getdocf983. 246KB Jun 04 2011 12:05:13 AM

J
i
on
Electr

c

o

u

a
rn l

o

f
P

r

ob
abil
ity

Vol. 7 (2002) Paper No. 1, pages 1–24.
Journal URL
http://www.math.washington.edu/~ejpecp/
Paper URL
http://www.math.washington.edu/~ejpecp/EjpVol7/paper5.abs.html
NON-COLLIDING RANDOM WALKS, TANDEM QUEUES,
AND DISCRETE ORTHOGONAL POLYNOMIAL ENSEMBLES
Wolfgang K¨
onig1 and Neil O’Connell
BRIMS, Hewlett-Packard Laboratories, Filton Road
Stoke Gifford, Bristol BS34 8QZ, United Kingdom
koenig@math.tu-berlin.de and noc@hplb.hpl.hp.com
S´
ebastien Roch
´
Ecole Polytechnique, 91128 Palaiseau Cedex, France

sebastien.roch@polytechnique.org
Q
Abstract We show that the function h(x) = i n)

h(y)
,
h(x)

x, y ∈ W.

(2.12)

This construction is easily extended to continuous-time random walks X = (X(t))t∈[0,∞) . Consider the canonical embedded discrete-time random walk and assume that its step distribution
5

satisfies the assumptions of Theorem 2.4. Then the definition of the Doob h-transform of
the process is possible in the analogous way. In particular, (2.12) holds in this case for any
n ∈ [0, ∞) as well, where the definition (2.10) of the leaving time T has to be adapted to
T = inf{t ∈ [0, ∞) : X(t) ∈
/ W }.

3

Discrete Ensembles: Examples

In this section, we present two examples of well-known discrete ensembles which can be viewed
as the distribution at a fixed time of a suitable random walk conditioned to stay in W . Our
examples are:
(i) the binomial random walk (leading to the Krawtchouk ensemble),
(ii) the Poisson random walk (leading to the Charlier ensemble) and its de-Poissonised version,
Throughout this section we shall use the notation of Section 2.

3.1

The Krawtchouk process.

The main aim of this subsection is to show that the Krawtchouk ensemble (on W ),
KrW
k,n,p (x)

k
Y
1
2
Bin,p (xi ),
= h(x)
Z
i=1

x = (x1 , . . . , xk ) ∈ W ∩ {0, . . . , n}k ,

(3.13)

appears as the distribution at fixed time of independent binomial walks conditioned to never
collide.
Fix a parameter p ∈ (0, 1) and let the components of X be independent random walks on
N 0 whose steps take the value one with probability p and
zero else. Hence, at each time n,
n l
n−l
each component has the binomial distribution Bin,p (l) = l p (1 − p) , l ∈ {0, . . . , n}, with

parameters n and p, if the process starts at the origin. Clearly, this step distribution satisfies
the assumptions of Theorem 2.4.

We conceive KrW
k,n,p as a probability measure on W . We can identify the distribution of the
conditioned binomial walk as follows. Abbreviate x∗ = (0, 1, . . . , k − 1) ∈ W .
Proposition 3.1 For any n ∈ N 0 and any y ∈ W ,
bx∗ (Xn = y) = KrW
P
k,n+k−1,p(y).
Proof. First observe that, for any y ∈ W ,

Pbx∗ (Xn = y) > 0

⇐⇒

∀i = 1, . . . , k : i − 1 ≤ yi ≤ n + i − 1

⇐⇒

KrW
k,n+k−1,p (y) > 0,

⇐⇒

0 ≤ y1 < y2 < · · · < yk ≤ n + k − 1

6

(3.14)

i.e., the two distributions on both sides of (3.14) have the same support. Assume that y lies in
that support.
Use the Karlin-McGregor theorem [KM59] to rewrite, for any x, y ∈ W ,


Px (Xn = y, T > n) = det (Bin,p (yj − xi ))i,j=1,...,k .

(3.15)

Here we trivially extend Bin,p to a probability measure on Z.

Observe that


 
bi,j
n
n+k−1
,
=
Qk−1
yj − i + 1
yj
l=1 (n + l)

where bi,j =

i−2
Y

l=0

(yj − l) ×

k−1
Y
l=i

(n − yj + l), (3.16)

and further observe that bi,j is zero if yj −i+1 is not in {0, . . . , n}. Hence, in view of (2.12),using
the multilinearity of the determinant, we obtain that
k
Y


∗
det (Bin,p (yj − xi ))i,j=1,...,k = K det(B)
Bin+k−1,p (yi ),

(3.17)

i=1

where B = (bi,j )i,j=1,...,k with bi,j as in (3.16). Here K denotes a positive constant which depends
on n, k and p only. Hence, the proof of the lemma is finished as soon as we have shown that
e
e which does not depend on y. Observe that bi,j is a
det(B) = Kh(y)
for some positive constant K
polynomial of degree k−1 in yj with coefficients c1,i , . . . , ck,i that do not depend on j or y1 , . . . , yk ,
P
i.e., bi,j = km=1 yjm−1 cm,i . Hence, B is equal to the matrix product B = (yjm−1 )j,m=1,...,k C
with
= (cm,i
 )m,i=1,...,k . According to the determinant multiplication theorem, det(B) =
 Cm−1
det (yj )j,m × det(C) = h(y) det(C). It is clear that det(C) 6= 0 since both sides of (3.14)
are strictly positive.
Substituting det(B) = h(y) det(C) in (3.17) and this in (3.15), and noting that the supports of

the two probability measures in (3.14) are identical, we arrive at the assertion.
With some more efforts, one also can show that det(C) = (n+1)k−1 (n+2)k−2 ×· · ·×(n+k−1)1 .



Introduce the Green’s function for the walk before the first time of a collision, Γ : W × W →
[0, ∞), by
X
x, y ∈ W.
(3.18)
Px (X(n) = y, T > n),
Γ(x, y) =
n∈N0
and the corresponding Martin kernel K : W × W → [0, ∞) by
K(x, y) =

Γ(x, y)
,
Γ(x∗ , y)

x, y ∈ W,

(3.19)

where we recall that x∗ = (0, 1, . . . , k − 1) ∈ W . For future reference, we state a result on the
asymptotic behaviour of the Martin kernel. By 1l we denote the vector (1, . . . , 1) ∈ ∂W .
Lemma 3.2 For any x ∈ W ∩ N k0 ,
K(x, y) →

h(x)
h(x∗ )

as y → ∞ through W such that y/|y| → 1l/k.
7

(3.20)

Proof. All following limit assertions refer to the limit as y → ∞ through W such that y/|y| →
1l/k. In order to prove the lemma, it is sufficient to prove the following. There is a constant
C > 0 (depending on k and p only) such that, for any x ∈ W ,
y

Γ(x, y) = Ch |y|
h(x)(1 + o(1)).
(3.21)
We again use Karlin-McGregor’s formula in (3.15). We may assume that yj − xi ≥ 0 for
any i and P
j. We split
P on
n ∈ N 0 in the definition of Γ(x, y) into the three sums
P the sum
+
+
Γ(x, y) =
n∈IIIy Px (X(n) = y, T > n) where Iy , IIy and IIIy , respectively,
n∈IIy
n∈Iy
are the subsets of N 0 in the three regions left of, between and right of |y|/(kp) ± |y|3/4 .
For n ∈ IIy ∪ IIIy , we use that n ≫ yj for any j to get that


n
y j − xi



 
 Y
xi
yj xi
yj − l + 1
n
n
(1 + o(1)),
=
=
yj
yj
n − yj + l
n − yj

(3.22)

l=1

uniformly in n. Hence, uniformly in n ∈ IIy ∪ IIIy , we have, using the multilinearity of the
determinant,
k
i
h y xi 

  p −|x| Y
j
det (Bin,p (yj − xi ))i,j=1,...,k =
(1 + o(1)).
Bin,p (yj ) det
1−p
n − yj
i,j=1,...,k
j=1

(3.23)
In order to evaluate the latter determinant, we introduce the Schur function Schurx (z) which
is a certain polynomial in z1 , . . . , zk whose coefficients are non-negative integers and may be
defined combinatorially. It is homogeneous of degree |x| − k2 (k − 1) and satisfies Schurx (1l) =
h(x)/h(x∗ ). Furthermore, it satisfies the generalized Vandermonde identity det[(zjxi )i,j=1,...,k ] =
h(z) Schurx (z). Applying this to the vector z = (yj /(n − yj ))j=1,...,k , we obtain from the last
display that
k
Y
  p −|x|

Bin,p (yj ).
h(z) Schurx (z)
det (Bin,p (yj − xi ))i,j=1,...,k ∼
1−p

(3.24)

j=1

We turn now to the second region of summation, i.e., |y|/(kp) − |y|3/4 ≤ n ≤ |y|/(kp) + |y|3/4 .
p
p
Observe that z converges to 1−p
1l, uniformly in all n ∈ IIy . Hence, Schurx (z) → Schurx ( 1−p
1l) =
k
p
|x|− 2 (k−1)
h(x)/h(x∗ ). Furthermore, by using the definition of h in (1.2) and recalling that
1−p
zj = yj /(n − yj ), one obtains that
− k (k−1)

h(y)
h(y)kyk∞2
 = (1 + o(1)) Q
h(z) = Q  1
= (const + o(1))h
2
i n) over n in Iy and IIIy are negligible
with respect to the sum over n ∈ IIy . In order to do that, we have to further divide Iy and
+˙
−
+
˙ −
IIIy into Iy = I+
y and IIIy = IIIy ∪III
y ∪I
y where Iy = {n ∈ Iy : n ≥ (1 + ε)yj , ∀j} and
+
1
IIIy = {n ∈ IIIy : n ≥ ε yj , ∀j}, for some small ε > 0. In order to handle the sum over
the extreme sets, use Stirling’s formula to estimate, for sufficiently small ε > 0 (the smallness
depends on p only):
+
Bin,p (yj − xi ) ≤ e−const |y| ,
n ∈ I−
y ∪ IIIy , for some j,


P
from which it follows that n∈I−
+ det (Bin,p (yj − xi ))i,j=1,...,k decays even exponentially in
y ∪IIIy
P
k
|y|. Since h(y)|y|− 2 (k−1) decays only polynomially, we have seen that n∈I−
+ Px (X(n) =
y ∪IIIy
y, T > n) is negligible with respect to the sum on n ∈ IIy .
−
We turn to the summation on n ∈ I+
y ∪ IIIy . Here (3.22) holds as well, and therefore also
k

(3.24) does. Use the first equation in (3.25) to estimate h(z) ≤ h(y)|y|− 2 (k−1) const (the latter
constant depends on ε only). Furthermore, note that y 7→ z is a bounded function, and since the
coefficients of the Schur function are positive, we may estimate Schurx (z) ≤ const Schurx (1l) ≤
const h(x). In order to handle the√sum of the binomial probability term, use Stirling’s formula
−
to deduce that Bin,p (yj ) ≤ e−const |y| uniformly in n ∈ I+
y ∪ IIIy . This in turn yields that
X

lim

y→∞
y/|y|→1l/k

k
Y

Bin,p (yj ) = 0.

− j=1
n∈I+
y ∪IIIy

−
This shows that the sum on n in I+
y ∪ IIIy is asymptotically negligible with respect to the right
hand side of (3.21). This ends the proof.


3.2

The Charlier process.

In this subsection, we show how the Charlier ensemble (on W ),
k

ChW
k,α (x) =

Y αxi
1
h(x)2
,
Z
xi !

x = (x1 , . . . , xk ) ∈ W ∩ N k0 ,

i=1

(3.26)

arises from the non-colliding version of the Poisson random walk.
We consider a continuous-time random walk X = (X(t))t∈[0,∞) on N k0 that makes steps after
independent exponential times of parameter k, and the steps are uniformly distributed on the
set of the unit vectors e1 , . . . , ek where ei (j) = δij . In other words, the components of X are
the counting functionsP
of independent unit-rate Poisson processes. The generator of this walk
is given as Gf (x) = k1 ki=1 [f (x + ei ) − f (x)]. By Corollary 2.2, h defined in (1.2) is harmonic
for this walk. Since the step distribution satisfies the conditions of Theorem 2.4, h is a strictly
positive harmonic function for the walk killed when it exits W . In other words, the embedded
9

discrete-time random walk satisfies the condition of Theorem 2.4. Hence, we may consider the
conditioned process given that the walker never leaves the Weyl chamber W , which is the Doob
h-transform of the free walk defined by
bx (X(t) = y) =
P

h(y)
Px (X(t) = y, T > t),
h(x)

x, y ∈ W, t > 0,

(3.27)

where T = inf{t > 0 : X(t) ∈
/ W }.

The distribution of the free Poisson walk at fixed time t is given by Px (X(t) = y) = P0 (X(t) =
y − x) and
k
Y
tyi e−t
(kt)|y| −kt
e
Mu|y| (y) =
,
P0 (X(t) = y) =
|y|!
yi !
j=1

y ∈ N k0 , t ∈ [0, ∞),

(3.28)



n
where Mun (y) = k−n y1 ,...,y
1l{|y| = n} denotes the multinomial distribution on N k0 , and | · |
k
denotes the lattice norm.
We conceive the Charlier ensemble defined in (3.26) as a probability measure on W . Let us now
show that the distribution of the conditioned walk at fixed time is a Charlier ensemble, if the
walker starts from the point x∗ = (0, 1, 2, . . . , k − 1).
Proposition 3.3 For any y ∈ W and any t > 0,
bx∗ (X(t) = y) = ChW (y).
P
k,t

(3.29)

Proof. We use Karlin-MacGregor’s [KM59] formula, which in this case reads:

Px (X(t) = y, T > t) = e−kt

X

Sk

sign(σ)

k 

Y
tyσ(j) −xj
1l{yσ(j) ≥ xj } ,
(yσ(j) − xj )!

(3.30)

j=1

σ∈

S

where k denotes the set of all permutations of 1, . . . , k, and sign(σ) denotes the signum of a
permutation σ, and yσ = (yσ(1) . . . , yσ(k) ). We may summarize this as
h
i


Px (X(t) = y, T > t) = t|y|−|x|e−kt det (yj − xi )!−1 1l{yj ≥ xi } i,j=1,...,k
= e−kt t−|x|

k
i −1
hxY

i
Y
t yj
det
(yj − l)
.
yj !
i,j=1,...,k

j=1

(3.31)

l=0

Applying (3.31) for x∗ = (0, 1, 2, . . . , k − 1), we note that the matrix on the right hand side may
be written as a product of the Vandermonde matrix (yjm−1 )j,m=1,...,k with some lower triangle
matrix C = (cm,i )m,i=1,...,k with diagonal coefficients ci,i = 1. Hence, the latter determinant is

equal to h(y). Now recall (3.27) to conclude the proof.
Remark 3.4 The imbedded discrete-time random walk of the Poisson walk leads to what is
known as the de-Poissonised Charlier ensemble. If X = (X(n))n∈N0 is a discrete-time random
walk on N k0 whose step distribution is the uniform distribution on the unit vectors 1 , . . . , k , then

e

10

e

its distribution at fixed time is given by Px (X(n) = y) = Mun (y − x). We call X the multinomial
walk. The step distribution satisfies the conditions of Theorem 2.4. The h-transform of the
multinomial walk satisfies

Pbx∗ (X(n) = y) =

1
h(y)2 Mun−k(k−1)/2 (y),
Z

y ∈ N k0 , n ≥

k
(k − 1),
2

(3.32)

where Z = 2−k(k−1)/2 (n + 1) . . . (n + k(k − 1)/2) denotes the normalizing constant.
Let us go back to the h-transform of the continuous-time Poisson random walk. We now identify
h with a particular point on the Martin boundary of the restriction of X to the Weyl chamber
W . Analogously
R ∞ to (3.18) and (3.19), we define the Green kernel associated with X on W
by Γ(x, y) = 0 dt Px (X(t) = y, T > t) and the corresponding Martin kernel by K(x, y) =
Γ(x, y)/Γ(x∗ , y). Recall from the proof of Lemma 3.2 (see below (3.23)) the Schur function
Schurx (v) for x ∈ W and v ∈ Rk . The following lemma implies that x 7→ Schurx (kv) is a
strictly positive regular function for PW , for any v ∈ (0, ∞)k ∩ W with |v| = 1; in particular, the
function h is not the only function that satisfies Theorem 2.4. We have thus identified infinitely
many ways to condition k independent Poisson processes never to collide. However, in [OY01b]
it is shown that, in some sense, h is the most natural choice, since the h-transform appears as
limit of conditioned processes with drifts tending to each other. Similar remarks apply to the
Krawtchouk case; in the proof of Theorem 4.6 we see that the h-transform has the analogous
interpretation. In this case we also expect infinitely many positive harmonic functions on the
Weyl chamber which vanish on the boundary. We remark that in the Brownian case h is the
only positive harmonic function on W .
Lemma 3.5 Fix x ∈ W ∩ N k0 and v ∈ (0, ∞)k ∩ W with |v| = 1. Then, in the above context,
K(x, y) → Schurx (kv)

as y → ∞ through W such that y/|y| → v.

(3.33)

as y → ∞ through W such that y/|y| → 1l/k.

(3.34)

In particular,
K(x, y) →

h(x)
h(x∗ )

Proof. We use (3.31) to see that, for any y ∈ W with |y| ≥ |x|,
i −1

i
hxY
(|y| − |x|)!
Γ(x, y) = k|x|−|y|−1 Qk
(yj − l)
.
det
i,j=1,...,k
j=1 yj !
l=0

(3.35)

Recall from the proof of Proposition 3.3 that the latter determinant is equal to h(y) for x = x∗ .
Also note that xi ≥ x∗i for any i, and hence |x − x∗ | = |x| − |x∗ |. Hence we obtain from (3.35)
that, as the limit in (3.33) is taken,
i −1
hxY

i
|y − x|! 1
det
(y
−
l)
j
|y − x∗ |! h(y)
i,j=1,...,k
l=0
h
i

1


∗
∗
= k|x−x | |y|−|x−x | + o(1)
det yjxi (1 + o(1)) i,j=1,...,k .
h(y)

K(x, y) = k|x−x

∗|

11

(3.36)





Recall that det yjxi i,j=1,...,k = h(y) Schurx (y). Use the continuity of the determinant and of
the Schur function and the fact that Schurx (·) is homogeneous of degree |x − x∗ | to deduce that

(3.33) holds.

4

A representation for the Krawtchouk process

In [OY01b], a representation is obtained for the Charlier process by considering a sequence of
M/M/1 queues in tandem. In this section we will present the analogue of this representation
theorem for the Krawtchouk process, by considering a sequence of discrete-time M/M/1 queues
in tandem. We will use essentially the same arguments as those given in [OY01b], but need
to take care of the added complication that, in the discrete-time model, the individual walks
can jump simultaneously. The Brownian analogue is also presented in [OY01b], in an attempt
to understand the recent observation, due to Baryshnikov [Ba01] and Gravner/Tracy/Widom
[GTW01], that the random variable
M=

inf

0=t0