A REPRESENTATION FOR NON-COLLIDING RANDOM WALKS

Let D 0 ( R + ) denote the space of cadlag paths f : R + ! R with f (0) = 0 . For f; g 2 D 0 ( R + ) , de(cid:12)ne f ⊗ g 2 D 0 ( R + ) and f (cid:12) g 2 D 0 ( R + ) by ; and Unless otherwise deleniated by parentheses, the default order of operations is from left to right; for example, when we write f ⊗ g ⊗ h , we mean ( f ⊗ g ) ⊗ h . De(cid:12)ne a sequence of mappings Γ k : D 0 ( R + ) k ! D 0 ( R + ) k by and, for k > 2 ,

both of these cases the conditioning is in the sense of Doob). This extends a recent observation, independently due to Baryshnikov (2001) and Gravner, Tracy and Widom (2001), that if B is a standard Brownian motion in R n , then (B 1 ⊗ · · · ⊗ B n )(1) has the same law as the smallest eigenvalue of a n × n GUE random matrix.

Introduction and Summary
Let B = (B 1 , . . . , B n ) be a standard n-dimensional Brownian motion and set The process R n was introduced in [14]. It has recently been observed [3,16] that:

Theorem 1
The random variable R n (1) has the same law as the smallest eigenvalue of a n × n GUE random matrix.
A n × n GUE random matrix A ∈ C n×n is constructed as follows: it is Hermitian, that is, A = A * (= (Ā) t ); the entries {A ij , i ≤ j} are independent; on the diagonal A ii are standard real normal random variables; below the diagonal, {A ij , i < j} are standard complex normal random variables, that is, the real and imaginary parts of A ij are independent centered real normal random variables, each with variance 1/2; above the diagonal we set A ji =Ā ij . Here, z = x − iy denotes the complex conjugate of z = x + iy. Hermitian Brownian motion is constructed in the same way as a GUE random matrix, but with Brownian motions instead of normal random variables. It is well-known (see, for example, [11,15,26]) that the eigenvalues of Hermitian Brownian motion evolve like independent Brownian motions started from the origin and conditioned (in the sense of Doob) never to collide. To make this more precise, the function is harmonic on R n , and moreover, is a strictly positive harmonic function for Brownian motion killed when it exits the Weyl chamber the conditioned process we refer to is the corresponding Doob h-transform, started at the entrance point (0, 0, . . . , 0). For related work on non-colliding diffusions and random matrices, see [4,9,18,22], and references therein. Thus, ifB is a realisation of this conditioned process, then the smallest eigenvalue of a n × n GUE random matrix has the same law asB 1 (1), and Theorem 1 states that R n (1) andB 1 (1) have the same law. Similar connections between directed percolation random variables, such as R n (1), and random matrix or discrete orthogonal polynomial ensembles have also been observed in [19,20]. See also [1,13]. These are all related to the amazing fact, recently discovered and proved by Baik, Deift and Johansson [2], that the asymptotic distribution of the longest increasing subsequence in a random permutation is the same as the asymptotic distribution of the largest eigenvalue in a GUE random matrix, which had earlier been identified by Tracy and Widom [33]. Before stating our main result we will introduce some notation. Let D 0 (R + ) denote the space of cadlag paths f : and Unless otherwise delineated by parentheses, the default order of operations is from left to right; for example, when we write f ⊗ g ⊗ h, we mean (f ⊗ g) ⊗ h. Define a mapping Γ : We now define a sequence of mappings Γ k : D 0 (R + ) k → D 0 (R + ) k recursively, as follows. Set Let ) be the counting functions of n independent Poisson processes on R + with respective intensities µ 1 < µ 2 < · · · < µ n . That is, N (µ k ) k (t) is the measure induced by the k th Poisson process on the interval (0, t], with the convention that N is the same as the unconditional law of Γ n (N (µ) ).
The proof of Theorem 2, presented in the next section, is based on some natural independence and reversibility properties of M/M/1 queues in series. At the heart of the proof is a generalisation of the celebrated theorem, due to Burke, which states that (in equilibrium) the output of a stable M/M/1 queue is Poisson. In Section 3, we recover the analogue of Theorem 2 for independent Poisson processes of equal rates. (This is interesting in its own right: in [23] the conditioned process is shown to be closely connected with the Charlier ensemble.) In Section 4, by carefully applying Dönsker's theorem, we deduce that the n-dimensional process Γ n (B) has the same law asB. To see that Theorem 1 follows, note that there is equality between the one-dimensional processes: where Γ n (B) 1 denotes the first component of the n-dimensional process Γ n (B). In the case n = 2, the fact that Γ n (B) has the same law asB is essentially equivalent to Pitman's representation [29,30] for the three-dimensional Bessel process; this connection is discussed in [28]. Note that, for n = 2 in the Poisson case, Theorem 2 (see also Theorem 5) yields the following discrete analogue of Pitman's theorem if X t is a simple random walk with non-negative drift (in continuous or discrete time) and M t = max 0≤s≤t X s , then 2M − X has the same law as that of X conditioned to stay positive (in the case of a symmetric random walk, this conditioning is in the sense of Doob). This result was obtained in [29] for the symmetric random walk; Pitman's original proof for Brownian motion used Dönsker's theorem and this simple random walk result. Finally, we mention that Bougerol and Jeulin [5] have recently found a proof of Theorem 1 by considering Brownian motion on symmetric spaces and applying a kind of Laplace method.

Proof of Theorem 2
We will first state and prove a generalisation of Burke's theorem [8], which states that the output of a stationary M/M/1 queue is Poisson. As was observed by Reich [31], there is an elementary proof of Burke's theorem using reversibility. We will state a slightly stronger result and prove it using essentially the same reversibility argument. The stationary M/M/1 queue can be constructed as follows. Let A and S be independent Poisson processes on R with respective intensities 0 < λ < µ. For intervals I, open, half-open or closed, we will denote by A(I) the measure of I with respect to dA; for I = (0, t] we will simply write A(t), with the convention that A(0) = 0. Similarly for S and any other point process we introduce. For t ∈ R, set and for s < t, In the language of queueing theory, A is the arrivals process, S is the service process, Q is the queue-length process, and D is the departure process. With this construction it is also natural (and indeed very important for what follows) to define the unused service process by We will use the following notation for reversed processes. For a point process X, the reversed processX is defined byX(s, t) = X(−t, −s). The reversed queue-length processQ is defined to be the right-continuous modification of {Q(−t), t ∈ R}.
Burke's theorem states that D is a homogeneous Poisson process with intensity λ. On a historical note, this fact was anticipated by O'Brien [27] and Morse [24], and proved in 1956 by Burke [8]. In 1957, Reich [31] gave the following very elegant proof which uses reversibility. The process Q is reversible (in fact, all birth and death processes are reversible). It follows that the joint law of A and D is the same as the joint law ofD andĀ. In particular,D, and hence D, is a Poisson process with intensity λ. Burke also proved that, for each t, {D(s, t], s ≤ t} is independent of Q(t). This property is now called quasi-reversibility. Note that it also follows from Reich's reversibility argument. Discussions on Burke's theorem and related material can be found in the books of Brémaud [6,7], Kelly [21] and Robert [32]. Some Remarks. The analogue of Theorem 3 holds for Brownian motions with drift-several proofs of this fact are given in [28]. (In fact, there is also a version given there which holds for exponential functionals of Brownian motion. ) It is closely related to Pitman's representation for the three-dimensional Bessel process [29] and Williams' path-decomposition [35]. See also [17,25], for related work. Note that We also have, on {Q(0) = 0}, Theorem 3 has the following multi-dimensional extension, which relates to a sequence of M/M/1 queues in tandem. Let A, S 1 , . . . , S n be independent Poisson processes with respective intensities λ, µ 1 , . . . , µ n , and assume that λ < min i≤n µ i . Set D 0 = A and, for k ≥ 1, t ∈ R, set and for s < t,
Proof. By Theorem 3, D 1 , T 1 and S 2 are independent Poisson processes with respective intensities λ, µ 1 and µ 2 . Applying Theorem 3 again we see that D 2 and T 2 are independent Poisson processes with respective intensities λ and µ 2 , and since D 2 and T 2 are determined by D 1 and S 2 they are independent of T 1 . Thus D 2 , T 1 , T 2 and S 3 are independent Poisson processes with respective intensities λ, µ 1 , µ 2 and µ 3 . And so on. The condition λ < min i≤n µ i ensures that this procedure is well-defined. 2 Remark. Again, the analogue of Theorem 4 can be shown to hold for Brownian motions with drifts, by exactly the same argument. By repeated iteration of (16) and (17), we obtain (almost surely) To see this, first recall thatĀ(s) = A(−s, 0), and This yields (19) for n = 1. For n = 2, almost surely, And so on. In particular, Q 1 (0) + · · · + Q n (0) depends only on the restriction of A, S 1 , . . . , S n to (−∞, 0]. Iterating (17) we obtain, for each k ≤ n, We also have, by (14), Applying this repeatedly (as in the derivation of (19) above) we obtain Note that, on {Q 1 (0) + · · · + Q n (0) = 0}, and for t ≥ 0, k ≤ n. We will prove Theorem 2 by induction on n.

The case of equal rates
Let N = (N 1 , . . . , N n ) be a collection of independent unit-rate Poisson processes, with N (0) = (0, . . . , 0). The function h given by (2) is a strictly positive harmonic function for the restriction of the transition kernel of N to the discrete Weyl chamber E = W ∩ Z n (this follows from a more general result presented in [23]). LetN be a realisation of the corresponding Doob h-transform of N , started at x * = (0, 1, . . . , n − 1) ∈ E. Apart from providing a convenient framework in which we can apply Dönsker's theorem and deduce the Brownian analogue of Theorem 2-this will be presented in the next section-the processN is interesting in its own right. In [23] it is shown (see the identity (40) below) that the random vectorN (1) is distributed according to the Charlier ensemble, a discrete orthogonal polynomial ensemble. Thus, the next result, which follows from Theorem 2, yields a representation for the Charlier ensemble. For more on discrete orthogonal polynomial ensembles, see [20].

Theorem 5
The processesN − x * and Γ n (N ) have the same law.
Proof. Let D(R + ) denote the space of cadlag paths f : R + → R, equipped with the Skorohod topology. Let D(R + ) n be equipped with the corresponding product topology, and M 1 (D(R + ) n ), the space of probability measures on D(R + ) n , with the corresponding weak topology. In this section, all weak convergence statements for processes will be with respect to this topology. Note that we can restate Theorem 2 as follows. Let x * = (0, 1, . . . , n − 1). Theorem 2 states that the conditional law of x * + N (µ) , given that x * + N (µ) (t) ∈ E, for all t ≥ 0, is the same as the unconditional law of x * + Γ n (N (µ) ). It is easy to see that the operations ⊗ and are continuous (with respect to the Skorohod topology); it follows that Γ n is continuous. The statement of Theorem 5 therefore follows from Lemma 6 below. It follows that the conditional law of x * + N (µ(m) ) , given that x * + N (µ(m) ) (t) ∈ E, for all t ≥ 0, converges to the Doob transform of N by the strictly positive harmonic function g on E, started at x * ; let N g be a realisation of this Doob transform. Now we apply Theorem 2, from which it follows that Γ n (N ) has the same law as N g . In particular, the limiting function g must be the same for any choice of sequence µ(m). It remains to show that g = h/h(x * ). The Martin boundary associated with Π is analysed in [23]. If k(x, y) is the Martin kernel associated with Π, and y → ∞ in such a way that y/ i y i → (1/n, . . . , 1/n), then k(x, y) → h(x)/h(x * ). Thus, by standard Doob-Hunt theory (see, for example, [10,34]) if we can show that, with probability one, N g (t)/t → (1, . . . , 1) as t → ∞, we are done. Since Γ n (N ) has the same law as N g , we need only check that

The corresponding result for Brownian motion
In this section we recover the analogous result for Brownian motion. For x ∈ R n , let P x denote the law of B started at x, and, for x ∈ W , letP x denote the law of the h-transform of B started at x, where h is given by (2). The lawsP x and P x are related as follows. If T denotes the first exit time of B from W , and F t the natural filtration of B, then for A ∈ F t , The point (0, . . . , 0) is an entrance point forP; we denote the corresponding law byP 0+ . The lawP 0+ is defined, for A ∈ T t = σ(B u , u ≥ t), t > 0, bŷ where θ is the shift operator (so that θ t A ∈ T 0 ) and is a normalisation constant. To see that this makes sense, we recall the following well-known connection betweenP and the GUE ensemble, as remarked upon in the introduction: (See, for example, [22].) LetB be a realisation ofP 0+ .

Theorem 7
The processesB and Γ n (B) have the same law.
Proof. We will use Dönsker's theorem. It is convenient to switch topologies: we now equip D(R + ) with the topology of uniform convergence on compacts, D(R + ) n with the corresponding product topology, and M 1 (D(R + ) n ) with the corresponding weak topology. In this section, all weak convergence statements for processes will be with respect to this topology. Note that the mapping Γ n is still continuous in this setting.
In the the context of the previous section, for m ∈ Z, set X m (t) = [N (mt) − mt]/ √ m and X m (t) = [N (mt) − mt]/ √ m. The theorem will be proved if we can show that X m converges in law toB. It is convenient to introduce general initial positions for the Markov processes X m andX m . Denote by P x . In this notation, all we need to show is that By an appropriate version of Dönsker's theorem (see, for example, [12,Section 7.5]), if x m → x in W , then P (m) xm → P x . Using the easy fact that P x * / √ m h(X m (t)) 2 is uniformly bounded in m, we deduce that for x m → x in W , we havê To deduce (38), we use the formula (see [23]): where C t is the same normalisation constant as in the Brownian case, given by (36). Since m n(n−1)/2 C mt = C t , this translates as: for y ∈ E/ √ m, Thus, we need to show that, for any bounded continuous function φ : D(R + ) n → R, To do this, we simply take an almost-sure realisation of the convergence P (m) x * / √ m → P 0 , appeal to (39), and use the easy fact that P x * / √ m h(X m (t)) 4 is uniformly bounded in m. 2