ ] for more background on Mathematics Subject Classification.
Primary 60J99, 60B15, 82B21; Secondary 05E10. the combinatorics. One of the Markov chains we consider gives rise to a family ofGibbs measures on ‘bead configurations’ on the infinite cylinder. This is relatedto recent work [
5, 6, 24 ] on planar and toroidal models. We will show that thesemeasures have determinantal structure and compute the corresponding space-timecorrelation kernel.We start with some motivation, and a flavour of some of the main results inthis paper. The following construction is closely related to the RSK algorithm [ ]. Let E = Z and, for n ≥ E n = { x ∈ Z n : x < · · · < x n } . The reader should think of an element x ∈ E n as a configuration of n particleson the integers located at positions x < · · · < x n . We say that a pair ( x, y ) ∈ E m × E m +1 are interlaced , and write x (cid:22) y , if y j < x j ≤ y j +1 for all j ≤ m . A(discrete) Gelfand-Tsetlin pattern of depth n is a collection ( x , x , . . . , x n ) suchthat x m ∈ E m for m ≤ n and x m (cid:22) x m +1 for 1 ≤ m < n .Fix n ≥ GT n the set of Gelfand-Tsetlin patterns of depth n . Let w , w , . . . be a sequence of independent random variables, each chosen according tothe uniform distribution on { , , . . . , n } . Using these, we will construct a Markovchain ( X ( k ) , k ≥
0) with state space GT n , which evolves according to X ( k + 1) = g ( X ( k ) , w k +1 ) , k ≥ g : GT n × { , . . . , n } → GT n is defined recursively as follows. Fix m < n and let ( x, y ) ∈ E m × E m +1 such that x (cid:22) y . Let x ′ ∈ E m such that, for some j ≤ m , x ′ i = x i + δ ij . The reader should have in mind m particles located atpositions x < · · · < x m , interlaced with another set of m + 1 particles locatedat positions y < · · · < y m +1 . The first configuration x is updated by movingthe particle at position x j one step to the right, that is, to position x j + 1, givinga new configuration x ′ . This can be used to obtain an update y ′ to the secondconfiguration y , obtained by moving the first available particle, strictly to the rightof position y j , which can be moved without breaking the interlacing constraint ,so that x ′ (cid:22) y ′ . Such a particle is guaranteed to exist because the interlacingconstraint cannot be broken by the rightmost particle. In other words, the updatedconfiguration is given by y ′ i = y i + δ ik where k = inf { l > j : y l + 1 < x l } . Let NTERLACED PROCESSES ON THE CIRCLE 3 us write y ′ = φ ( x, y, x ′ ) where φ is defined on an appropriate domain. Now, given m ≤ n and a pattern ( x , . . . , x n ) ∈ GT n we define a new pattern( y , . . . , y n ) ≡ g (( x , . . . , x n ) , m )as follows. First we set y j = x j for j < m . Then we obtain y m from the configura-tion x m by moving the first available particle, starting from the particle at position x m , which can be moved one step to the right without breaking the interlacingconstraint. Finally, we define, for m ≤ l < n , y l +1 = φ ( x l , x l +1 , y l ).The Markov chain X has remarkable properties. For example, if we start fromthe initial pattern(1) X (0) = ((0) , ( − , , ( − , − , , . . . , ( − n + 1 , . . . , , then ( X n ( k ) , k ≥
0) is a Markov chain (with respect to its own filtration) with statespace E n and transition probabilities given by(2) P n ( x, y ) = n h ( y ) h ( x ) x ր y h is the Vandermonde function h ( x ) = Q i 30, 31 ]): Proposition 1.1. Given x ∈ E n , if X (0) is chosen uniformly at random from theset of patterns ( x , . . . , x n ) with x n = x , then ( X n ( k ) , k ≥ is a Markov chainstarted from x with transition probabilities given by P n . Moreover, for each T > ,the conditional law of X ( T ) , given ( X n ( k ) , T ≥ k ≥ , is uniformly distributed onthe set of patterns ( x , . . . , x n ) with x n = X n ( T ) . The connection with the RSK algorithm is the following: if we start from theinitial pattern (1), then the integer partition( X nn ( k ) , X nn − ( k ) + 1 , X nn − ( k ) + 2 , . . . , X n ( k ) + n − ANTHONY P. METCALFE, NEIL O’CONNELL, AND JON WARREN is precisely the shape of the tableau obtained when one applies the RSK algorithm,with column insertion, to the word w w · · · w k . We refer to [ ] for more details.This construction clearly has a nested structure. If we consider the evolution ofthe last two rows X n − and X n we see that we can construct a Markov chain withtransition probabilities P n from a Markov chain with transition probabilities P n − plus a ‘little bit of extra randomness’.In the above construction we are thinking in terms of particles moving on aline. In random matrix theory, there are often strong parallels between naturalmeasures on configurations of particles (or ‘eigenvalues’) on the line, and configura-tions of particles on the circle. It is therefore natural to ask if there is an analogueof the above construction for configurations of particles on the circle. The notionof interlacing carries over in the obvious way. However, in this setting, interlacedconfigurations should have the same number of particles, and the analogue of aGelfand-Tsetlin pattern could potentially be an infinite object. Despite this fun-damental difference between the two settings, there is indeed a natural analogueof the above ‘RSK dynamics’ and a natural analogue of Proposition 1.1. Considerthe discrete circle with N positions which we label { , , . . . , N − } in the anti-clockwise direction. The analogue of the Markov chain with transition matrix P n is a Markov chain on the set C Nn of configurations of n indistinguishable particleson the discrete circle with transition probabilities given by(3) Q ( x, y ) = cn ∆( y )∆( x ) x ր y x ր y means that the configuration y canbe obtained from x by moving one particle one step anti-clockwise, and the function∆ is again a Vandermonde function defined, for a configuration x which consists ofa particles located at positions k , . . . , k n , by(4) ∆( x ) = Y i 0. A formula for the eigenvalue λ can be found in [ ].In this setting we will say that a pair of configurations ( x, y ) are interlaced,and write x (cid:22) y as before, if there is a labelling of the particles such that x consistsof a particles located at positions k , . . . , k n , y consists of a particles located atpositions l , . . . , l n , k j < l j ≤ k j +1 for j < n and either k n < l n ≤ k + N or k n < l n + N ≤ k + N .Let ( X ( k ) , k ≥ 0) be a Markov chain with state space C Nn and transitionmatrix Q . On the same probability space, without using any extra randomness,we can construct a second process ( Y ( k ) , k ≥ C Nn , suchthat X ( k ) (cid:22) Y ( k ) for all k . This is given as follows. For each k > 0, given X ( k ), Y ( k ) and X ( k + 1) we obtain the configuration Y ( k + 1) from Y ( k ) by moving aparticle one step anti-clockwise; this particle is chosen as follows. The transitionfrom X ( k ) to X ( k +1) involves one particle moving one step anti-clockwise; startingat the position of this particle, choose the first particle in the configuration Y ( k )that we come to, in an anti-clockwise direction, which can be moved by one stepanti-clockwise without breaking the interlacing constraint.The function ∆ defined by (4) is also a positive (left and right) eigenvector ofthe matrix 1 x (cid:22) y . This follows, for example, from the discussion given at the end ofSection 3. In particular, X x (cid:22) y ∆( x ) = γ ∆( y )for some γ > C Nn by(5) M ( y, x ) = 1 γ ∆( x )∆( y ) 1 x (cid:22) y . The analogue of Proposition 1.1 in this setting is the following: Proposition 1.2. If Y (0) = y and X (0) is chosen at random according to thedistribution M ( y, · ) , then ( Y ( k ) , k ≥ is a Markov chain started at y with tran-sition matrix Q . Moreover, for each T > , the conditional law of X ( T ) , given ( Y ( k ) , T ≥ k ≥ , is given by M ( Y ( T ) , · ) . ANTHONY P. METCALFE, NEIL O’CONNELL, AND JON WARREN In the sequel we will present a number of variations of this result, firstly involv-ing random walks with jumps in a continuous state space and secondly involvingBrownian motion. Proposition 1.2 follows from results presented in Section 3 (seediscussion towards the end of that section).We will also study continuous analogues of the Markov chain with transitionmatrix M . These Markov chains also arise naturally in the context of a certainrandom walk on the unitary group which is obtained by taking products of certainrandom (complex) reflections, as studied for example in [ 15, 35 ]. This is describedin Section 2 and taken as a starting point for the exposition that follows. The mainpoint is that these Markov chains commute with each other and with the DirichletLaplacian on the set of conjugacy classes of the unitary group. In Sections 3 and 4,we present couplings which realise these commutation relations, first between inter-laced random walks on the circle and later between interlaced Brownian motions.These couplings are precisely the variations of Proposition 1.2 mentioned above.Actually there are two natural couplings for the random walks and these corre-spond to dynamical rules inspired by the RSK algorithm with row, and columninsertion, respectively. The couplings between interlaced Brownian motions can bethought of as a limiting case where the two types of coupling become equivalent.In Section 5, we consider a family of Markov chains which can be thought of asperurbations of a continuous analogue of the Markov chain with transition matrixgiven by (5). These give rise to a natural family of Gibbs measures on ‘bead config-urations’ on the infinite cylinder. This is a cylindrical analogue of the planar beadmodel studied in [5] and, in one special case (the ‘unperturbed’ case), can also beregarded as a cylindrical analogue of some natural measures on Gelfand-Tsetlinpatterns related to ‘GUE minors’ [ 2, 23, 33, 14 ] (see also [ ] for extensions to theother classical complex Lie algebras). We show that these measures have determi-nantal structure by first writing the restrictions of these measures to cylinder setsas products of determinants and then following the methodology of Johansson (see,for example, [ ]) to compute the space-time correlation functions. 2. Markov processes on the conjugacy classes of the unitary group Consider the group U ( n ) of n × n unitary matrices, and denote by C n theset of conjugacy classes in U ( n ). Each element of C n can be identified with an NTERLACED PROCESSES ON THE CIRCLE 7 unlabelled configuration of n points (eigenvalues) on the unit circle, which in turncan be identified with the Euclidean set D n = { θ ∈ R n : 0 ≤ θ ≤ · · · ≤ θ n < π } .Denote by dx the image of Lebesgue measure under the latter identification, andby µ the probability measure on C n induced from Haar measure on U ( n ). Then µ ( dx ) = (2 π ) − n ∆( x ) dx , where ∆( x ) is defined, for x = { e iθ , . . . , e iθ n } , by(6) ∆( x ) = Y ≤ l Such random walks on U ( n ), and other classical compact groups, have been studiedextensively in the literature (see, for example, [ 9, 8, 38, 34, 35 ]).The case of interest in this paper is the random walk obtained by taking prod-ucts of certain random (complex) reflections in U ( n ) (see, for example, [ 15, 35 ]).More precisely, we take x = { e ir , , , . . . , } in the above kernel, where r ∈ (0 , π ).Let us write(9) p r := p x for this case. A concrete description of this kernel can be given as follows (see [ ]for details). For a, b ∈ D n , write a (cid:22) b if a ≤ b ≤ a ≤ · · · ≤ a n ≤ b n . For y = { e ia , . . . , e ia n } and z = { e ib , . . . , e ib n } , where a, b ∈ D n , write y (cid:22) r z ifeither a (cid:22) b and P j ( b j − a j ) = r , or b (cid:22) a and P j ( b j − a j ) + 2 π = r . The measure p r ( y, · ) is supported on the set F r ( y ) = { z ∈ C n : y (cid:22) r z } . This set can be identified with the disjoint union of a pair of ( n − { b ∈ D n : a (cid:22) b, X j ( b j − a j ) = r } ∪ { b ∈ D n : a (cid:23) b, X j ( b j − a j ) = r − π } , each of which can be endowed with ( n − F r ( y ) via this identifi-cation can be extended to a measure ν r ( y, · ) on C n by setting ν r ( y, C n \ F r ( y )) = 0.The following identity can be deduced from [ , Lemma 2]. Proposition 2.1. (10) p r ( y, dz ) = 1 γ r ∆( z )∆( y ) ν r ( y, dz ) , where γ r = | − e ir | ( n − / ( n − . In Section 5, we will present a determinantal formula for R π q r ν r ( y, dz ) dr ,where q > NTERLACED PROCESSES ON THE CIRCLE 9 Another operator which will play a role in this paper is the Dirichlet Laplacianon the (closed) alcove(11) A n = { θ ∈ R n : θ ≤ θ ≤ · · · ≤ θ n ≤ θ + 2 π } . Let ( Q t ) denote the transition semigroup of a standard Brownian motion condi-tioned never to exit A n . This is a Doob transform of the Brownian motion whichis killed when it exits A n , via the positive eigenfunction(12) h ( θ ) = Y ≤ l 0. Notethat the corresponding process on C n can be thought of as n standard Brownianmotions on the circle conditioned never to collide.We will also consider discrete analogues of the above processes. Set A Nn = ( N A n / π ) ∩ Z n , C Nn = exp(2 πiA n ( N ) /N ) / C n , Ω Nn = { λ ∈ Ω n : λ ≤ N − } . Much of the above discussion can be replicated in this setting, but for our purposesit suffices to make the following remark. Think of C Nn as the set of configurationsof n particles at distinct locations on the discrete circle with N positions. Considerthe random walk in C Nn where at each step a particle is chosen at random andmoved one position anti-clockwise if that position is vacant; if it is not vacant theprocess is killed. It is known that the restriction of the function ∆ to C Nn is thePerron-Frobenius eigenfunction for this sub-Markov chain. In fact, a complete setof eigenfunctions (with respect to the measure ∆( x ) ) is given by the restrictionsof the characters { χ λ , λ ∈ Ω Nn } to C Nn . (See, for example, [ ].) As we shall seelater, the discrete analogues of the ν r (thought of as operators) commute with thetransition kernel of this killed random walk and therefore share these eigenfunctions. 3. Couplings of interlaced random walks The Markov chain on C n with transition probabilities p r , defined by (9), can belifted to a Markov chain on A n , which is better suited to the constructions of this section. To make this precise let us say x and x ′ belonging to A n are r -interlaced,for some r ∈ (0 , π ), if(13) x ′ i ∈ [ x i , x i +1 ] for i = 1 , , . . . , n and n X i =1 ( x ′ i − x i ) = r, when we adopt the convention that x n +1 = x + 2 π . In this case we will write x (cid:22) r x ′ . Define π : A n → C n by π ( x ) = { e ix , . . . , e ix n } . Denote by l r ( x, dx ′ ) the( n − G r ( x ) = { x ′ ∈ A n : x (cid:22) r x ′ } . Clearly the restriction of π to G r ( x ) is injective, with π ( G r ( x )) = F r ( π ( x )). More-over, for measurable B ⊂ A n , l r ( x, B ) = ν r ( π ( x ) , π ( B ∩ G r ( x ))). Thus, if we define,for measurable B ⊂ A n ,(14) q r ( x, B ) = p r ( π ( x ) , π ( B ∩ G r ( x )))then, by Proposition 2.1, q r ( x, B ) = Z π ( B ∩ G r ( x )) p r ( π ( x ) , dz )= Z π ( B ∩ G r ( x )) γ r ∆( z )∆( π ( x )) ν r ( π ( x ) , dz )= Z B γ r ∆( π ( x ′ ))∆( π ( x )) l r ( x, dx ′ ) , and hence,(15) q r ( x, dx ′ ) = 1 γ r ∆( π ( x ′ ))∆( π ( x )) l r ( x, dx ′ ) . We will refer to a Markov chain with values in A n and transition probabilities q r as an r -interlacing random walk. As far as we are aware, such processes havenot previously appeared in the literature. Note that since p r p s = p s p r we have q r q s = q s q r or, equivalently, l r l s = l s l r .The goal of this section is to construct, for given r, s ∈ (0 , π ), two differentMarkovian couplings (cid:0) X ( k ) , Y ( k ); k ≥ 0) of a pair of r -interlacing random walkson A n , having the property that X ( k ) and Y ( k ) are s -interlaced for each n andmoreover, for each l ≥ 0, the trajectory (cid:0) Y ( k ); 0 ≤ k ≤ l ) will be a deterministicfunction of Y (0) together with the trajectory (cid:0) X ( k ); 0 ≤ k ≤ l (cid:1) . NTERLACED PROCESSES ON THE CIRCLE 11 The existence of such couplings is suggested by the commutation relation q r q s = q s q r , equivalently l r l s = l s l r . For any u, v ∈ A n consider the two sets τ u,v = { x ∈ A n : u (cid:22) s x (cid:22) r v } and τ ′ u,v = { y ∈ A n : u (cid:22) r y (cid:22) s v } . If either is non-empty,then they both are, and in this case they are ( n − l r l s = l s l r can be interpreted as saying these two polygons have the same( n − y = φ u,v ( x )via(17) y i = min( u i +1 , v i ) + max( u i , v i − ) − x i . It is easy to see that φ u,v is an isometry from τ u,v to τ ′ u,v using the facts that y i ∈ [max( u i , v i − ) , min( u i +1 , v i )] if and only if x i ∈ [max( u i , v i − ) , min( u i +1 , v i )] , and n X i =1 y i = n X i =1 (min( u i +1 , v i ) + max( u i , v i − ) − x i ) = n X i =1 ( u i + v i − x i ) . Proposition 3.1. Let ( X ( k ); k ≥ be an r -interlacing random walk, starting from X (0) having the distribution q s ( y, dx ) for some given y ∈ A n , where q s is definedby (14). Let the process ( Y ( k ); k ≥ be given by Y (0) = y and Y ( k + 1) = φ Y ( k ) ,X ( k +1) ( X ( k )) , for k ≥ , where φ is defined by (16). Then ( Y ( k ); k ≥ is distributed as an r -interlacingrandom walk starting from y . Proof. We prove by induction on m that the law of (cid:0) Y (1) , . . . Y ( m ) , X ( m ) (cid:1) isgiven by q r ( y, dy (1)) . . . q r ( y ( m − , dy ( m )) q s ( y ( m ) , dx ( m )). Suppose this holds forsome m . Then, since Y (1) . . . Y ( m ) are measurable with respect to X (0) , X (1) , . . . X ( m )the joint law of (cid:0) Y (1) , . . . Y ( m ) , X ( m ) , X ( m + 1) (cid:1) is given by q r ( y, dy (1)) . . . q r ( y ( m − , dy ( m )) q s ( y ( m ) , dx ( m )) q r ( x ( m ) , dx ( m + 1)) . Equivalently we may say that the law of (cid:0) Y (1) , . . . Y ( m ) , X ( m + 1) (cid:1) is q r ( y, dy (1)) . . . q r ( y ( m − , dy ( m ))( q s q r )( y ( m ) , dx ( m + 1)) and that the conditional law of X ( m ) given the same variables is uniform on τ Y ( m ) X ( m +1) . From the measure preserving properties of the maps φ u,v , it fol-lows that, conditionally on (cid:0) Y (1) , . . . Y ( m ) , X ( m + 1) (cid:1) , Y ( m + 1) is distributeduniformly on τ ′ Y ( m ) ,X ( m +1) , and the inductive hypothesis for m + 1 follows fromthis and the commutation relation q s q r = q r q s . (cid:3) The dynamics of the coupled processes (cid:0) X ( k ) , Y ( k ); k ≥ (cid:1) are illustrated in thefollowing two diagrams, in which interlacing configurations x = X ( k ) and y = Y ( k )are shown together with updated configurations x ′ = X ( k + 1) and y ′ = Y ( k + 1).It is natural to think of these as particle positions on ( a portion of ) the circle. Forsimplicity we consider an example where x ′ and x differ only in the i -th co-ordinate. y ′ i − y i − ❤❤ ❄ x ′ i − x i − ①① ❄ y ′ i y i ❤❤ ❅❅❅❘ x ′ i x i ①① ❅❅❅❘ y ′ i +1 y i +1 ❤❤ ❄ x ′ i +1 x i +1 ①① ❄ The configuration y ′ is of course determined by y , x and x ′ together. Thesimplest possiblity is shown above, in this case i th y particle advances by the sameamount as the i th x -particle. However should the i th x -particle advance beyond y i +1 then it it pushes the ( i + 1)th y particle along, whilst the increment in theposition of the i th y particle is limited to y i +1 − x i . y ′ i − y i − ❤❤ ❄ x ′ i − x i − ①① ❄ y ′ i y i ❤❤ ❅❅❅❘ x ′ i = y ′ i +1 x i ❤①① ❍❍❍❍❍❥ y i +1 ❤ ❆❆❆❯ x ′ i +1 x i +1 ①① ❄ The proof of Propostion 3.1 made use of only the measure preserving propertiesof the family of maps φ uv , and consequently we can replace it by another familyof maps having the same property and obtain a different coupling of the sameprocesses. The pushing interaction in the coupling constructed above is also seenin the dynamics induced on Gelfand-Tsetlin patterns by the RSK correpondence.There exists a variant of the RSK algorithm (with column insertion replacing themore common row insertion), in which pushing is replaced by blocking. We will NTERLACED PROCESSES ON THE CIRCLE 13 next describe a family ψ uv of measure-preserving maps that lead to a coupling withsuch a blocking interaction.We recall first a version of the standard Skorohod lemma for periodic sequences. Lemma 3.2 (Skorohod) . Suppose that ( z i ; i ∈ Z ) is n -periodic and satisfies P ni =1 z i < . Then there exists a unique pair of n -periodic sequences ( r i ; i ∈ Z ) and ( l i ; ∈ Z ) such that r i +1 = r i + z i + l i +1 for all i ∈ Z , with the additional properties r i ≥ , l i ≥ , and l i > ⇒ r i = 0 for all i ∈ Z . A configuration ( x , x . . . x n ) of n points on the circle will be implicitly ex-tended to a sequence ( x i ; i ∈ Z ) satisfying x i + n = x i + 2 π . Proposition 3.3. Suppose that u , v , and x are three configurations on the circlewith u (cid:22) s x (cid:22) r v , where s > r . Define an n -periodic sequence via z i = v i − x i − x i +1 + u i +1 for i ∈ Z , and let ( r, l ) be the associated solution to the Skorohod problem. Set y i = x i − l i ,then ( y , . . . , y n ) is configuration on the circle such that u (cid:22) r y (cid:22) s v . Proof. Notice that P ni =1 z i = r − s < P ni =1 z i = − P ni =1 l i . Consequently,using the definition of y , n X i =1 ( y i − u i ) = n X i =1 ( x i − l i − u i ) = s − n X i =1 l i = r. Similarly P ni =1 ( v i − y i ) = s .To conclude we will verify the inequalitiesmax( u i , v i − ) ≤ y i ≤ x i ≤ min( u i +1 , v i ) . It is easily checked that z − i − ≤ min( x i − u i , x i − v i − ). The solution of the Skorohodproblem satisfies 0 ≤ l i ≤ z − i − . Together with the definition of y i , this gives thedesired inequalities. (cid:3) By virtue of the preceeding result we may define ψ u,v : τ uv → τ ′ uv via ψ uv ( x ) = y . Proposition 3.4. ψ uv is a measure-preserving bijection. Proof. For x ∈ A n , extended as before to a sequence ( x i ; i ∈ Z , let x † bedefined by x † i = − x − i for i ∈ Z . Observe that if x ≺ r y , then y † ≺ r x † . For u, v, x and y as in the previous proposition we will show that the application( u, x, v ) ( v † , y † , u † )is an involution, and this implies in particular that ψ u,v is invertible with x † = ψ v † u † ( y † ). To this end first note that v † ≺ s y † ≺ r u † , and hence it is meaningful toapply ψ v † u † to y † . We must proof that the resulting conguration ˜ x say, is equal to x † . We have ˜ x i = y † i − ˜ l i , where (˜ r, ˜ l ) solves the Skorohod problem with data˜ z i = u † i − y † i − y † i +1 + v † i +1 . Since ˜ x i = y † i − ˜ l i = − x − i + l − i − ˜ l i verifying that ˜ x = x † boils down to checkingthat ˜ l i = l − i . For this it is suffices, by the uniqueness property of solutions tothe Skorohod problem, to confirm that r ′ i = r − i and l ′ i = l − i solve the Skorohodproblem with data ˜ z . Now r ′ and l ′ are non-negative and satisfy l ′ i > ⇒ r ′ i = 0because r and l have these properties. We just have to compute r ′ i + ˜ z i + l ′ i +1 = r − i + u † i − y † i − y † i +1 + v † i +1 + l − ( i +1) = r − i − u − i + y − i + y − ( i +1) − v − ( i +1) + l − ( i +1) = r − i − u − i + x − i − l − i + x − ( i +1) − l − ( i +1) − v − ( i +1) + l − ( i +1) = r − i − z − ( i +1) − l − i = r − ( i +1) = r ′ i +1 . This proves the involution property.It remains to verify the measure-preserving property. The construction of ψ uv is such that it is evident that it is a piecewise linear mapping, and that its Jacobianis almost everywhere integer valued. Since the same applies to the inverse mapconstructed from ψ v † u † we conclude the Jacobian is almost everywhere ± (cid:3) Finally the following propostion follows by exactly the same argument as Propo-sition 3.1. NTERLACED PROCESSES ON THE CIRCLE 15 Proposition 3.5. Suppose that s > r . Let ( X ( k ); k ≥ be an r -interlacingrandom walk, starting from X (0) having the distribution q s ( y, dx ) for some given y ∈ A n . Let the process ( Y ( k ); k ≥ be given by Y (0) = y and Y ( k + 1) = ψ Y ( k ) ,X ( k +1) ( X ( k )) . for k ≥ . Then ( Y ( k ); k ≥ is distributed as an r -interlacing random walk starting from y . Let us illustrate this coupling in a similar fashion to before. y ′ i y i ❤❤ ❄ x ′ i x i ①① ❅❅❅❘ y ′ i +1 y i +1 ❤❤ ❅❅❅❘ x ′ i +1 x i +1 ①① ❄ y ′ i +2 y i +2 ❤❤ ❄ x ′ i +2 x i +2 ①① ❄ In the simplest case shown above, the ( i + 1)th y particle advances by the sameamount as the i th x -particle. However in the event that this would result in itpassing the position x i , then it is blocked at that point, and the unused part of theincrement ( x ′ i − x i ) is passed to the ( i + 2)th y particle as is shown beneath. y ′ i y i ❤❤ ❄ x ′ i x i ①① ❍❍❍❍❍❥ y ′ i +1 = x ′ i +1 y i +1 ①❤❤ ◗◗◗◗s x i +1 ① ❄ y ′ i +2 y i +2 ❤❤ ❆❆❆❯ x ′ i +2 x i +2 ①① ❄ We can also consider discrete analogues of these constructions. Set A Nn = A n ∩ Z n / (2 πN ). Then, for r ∈ { / (2 πN ) , / (2 πN ) , . . . , ( N − / (2 πN ) } we maydefine a Markovian transition kernel q Nr on A Nn as follows. Define l Nr ( x, x ′ ) = z (cid:22) r x ′ , h N ( x ) = Y ≤ l 0) with these transitionprobabilities, an r -interlacing random walk on on A Nn .In the special case r = 1 / (2 πN ) the r -interlacing random walk on on A Nn isclosely related to non-coliding random walks on the circle, as considered in [ ] forinstance. Specifically if ( X ( k ); k ≥ 0) is the walk on A Nn then the process given by (cid:0) e i ¯ X ( k ) , e i ¯ X ( k ) , . . . , e i ¯ X n ( k ) (cid:1) , where ¯ X j ( k ) = NN + n ( X j ( k ) + j/ (2 πN )), gives a process of n non-coliding randomwalks on the circle { e ik/ (2 π ( N + n ) ; k = 0 , , , . . . , N + n − } .Fix r and s ∈ { / (2 πN ) , / (2 πN ) , . . . , ( N − / (2 πN ) } . For u, v ∈ A Nn we mayconsider the two sets τ Nu,v = τ u,v ∩ Z n / (2 πN ) and τ N ′ u,v = τ ′ u,v ∩ Z n / (2 πN ), where τ u,v and τ ′ u,v are the polygons defined previously. The map φ u,v carries Z n / (2 πN ) into Z n / (2 πN ), and since we know it to be an isometry between τ u,v and τ ′ u,v it musttherefore restrict to a bijection between τ Nu,v and τ N ′ u,v . In particular the cardinalitiesof these two sets are equal. Since l Ns l Nr ( u, v ) = | τ Nu,v | = | τ N ′ u,v | = l Nr l Ns ( u, v ) wededuce that l Nr and l Ns , and so q Nr and q Ns commute. This is useful in verifyingour previous assertion that h N is a eigenfunction of l Nr . The case r = 1 may beeasily deduced by direct calculation, see [ ] for a similar calculation. Moreoverthe statespace A Nn is irreducible for l Nr for r = 1 (though not in general), and thusthe Perron-Frobenius eigenfunction of l N is unique up to multiplication by scalars.Consequently it follows from the commutation relation that h N is an eigenfunctionof l Nr for every r .Using the fact that φ u,v is a bijection between τ Nu,v and τ N ′ u,v , we see that,substuting τ Nu,v for τ u,v , q Nr for q r and so on, the proof of Proposition 3.1 appliesverbatum in the case that X is an r interlacing random walk on A Nn rather than A n . Similarly ψ u,v is a bijection between τ Nu,v and τ N ′ u,v , and so Proposition 3.5 holdsalso in the discrete case. NTERLACED PROCESSES ON THE CIRCLE 17 We can extend the above constructions, by Kolmogorov consistency, to definea Markov chain ( { X ( m ) ( k ) , m ∈ Z } , k ∈ Z ) on the state space { x ∈ A Z n : x ( m ) (cid:22) r x ( m +1) , m ∈ Z } with the following properties:(1) For each m ∈ Z , ( X ( m ) ( k ) , k ∈ Z ) is an s -interlacing random walk(2) For each k ∈ Z , ( X ( m ) ( k ) , m ∈ Z ) is an r -interlacing random walk(3) For each m, k ∈ Z , X ( m ) ( k + 1) = φ X ( m +1) ( k ) ,X ( m ) ( k +1) ( X ( m ) ( k )) . In the above, we can replace φ by ψ , and also consider the discrete versions. It isinteresting to remark on the non-colliding random walk case, that is, the discretemodel on A Nn with r = s = 1 / πN . In this case, the evolution of the above Markovchain is completely deterministic. Indeed, by ergodicity of the non-colliding randomwalk, for each k ∈ Z , with probability one, there exist infinitely many m such that X ( m ) ( k ) ∈ { ((1 + c ) / πN, . . . , ( n + c ) / πN ) , c ∈ Z } and for these m there isonly one allowable transition, namely to X ( m ) j ( k + 1) = X ( m ) j ( k ) for j ≤ n − X ( m ) n ( k + 1) = X ( m ) n ( k ) + 1 / πN . By property (3) above, this determines X ( m ) ( k + 1) for all m ∈ Z . 4. Couplings of interlaced Brownian motions In this section we describe a Brownian motion construction that can consideredas a scaling limit of the coupled random walks of the previous section.The Laplacian on A n with Dirichlet boundary conditions admits a unique non-negative eigenfunction, the function h defined previously at (12) which correspondsto the greatest eigenvalue λ = − n ( n − n + 1) / 12. A h -Brownian motion on A n ,is a diffusion with transition densities Q t given by(20) Q t ( x, x ′ ) = e − λ t h ( x ′ ) h ( x ) Q t ( x, x ′ ) , where Q t are the transition densities of standard n -dimensional Brownian mo-tion killed on exiting A n , given explicitly by a continuous analogue of the Gessel-Zeilberger formula, [ 17, 21 ]. If ( X ( k ); k ≥ 0) is an h -Brownian motion in A n then (cid:0) e iX ( t ) , e iX ( t ) , . . . , e iX n ( t ) (cid:1) gives a process of n non-coliding Brownian motions on the circle, the same as arises as the eigenvalue process of Brownian motion in theunitary group, see [ ].We wish to construct a bivariate process ( X, Y ) with each of X and Y dis-tributed as h -Brownian motions in A n , and with Y ( t ) (cid:22) s X ( t ), for some fixed s ∈ (0 , π ). The dynamics we have in mind is based on the map φ of the previ-ous section. Y should be deterministically constructed from its starting point andthe trajectory of X , with Y i ( t ) tracking X i ( t ) except when this would cause theinterlacing constraint to be broken. In the following we set θ ( t ) = θ n ( t ). Proposition 4.1. Given a continuous path (cid:0) x ( t ); t ≥ (cid:1) in A n , and a point y (0) ∈ A n such that y (0) (cid:22) s x (0) , there exist unique continuous paths (cid:0) y ( t ); t ≥ (cid:1) in A n and (cid:0) θ ( t ); t ≥ (cid:1) in R n such that (i) y i ( t ) (cid:22) s x i ( t ) for all t ≥ ; (ii) y i ( t ) − y i (0) = x i ( t ) − x i (0) + θ i − ( t ) − θ i ( t ) ; (iii) for i = 1 , , . . . n , the real-valued process θ i ( t ) starts from θ i (0) = 0 , isincreasing, and the measure dθ i ( t ) is supported on the set { t : y i +1 ( t ) = x i ( t ) } . Our principal tool in proving this proposition is another variant of the Skorohodreflection lemma. For l > 0, let A n ( l ) = (cid:8) x ∈ R n : x ≤ x ≤ . . . ≤ x n ≤ x + l (cid:9) .We adapt our usual convention, letting x n +1 = x + l and x = x n − l . Lemma 4.2. Given a continuous path (cid:0) u ( t ); t ≥ (cid:1) in R n starting from u (0) = 0 ,and a point v (0) ∈ A n ( l ) then there exists a unique pair of R n -valued, continuouspaths ( v, θ ) starting from ( v (0) , with (i) v ( t ) ∈ A n ( l ) for every t ≥ , (ii) each component θ i ( t ) increasing with the measure dθ i ( t ) supported on theset { t : v i ( t ) = v i +1 ( t ) } , (iii) v i ( t ) = v i (0) + u i ( t ) − θ i ( t ) . Proof. Fix T > 0. The path u is uniformly continuous on [0 , T ], and sothere exists ǫ > P i | u i ( t ) − u i ( s ) | < l/n for all | s − t | < ǫ . We proveexistence and uniqueness over the time interval [0 , ǫ ], and then repeat the argumentover consecutive time intervals [ ǫ, ǫ ] etc. NTERLACED PROCESSES ON THE CIRCLE 19 Observe that there must exist k ∈ { , , . . . , n } such that v k +1 (0) − v k (0) ≥ l/n .Set v k ( t ) = v k (0) + u k ( t ) and θ k ( t ) = 0 for t ∈ [0 , ǫ ]. Then we the usual Skorohodlemma applied sucessively to give v k − , v k − , . . . , v , v n , . . . v k +1 , in particular welet θ i ( t ) = sup s ≤ t (cid:0) v i +1 ( t ) − u i ( t ) − v i (0) (cid:1) − . The choice of ǫ is such that the v k − so constructed satisfies v k ( t ) < v k +1 ( t ) forall t ∈ [0 , ǫ ], and consequently all the desired properties of v and θ hold. Foruniqueness, we note that any ( v, θ ) for which the desired properties holds must beequal to the one just constructed, which follows from the uniqueness for the usualSkorohod construction. (cid:3) Proof of Proposition 4.1. This is based on applying the Skorohod map-ping to the path u specified from x by u i ( t ) = x i ( t ) − x i (0).Choose v (0) ∈ A n (2 π − s ) so that v i (0) − v i − (0) = y i (0) − x i − (0), and let (cid:0) v, θ (cid:1) be given by the Skorohod mapping for the domain A n (2 π − s ) with data u and v (0). Then for t ≥ y i ( t ) = y i (0) + u i ( t ) + θ i − ( t ) − θ i ( t ) . We claim that(i) y ( t ) ∈ A n with y ( t ) (cid:22) s x ( t ) for every t ≥ dθ i ( t ) is supported on the set { t : y i +1 ( t ) = x i ( t ) } .Calculate as follows. y i ( t ) − x i − ( t ) = (cid:0) y i (0) + u i ( t ) + θ i − ( t ) − θ i ( t ) (cid:1) − (cid:0) x i − (0) + u i − ( t ) (cid:1) = (cid:0) v i (0) + u i ( t ) − θ i ( t ) (cid:1) − (cid:0) v i − (0) + u i − ( t ) − θ i − ( t ) (cid:1) = v i ( t ) − v i − ( t ) . This is valid even if i = 0 when we adhere to our conventions regarding v etc.Assertion (ii) above follows since we see that { t : y i ( t ) = x i − ( t ) } = { t : v i ( t ) = v i − ( t ) } , and this latter set carries dθ i − . Also since v i ( t ) ≥ v i − ( t ) we obtainone part of the interlacing condition, namely, y i ( t ) ≥ x i − ( t ). For the other part,consider the equality x i ( t ) − y i ( t ) = x i (0) − y i (0) + θ i ( t ) − θ i − ( t ) . Since this quantity is initially x i (0) − y i (0) > θ i − increases it follows that if there exists an instant t for which x i ( t ) − y i ( t ) < t , for which x i ( t ) − y i ( t ) < dθ i − . The latter implies that y i ( t ) = x i − ( t ), and thus x i ( t ) < y i ( t ) = x i − ( t ) which contradicts x ( t ) ∈ A n . This proves existence.Uniqueness follows from the uniqueness statement in Proposition 4.2. (cid:3) By virtue of this result we may make the following definition. For y (0) ∈ A n ,let Γ y (0) be the application which applied to an A n -valued path (cid:0) x ( t ) ≥ (cid:1) returnsthe path (cid:0) y ( t ); t ≥ (cid:1) specified by Propostion 4.1. The main result of this sectionis the following. Proposition 4.3. Let ( X ( t ); t ≥ be an h -Brownian motion in A n , starting froma point X (0) having the distribution q s ( y, dx ) for some given y = y (0) ∈ A n . Thenthe process Y = Γ y (0) ( X ) is distributed as a h -Brownian motion in A n . The domain A n is unbounded, which is a nuisance when we come to the prob-ability, where life is easier if we have a finite invariant measure on the state space.Everything we do is invariant to shifts along the diagonal of R n , and it is useful toproject onto the hyperplane H = (cid:8) x ∈ R n : P x i = 0 (cid:9) . For x ∈ R n , the orthogo-nal projection of x onto H is given by x x − ¯ x where ∈ R n is the vector withevery component equal to 1, and ¯ x = n − P i x i . If (cid:0) X ( t ); t ≥ (cid:1) is an h -Brownianmotion in A n , then its projection onto H is itself a diffusion process and indeed canbe described as the h -transform of an ( n − H ,killed on exiting A n ∩ H . Introducing more generally H s = (cid:8) x ∈ R n : P x i = s (cid:9) ,and the notion of an h -Brownian motion on A n ∩ H s , we have the following variantof Propostion 4.3. Proposition 4.4. Let ( X ( t ); t ≥ be an h -Brownian motion in A n ∩ H s/ , startingfrom a point X (0) having the distribution q s ( y, dx ) for some given y = y (0) ∈ A n ∩ H − s/ . Then the process Y = Γ y (0) ( X ) is distributed as a h -Brownian motionin A n ∩ H − s/ . Proposition 4.3 is easily deduced from this variant using the fact that if (cid:0) X ( t ); t ≥ (cid:1) is an h -Brownian motion in A n , then the projection of X onto H is independentof the process n − / P i X i ( t ), which is a one-dimensional Brownian motion. NTERLACED PROCESSES ON THE CIRCLE 21 The results in the previous section were proved using the measure-preservingproperties associated with the dynamics for a single update. Since here we are work-ing with continuous time processes such one time-step methods are not applicable.The use of the Skorohod lemma in the construction of Γ suggests that there may bea role to be played by a certain reflected Brownian motion. We describe next how,adapting the idea of Proposition 4.1 slightly, we can construct interlaced processes X and Y from a reflected Brownian motion R in the domain H ∩ A n (2 π − s ). Thenit will turn out that time reversal properties of R can be used to prove Proposition4.4. Let E ( s ) = { ( x, y ) ∈ ( A n ∩ H s/ ) × ( A n ∩ H − s/ ) : y (cid:22) s x } . We can introducenew co-ordinates on E ( s ) as follows. For ( x, y ) ∈ E ( s ) we let f ( x, y ) be the unique( r, l ) ∈ H × H such that r i +1 − r i = y i +1 − x i , (21) l i +1 − l i = x i +1 − y i +1 , (22)where r n +1 = r + (2 π − s ) and l n +1 = l + s . It is easily seen that f : E ( s ) → ( H ∩ A n (2 π − s )) × ( H ∩ A n ( s )) is bijective.Now we construct a process in E ( s ) via these alternative co-ordinates. Begin byletting (cid:0) U ( t ); t ≥ (cid:1) be a standard Brownian motion in R n starting from zero, andlet V (0) be an independent random variable, uniformly distributed in H ∩ A n (2 π − s ). Let (cid:0) V, Θ (cid:1) be determined from U and V (0) by applying the Skorohod mappingfor A n (2 π − s ) as given in Proposition 4.2. Finally let R ( t ) be the projection of V ( t )onto H , so R ( t ) = V ( t ) − ¯ V ( t ) . The process (cid:0) R ( t ); t ≥ (cid:1) is, by construction, asemimartingale reflecting Brownian motion in the polyhedron A n (2 π − s ) ∩ H . See[ ] for the general theory of such processes. Next we introduce a process L alsotaking value in H , which is constructed out a random initial value L (0) togetherwith the increasing processes Θ i for i = 1 , , . . . , n . Choose L (0) independent of R and uniformly distributed on A n ( s ) ∩ H , and let L ( t ) be given by L ( t ) = L i (0) − π Θ( t ) , where π : R n → H denotes the projection onto H defined by πx = x − ¯ x . Definethe stopping time τ = inf { t ≥ L ( t ) A n ( s ) } . Then for t ≤ τ , we define X ( t ) and Y ( t ) by (cid:0) R ( t ) , L ( t ) (cid:1) = f (cid:0) X ( t ) , Y ( t ) (cid:1) . The joint law of (cid:0) X ( t ∧ τ ) , Y ( t ∧ τ ); t ≥ (cid:1) may be described as follows.(1) ( X (0) , Y (0)) is uniformly distributed on E ( s ).(2) (cid:0) X ( t ∧ τ ); t ≥ 0) is distributed as a Brownian motion in H stopped at theinstance it first leaves A n , and conditionally independent of Y (0) given X (0).(3) Y = Γ Y (0) X .We now turn to the time reversibility of R . For any vector x ∈ R n we willdenote by x † the vector given by x † i = − x n − i +1 . Note that if x ∈ H ∩ A n (2 π − s )then x † ∈ H ∩ A n (2 π − s ) also. We also define, for any x ∈ R n , the vector x ‡ via x ‡ i = − x n − i − s/n for 1 = 1 , , . . . ( n − 1) and x ‡ n = − x n + s ( n − /n . Note that if x ∈ H ∩ A n ( s ) then x ‡ ∈ H ∩ A n ( s ) also. Proposition 4.5. Fix some constant T > . Let the processes R and L be as aboveand let Λ be the event { L ( t ) ∈ A n ( s ) for all t ∈ [0 , T ] } . Then conditionally on Λ , (cid:0) R ( t ) , L ( t ); t ∈ [0 , T ] (cid:1) law = (cid:0) R † ( T − t ) , L ‡ ( T − t ); t ∈ [0 , T ] (cid:1) . Proof. As remarked above R is a semimartingale reflected Brownian motionin the polyhedral domain A n (2 π − s ) ∩ H . Indeed it satisfies R ( t ) = R (0) + B ( t ) + 1 √ X Θ i ( t ) v i , where B is a standard Brownian motion in H and the vector v i describes thedirection of reflection associated with the face F i = { x ∈ A n (2 π − s ) ∩ H : x i = x i +1 } . Let n i be the inward facing unit normal to this face. Then an easy calculationshows that v i , which is normalized so that the inner product n i · v i = 1, is given by v i = n i + q i where the j th component of the vector q i is given by q ij = √ /n − / √ j = i, i + 1 , √ /n otherwise.We observe that the skew-symmetry condition, n i · q j + q i · n j = 0 , for all i = j , is met. Consequently by Theorem 1.2 of [ ], the reflected Brownianmotion R is in duality relative to Lebesgue measure to another reflected Brownian NTERLACED PROCESSES ON THE CIRCLE 23 motion on H ∩ A n (2 π − s ) with direction of reflection from the face F i being n i − q i .It is not difficult to check that R † is such a reflected Brownian motion. Thus (cid:0) R ( t ); t ∈ [0 , T ] (cid:1) law = (cid:0) R † ( T − t ); t ∈ [0 , T ] (cid:1) . The process 2Θ i ( t ) is the local time of R i +1 ( t ) − R i ( t ) at zero, and can be representedas lim ǫ ↓ ǫ Z t (cid:0) R i +1 ( s ) − R i ( s ) ≤ ǫ (cid:1) ds. Now note that, Z t (cid:0) R i +1 ( s ) − R i ( s ) ≤ ǫ (cid:1) ds law = Z t (cid:0) R † i +1 ( T − s ) − R † i ( T − s ) ≤ ǫ (cid:1) ds = Z TT − t (cid:0) R n − i +1 ( s ) − R n − i ( s ) ≤ ǫ (cid:1) ds From this we deduce that the time reversal property extends to (cid:0) R ( t ) , Θ( t ); t ∈ [0 , T ] (cid:1) law = (cid:0) R † ( T − t ) , Θ ♯ ( T ) − Θ ♯ ( T − t ); t ∈ [0 , T ] (cid:1) , where Θ ♯i ( t ) = Θ n − i for i = 1 , , . . . , ( n − 1) and Θ ♯n = Θ n .Let F be a bounded path functional. Let v n ( s ) denote the Lebesgue measureof H ∩ A n ( s ). Then using the time reversal property, E (cid:2) F (cid:0) R ( t ) , L ( t ); t ∈ [0 , T ] (cid:1) Λ (cid:3) =1 v n ( s ) E (cid:20)Z H dαF (cid:0) R ( t ) , α − π Θ( t ); t ∈ [0 , T ] (cid:1) { α − π Θ( t ) ∈ A n ( s ); t ∈ [0 , T ] } (cid:21) =1 v n ( s ) E "Z H dαF (cid:0) R † ( T − t ) , α − π Θ ♯ ( T ) + π Θ ♯ ( T − t ); t ∈ [0 , T ] (cid:1) × (cid:8) α − π Θ ♯ ( T ) + π Θ ♯ ( T − t ) ∈ A n ( s ); t ∈ [0 , T ] (cid:9) . Now we make the substitution ˆ α = α − π Θ ♯ ( T ) to obtain1 v n ( s ) E "Z H d ˆ αF (cid:0) R † ( T − t ) , ˆ α + π Θ ♯ ( T − t ); t ∈ [0 , T ] (cid:1) × { ˆ α + π Θ ♯ ( T − t ) ∈ A n ( s ) , t ∈ [0 , T ] } = E (cid:2) F (cid:0) R † ( T − t ) , L ‡ ( T − t ); t ∈ [0 , T ] (cid:1) Λ (cid:3) , where we have used L ‡ ( t ) = L ‡ (0) + π Θ ♯ ( t ). (cid:3) Proof of Proposition 4.4. Let R , Θ and L be as above, and once again letΛ be the event that (cid:8) L ( t ) ∈ A n ( s ) for all t ∈ [0 , T ] (cid:9) . Recall the mapping f suchthat (cid:0) R ( t ) , L ( t ) (cid:1) = f (cid:0) X ( t ) , Y ( t ) (cid:1) . It is easily verified that (cid:0) Y † ( t ) , X † ( t ) (cid:1) ∈ E ( s )and that (cid:0) R † ( t ) , L ‡ ( t ) (cid:1) = f (cid:0) Y † ( t ) , X † ( t ) (cid:1) . Thus the preceeding time reversal result implies that, conditionally on Λ, (cid:0) X ( t ) , Y ( t ); t ∈ [0 , T ] (cid:1) law = (cid:0) Y † ( T − t ) , X † ( T − t ); t ∈ [0 , T ] (cid:1) . For the final step of the argument we consider X and Y be as above and denotethe governing measure by P . Then we let˜ P = e − λ T γ r Λ h ( Y (0)) h ( X ( T )) · P . Under ˜ P , the equality in law, (cid:0) X ( t ) , Y ( t ); t ∈ [0 , T ] (cid:1) law = (cid:0) Y † ( T − t ) , X † ( T − t ); t ∈ [0 , T ] (cid:1) . holds unconditionally. Finally we note that under ˜ P , the distribution of (cid:0) X ( t ) , t ∈ [0 , T ] (cid:1) and hence of (cid:0) Y † ( T − t ) , t ∈ [0 , T ] (cid:1) is that of a stationary h -Brownian motionon A n ∩ H s/ But this latter law is invariant under time reversal, and its imageunder the conjugation x x † is the law of a stationary h -Brownian motion on A n ∩ H − s/ . This is therefore the law of (cid:0) Y ( t ) , t ∈ [0 , T ] (cid:1) . Conditioning on Y (0)gives the statement of the proposition. (cid:3) In the case n = 2, the results of this section can be expressed in terms ofBrownian motion in a compact interval. Let X = ( X t , t ≥ 0) be a Brownianmotion conditioned, in the sense of Doob, never to exit the interval [ − p, p ], where p > 0. Let y ∈ [ − p, p ], a ∈ [0 , p ] and suppose that the initial law of X is supportedin the interval [ | y + a − p |− p, p −| y + p − a | ] with density proportional to cos( πx/ p ).Let Z be the image of the path X + ( y − X + a ) / , a ]. In other words, Z t = X t + ( y − X + a ) / L t − U t , where L and U are the unique continuous, non-decreasing paths such that the pointsof increase of L occur only at times when Z t = 0, the points of increase of U occur NTERLACED PROCESSES ON THE CIRCLE 25 only at times when Z t = a , and Z t ∈ [0 , a ] for all t ≥ 0. Then the process Y t = y − X + X t + 2( L t − U t ) t ≥ − p, p ]. This is a special case of Proposition 4.4. Actually, in the statement ofthat proposition we have p = π , but this can be easily modified for general p . Itis interesting to consider this statement when y = 0 and p → ∞ . Then X is astandard Brownian motion, initially uniformly distributed on the interval [ − a, a ].The process Z is a reflected Brownian motion in [0 , a ], initially uniformly distributedon [0 , a ]. The conclusion in this case is that Y is a standard Brownian motionstarted from zero. We remark that in this setting, if instead we take X = − a ,then Y is a Brownian motion started from zero, conditioned (in an appropriatesense) to hit a before returning to zero. This is a straightforward consequence ofthe above result (for uniform initial law) and the fact (see [ ]) that, if we set T = inf { t ≥ Y t = a } , then the law of X T is uniform on [ − a, a ]. Note that if welet a → ∞ in this case we recover Pitman’s representation for the three-dimensionalBessel process. There are explicit formulae for the Skorohod reflection map for thecompact interval [0 , a ] and hence for the process Y in the above discussion. Let f ( t ) = X t + ( y − X + a ) / f ( s, t ) = f ( t ) − f ( s ). A discrete version of theSkorohod problem was considered in [ ], from which we deduce the expressions Z t = max (cid:26) sup ≤ r ≤ t min { f ( r, t ) , a + inf r ≤ s ≤ t f ( s, t ) } , min { f ( t ) , a + inf 5. A bead model on the cylinder In this section it will be convenient to work with a slightly weaker notion ofinterlacing, defined as follows. For a, b ∈ D n , write a ≺ b if a ≤ b < a ≤ · · · < a n ≤ b n , and a ≻ b if b < a ≤ b < · · · ≤ b n < a n . For y = { e ia , . . . , e ia n } and z = { e ib , . . . , e ib n } , where a, b ∈ D n , write y ≺ z ifeither a ≺ b or a ≻ b , and define l ( y, z ) = P j ( b j − a j ) if P j ( b j − a j ) ≥ P j ( b j − a j ) + 2 π otherwise.Consider the Markov kernels defined, for q > 0, by(23) m q ( y, dz ) = c − q Z π | − e ir | n − q r p r ( y, dz ) dr, where p r is defined by (9) and c q = Z π | − e ir | n − q r dr. By Proposition 2.1, if we define I q ( y, z ) = q l ( y,z ) if y ≺ z ,0 otherwise,then(24) m q ( y, dz ) = ˜ c − q ∆( z )∆( y ) I q ( y, z ) dz, where ˜ c q = c q / ( n − µ ( dx ) = (2 π ) − n ∆( x ) dx is the probabilitymeasure on C n induced from Haar measure on U ( n ). The Markov chain withtransition density m q has µ as an invariant measure and, with respect to µ , hastime-reversed transition probabilities m q ( z, dy ) = ˜ c − q ∆( y )∆( z ) I q ( y, z ) dy. We can thus construct a two-sided stationary version of this Markov chain to obtaina probability measure α on C Z n , supported on configurations · · · ≺ x − ≺ x ≺ x ≺ x ≺ · · · . We will show that α defines a determinantal point process on NTERLACED PROCESSES ON THE CIRCLE 27 [0 , π ) Z . By stationarity it suffices to consider the restrictions α m to the cylindersets C n,m := C { , ,...,m } n . Writing ¯ x = ( x , . . . , x m ) and d ¯ x = dx · · · dx m ,(25) α m ( d ¯ x ) = µ ( dx ) m q ( x , dx ) · · · m q ( x m − , dx m ) . Assume for the moment that q = 1. Define a function f : R → C by f ( u ) = ( qe i ( n − / ) u mod 2 π − ( − n − q π . Lemma 5.1. For y = { e ia , . . . , e ia n } and z = { e ib , . . . , e ib n } , where a, b ∈ D n , I q ( y, z ) = (1 − ( − n − q π ) e i n − P j ( a j − b j ) det ( f ( b k − a j )) ≤ j,k ≤ n . Proof. Let c = 1 and consider the n × n matrix W = ( w jk ) defined by w jk = a j ≤ b k c a j > b k . If a ≺ b , W consists of 1’s on and above the diagonal and c ’s below, so thatdet W = (1 − c ) n − . If a ≻ b , W consists of 1’s above the diagonal and c ’s on andbelow the diagonal, so that det W = c ( c − n − . If neither a ≺ b or a ≻ b , thenthere must exist an index j such that, either a j = a j +1 , or a j < a j +1 ≤ b k for all k , or b k < a j < a j +1 ≤ b k +1 for some k . In each of these cases, rows j and j + 1 of W are identical and hence det W = 0. Thus,(26) det W = (1 − c ) n − a ≺ bc ( c − n − a ≻ b c = ( qe i ( n − / ) π = ( − n − q π , we can write(1 − c ) e i n − P j ( a j − b j ) det ( f ( b j − a k )) ≤ j,k ≤ n = q P j ( b j − a j ) (1 − c ) − ( n − det W = q P j ( b j − a j ) a ≺ bq P j ( b j − a j )+2 π a ≻ b I q ( y, z ) , as required. (cid:3) For r = 1 , . . . , m − 1, define φ r,r +1 : [0 , π ) → C by φ r,r +1 ( a, b ) = f ( b − a ).Define φ , : R × [0 , π ) → C and φ m,m +1 : [0 , π ) × R → C by φ , ( a, b ) = e iab and φ m,m +1 ( a, b ) = e − iab . For r = 1 , . . . , m , write x r = { e ia r , . . . , e ia rn } , where a r ∈ D n , and set a j = a m +1 j = j − 1, for j = 1 , . . . , n . Theorem 5.2. For q = 1 , α m ( d ¯ x ) = Z − m m Y r =0 det (cid:0) φ r,r +1 ( a rj , a r +1 k ) (cid:1) ≤ j,k ≤ n d ¯ x, where Z m = ˜ c m − q (1 − ( − n − q π ) − ( m − (2 π ) n . Proof. By (24) we can write α m ( d ¯ x ) = ˜ c − ( m − q (2 π ) − n ∆( x )∆( x m ) I q ( x , x ) · · · I q ( x m − , x m ) d ¯ x. Using the formula∆( x )∆( x m ) = det (cid:16) e i ( j − ( n +1) / a k (cid:17) ≤ j,k ≤ n det (cid:16) e − i ( j − ( n +1) / a mk (cid:17) ≤ j,k ≤ n , and Lemma 5.1, we obtain α m ( d ¯ x ) = Z − m e i n − P j ( a j − a mj ) det (cid:16) e i ( j − ( n +1) / a k (cid:17) ≤ j,k ≤ n × det (cid:16) e − i ( j − ( n +1) / a m +1 k (cid:17) ≤ j,k ≤ n m − Y r =1 det (cid:0) φ r,r +1 ( a rj , a r +1 k ) (cid:1) ≤ j,k ≤ n d ¯ x = Z − m det (cid:16) e i ( j − a k (cid:17) ≤ j,k ≤ n det (cid:16) e − i ( j − a m +1 k (cid:17) ≤ j,k ≤ n × m − Y r =1 det (cid:0) φ r,r +1 ( a rj , a r +1 k ) (cid:1) ≤ j,k ≤ n d ¯ x = Z − m m Y r =0 det (cid:0) φ r,r +1 ( a rj , a r +1 k ) (cid:1) ≤ j,k ≤ n d ¯ x, as required. (cid:3) Corollary 5.3. For any q > , the measure α defines a determinantal point processon [0 , π ) Z with space-time correlation kernel given by K ( r, a ; s, b ) = π P n − k =0 g r − sk e i ( b − a ) k r ≥ s − π P k ∈ Z \{ ,...,n − } g r − sk e i ( b − a ) k r < s where g k = (cid:18)Z π f ( u ) e − iuk du (cid:19) − = i (cid:18) k − n − (cid:19) − log q. NTERLACED PROCESSES ON THE CIRCLE 29 Proof. For q = 1, this follows from Theorem 5.2, [ , Proposition 2.13] anda straightforward computation. The case q = 1 is obtained by continuity. (cid:3) Lemma 5.1 can be used to give a direct proof of (24), and hence Proposition 2.1. Proof of Proposition 2.1. The characters χ λ are given, for y = { e ia , . . . , e ia n } ,by χ λ ( y ) = i − ( n )∆( y ) − det (cid:0) e iµ j a k (cid:1) ≤ j,k ≤ n , where µ = λ + ρ and ρ = (cid:18) n − , n − − , . . . , − n − 12 + 1 , − n − (cid:19) . Using Lemma 5.1 and the Cauchy-Binet formula, we obtain Z ∆( z )∆( y ) I q ( y, z ) χ λ ( z ) dz = (1 − ( − n − q π ) Y j ( − iµ j − log q ) − χ λ ( y ) . On the other hand, writing x r = { e ir , , , . . . , } , an easy calculation shows that Z π | − e ir | q r χ λ ( x r ) d λ dr = ( n − − ( − n − q π ) Y j ( − iµ j − log q ) − and so, by (7), m q χ λ = ˜ c − q (1 − ( − n − q π ) Y j ( − iµ j − log q ) − χ λ . Since { χ λ , λ ∈ Ω n } is a basis for L ( C n , µ ), this implies (24). (cid:3) Analogous results to those presented in this section can be obtained for thediscrete version of this model, which is equivalent to considering a certain familyof Gibbs measures on rhombic tilings of the cylinder. For more details, see [ ].The couplings defined in Section 3 are quite useful in this setting, where the group-theoretic considerations of Section 2 no longer apply. For example, they can beused to prove that the discrete analogues of the interlacing operators {I q , q > } commute with each other. In this setting, the symmetric functions( q , . . . , q k ) 7→ I q · · · I q k ( y, z )are essentially the cylindrical skew Schur functions discussed in the papers [ 27, 36 ].Finally, we remark that, in the case q = 1, the probability measure α definedby (25) also arises naturally in random matrix theory. The probability measures on C n given by µ ( dx ) = (2 π ) − n ∆( x ) dx and A − n ∆( x ) dx , where A n is a normalisation constant, are known, respectively, as the circular unitary ensemble and circularorthogonal ensemble . It is a classical result, which was conjectured by Dyson [ ]and subsequently proved by Gunson [ ], that the set of alternate eigenvalues from asuperposition of two independent draws from the circular orthogonal ensemble, aredistributed according to the circular unitary ensemble. 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E-mail address : [email protected] Department of Statistics, University of Warwick, Coventry CV4 7AL, UK. E-mail address ::