Multidimensional Pitman’s theorem and non-colliding processes

In this section we discuss the extension of the Pitman’s theorem, both its classical and discrete versions, to higher dimensions. In notation of Chapter 3 this is theθ=∞and q = 0 cases respectively. We start by describing the discrete setting, from which the multidimensional extension of the Pitman’s theorem is obtained by taking the scaling limit.

Let X = (X(k);k ≥ 1) be an n-dimensional Markov chain taking values in

Nn = {0, 1, ...}n. Denote by o the origin of Rn and by b = {e1, ...,en} the canonical

orthonormal basis ofCn. SupposeX(0) =oand the transition matrix is given by

P(x,x+e_i) = 1

n, 1≤i≤n. (5.20)

We callX themultinomial random walk.

Define thediscrete Weyl chamberas the subset of_Nn

Wn={(x1, ...,xn)∈Nn:x1≥x2≥ · · · ≥xn} and a functionh:Wn→R+by h(x) =Y i<j ˜ x_i−˜x_j j−i , (5.21)

where ˜x_i = x_i−i. We note that by definitionh(x) =|Kx|, whereKx is the collection

of all Gelfand-Cetlin patterns with top rowx, as defined in Chapter 2 (see Thm. 2.9). Recall the branching rule for representations ofU(gl(n))(orU_q(gl(n))) (see Ch. 2.5). The dimensions of spaces on either side of the expression (2.16) are the same.

Using this and the fact thatd im(V_x) =|Kx|we obtain the identity

n_|Kx|= X

x:x%x0

|Kx0|, (5.22)

where x_%x0means thatx is interlaced with x0, i.e. x x0, and_|x_|+1=_|x0_|. Define

b

P(x,y) =P(x,y)1_{y∈W

n} forx− y∈b. Using (5.22) we have

n

X

i=1

b

P(x,x+ei)h(x+ei) =h(x), x∈ Wn,

which shows thath(x)is harmonic with respect toPrestricted toWn. Thus

Q(n)(x,x+e_i):=bP(x,x+e_i) h(x+ei) h(x) = 1 n |Kx+ei| |Kx| 1_{x+e i∈Wn}, x∈ Wn (5.23)

is a well defined transition matrix. The Markov processXb defined by these transition

probabilities and started atois a system of nsymmetric random walks conditioned to stay in the Weyl chamber_Wn, i.e. to maintain their order. We will refer to this process asDyson’s random walks.

In[59]O’Connell gives a construction of Dyson’s random walks by applying a certain transformation to the original processX. The construction produces a process in the Gelfand-Cetlin cone, such that the top row is evolving as the n-dimensional Dyson’s random walk. This transformation is closely related to the Robinson-Schensted algorithm and also has an interpretation in queueing theory. We discuss it next.

5.2.1

A representation for conditioned random walks

In this section we describe certain transformations for random walks introduced by O’Connell and Yor[60]. Continuous version of these transformations can be seen as a generalisation of the Pitman’s transform to the Weyl chambers and also have intimate connection to the eigenvalues of a GUE matrix.

define (x4y)(m) = min 0≤k≤m[x(k) +y(m)−y(k)], (x5y)(m) = max 0≤k≤m[x(k) +y(m)−y(k)]. (5.24)

Forn≥2 define the mappingsG(n):D0(N)n→D0(N)nas follows. Forn=2 let

G(2)(x,y) = (x4y,y5x),

and forn≥2 define inductively

G(n)(x1, ...,xn) = (x14 · · · 4xn,G(n−1)(x25x1,x35(x14x2), ...,xn5(x14 · · · 4xi−1))).

One can show that for eachn_≥2 andx_∈Nn,G1(n)(x)≤G

(n)

2 (x)≤ · · · ≤G(n)n (x)

(see O’Connell[59, Sec. 2]) and that, in particular,(G(n)(x1, ...,xn)∗, ...,G(1)(x1)∗)is a Gelfand-Cetlin pattern of depthn. Here we writex∗, forx _∈Rn, to indicatexi =x∗n−i+1, 1_≤i≤n. It follows that ifX is a random walk in_Nn _{defined at the beginning of this}

section, then the transformed process (G(i)(X1(k), ...,Xi(k))∗, 1 ≤ i ≤ n;k ≥ 0) is a

random walk in the Gelfand-Cetlin cone_Kn. Moreover,

Theorem 5.6. ([59, Cor. 6.2]) The transformed process G(n)(X) is distributed as the random walk X conditioned to stay in the Weyl chamberWn. In particular, the transition

matrix of G(n)(X)is given by(5.23).

Note that in case n = 2, Theorem 5.6 is equivalent to the discrete Pitman’s theorem. Indeed, if we letz= y−x andm∈N, then

G₂(2)(x,y)(m)−G₁(2)(x,y)(m) = (y5x)(m)−(x4y)(m) =2 max

1≤k≤mz(k)−z(m).

But, since the two components of G(2)(X,Y) are conditioned to stay ordered, G₂(2)(X,Y)−G₁(2)(X,Y)is distributed as a random walk conditioned to stay non-negative, i.e. as a discrete BES3 process. Finally note thatZ =Y₋X is just a simple symmetric random walk.

Just as in the classical Pitman’s theorem, intertwining plays an important role; transition functions of the transformed and the original processes are intertwined with respect to a certain Markov kernel (see[59, Cor. 6.5]).

The connection to the Robinson-Schensted algorithm is as follows. Consider a word a∈[n]k and let x_i(m) =|{1≤ j ≤m:a_j =i}|, for 1≤ i≤ nand 1≤m≤ k, i.e. xi(m)is the number of aj’s, up to and including am, equal to i. Denote byτ(m)

the semi-standard Young tableau obtained by applying the Robinson-Schensted algo- rithm with column insertion to (a1, ...,am). Then for each 1 ≤ m ≤ k, the GC pat-

tern((G(n)(x1, ...,xn)∗)(m), ...,(G(1)(x1)∗)(m))corresponds to the semi-standard Young tableauτ(m)(see[59, Thm. 3.1]).

Now consider a sequence of random variables(η_k,k≥0)such thatη0=0 and for eachi_≥1η_i is distributed uniformly over_{1, ...,n_}. Define

Xi(m) =|{1≤ j≤m:ηj=i}|, 1≤i≤n,m≥0 .

Then X = (X1, ...,Xn) is the familiar multinomial random walk. Moreover, at

any time k_≥1, X(k) is the type and G(n)(X(k))∗ is the shape of a randomly growing Young tableau constructed from a random word(η₁, ...,η_k)via column insertion. For two alternative dynamics related to the Robinson-Schensted-Knuth algorithm see[58]. In order to pass to the continuous version of the Theorem 5.6 and, thus, to a result about Hermitian Brownian motion, we define a Poissonized version of the Robinson-Schensted dynamics described above. Transformations (5.24), and so map- pings G(i), admit continuous versions. Denote by D0(R+) a space of càdlàg paths

f :R+→Rwith f(0) =0 and let

(f4g)(t) = inf 0≤s≤t[f(s) +g(t)−g(s)], (f 5g)(t) = sup 0≤s≤t [f(s) +g(t)−g(s)], forf,g∈D0(R+). Define Γ(2)_{(f,g) = (f} 4g,g5f)

and mappingsΓ(n):D0(R+)n→D0(R+), forn≥2, by

Γ(n)(f₁, ...,f_n) = (f_{14 · · · 4}f_n,Γ(k−1)(f₂₅f₁,f₃₅(f₁₄f₂), ...,f_k5(f_{14 · · · 4}f_n−1))).

LetN = (N₁, ...,N_n)withN(0) =obe the counting processes ofnindependent Poisson processes on _R+, each with intensity 1. Let Nb be the h-transform of N with

respect toh(x) given by (5.21). Then the law ofΓ(n)(N) is the same as the law ofNb

([59, Thm. 7.1],[60, Thm. 5]). As in the discrete case, generators of the two processes are intertwined with respect to a certain Markov kernel (see[59, Thm. 7.2]).

Both the discrete conditioned random walk and its Poissonised version are closely connected to discrete orthogonal polynomial ensembles and so to determinantal pro- cesses. In particular, the distribution of the Poisson random walk is connected to the Charlier ensemble, while the distribution of the multinomial walk – to thede-Poissonised Charlier ensemble, see [54] and also [44]. It is this determinantal structure of the measures involved that lies at the heart of the relation of both processes to the GUE matrices.

By applying an appropriate version of the Dönsker’s theorem one obtains a ver- sion of Theorem 5.6 for the Brownian motion. Let B be a standard n-dimensional Brownian motion started at the origin andBˆanh-transform ofBwithh(x)given by the Vandermonde determinant. Then

Theorem 5.7. (O’Connell, Yor[60, Thm. 7]) The processesB andˆ Γ(n)(B)have the same law.

Again whenn=2, this result corresponds to the Pitman’s construction of the 3- dimensional Bessel process. Since by Theorem 5.7Γ(2)(B(t))is a 2-dimensional Dyson’s BM, one verifies with the help of Itô’s lemma thatR(t):= (Γ(₂2)(B(t))−Γ(12)(B(t)))/

p

2 has the distribution of the BES3 process. At the same time X(t) := (Γ₂(2)(B(t)) +

Γ(2)

1 (B(t)))/

p

2 is a standard Brownian motion independent of R. But by definition ofΓ(1) R(t) =p1 2(Γ (2) 2 (B(t))−Γ (2) 1 (B(t))) =2 sup s≤t X(s)−X(t).

Brownian motions conditioned to stay in the Weyl chamber, we see that Theorem 5.7 yields an alternative representation for them. In particular, by showing thatΓ(_nn)(B) =

(B15· · ·5Bn), one finds an alternative proof of the result of Baryshnikov[6]and Tracy,

Gravner and Widom[37]which states that the random variable

λ1:= Γ(n)n (B(1)) = sup 0≤t1≤...≤tn=1 n X k=1 {Bk(tk)−Bk(tk−1)}

has the same distribution as the largest eigenvalue of a GUE matrix. What’s more, we see that O’Connell and Yor’s results generalise this formula to give a description of all the eigenvalues of a GUE matrix.

The same transformations were also described by Bougerol and Jeulin[14]in a purely representation theoretic context. Authors describe a transformation taking paths in a finite-dimensional Euclidian spaceato paths in the interior of the corresponding Weyl chamber a₊. a is taken to be the maximal Torus of the maximal compact sub- algebra of some algebra g. In case of a Hermitian Brownian motion g = gl_C(n) and

a' Rn is the space of all n×nreal diagonal matrices. By takingg to be other clas-

sical groups Bougerol and Jeulin find representation for eigenvalues of other types of random matrices. See also a paper by Biane, Bougerol and O’Connell[12].