An interacting particle model and a Pieri-type formula for the orthogonal group

0.21 MB·English

by  Manon Defosseux

AN INTERACTING PARTICLE MODEL
AND
A PIERI-TYPE FORMULA FOR THE ORTHOGONAL GROUP
2
1
0 MANONDEFOSSEUX
2

n Abstract. Weintroduceanewinteractingparticlemodelwithblockingand
a pushing interactions. Particles evolve on Z+ jumping on their own voli-
J
tion rightwards or leftwards according to geometric jumps with parameter

8 q ∈(0,1). We show that the model involves a Pieri-type formula for the or-
2 thogonal group. We prove that the two extreme cases - q = 0 and q = 1 -
leadrespectivelytoarandomtilingmodelstudiedin[1]andarandommatrix

] modelconsideredin[4].
R
P
.
h
t 1. introduction
a
m
In [1] A. Borodin and J. Kuan consider a random tiling model with a wall
[ which is related to the Plancherel measure for the orthogonal group and thus to

representation theory of this group. Similar connection holds for the interacting
2
v particle model and the random matrix model considered in [4]. The aim of this
7 paperis to establisha directlink betweenthe randomtiling model onone side and
1 the interacting particle model or the random matrix model on the other side. For
1
this we consider aninteracting particle model depending ona parameterand show
0
that these models correspond to different parameter values.
.
2 The paper is organized as follows. Definition of the set of Gelfand-Tsetlin pat-
1
terns for the orthogonal group is recalled in section 2. Section 3 is devoted to the
0
descriptionofthe particle model. We recallin section4 the descriptionofaninter-
1
: acting particle model equivalent to the random tiling model studied in [1]. Models
v
considered in that paper involve Markov kernels which can be obtained with the
i
X help of a Pieri-type formula for the orthogonal group. These Markov kernels are
r constructedin section 5 after recallingsome elements of representationtheory. We
a
describe the matrix model related to the particle model in section 6. Results are
stated in section 7 and proved in section 8.
Acknowlegments: The authorwouldlike tothank AlexeiBorodinforhis sugges-
tions and helpful explanations.
2. Gelfand-Tsetlin patterns
Let n be a positive integer. For x,y Rn such that x x and y
n 1 n
∈ ≤ ··· ≤ ≤
y , we write x y if x and y are interlaced, i.e.
1
···≤ (cid:22)
x y x x y .
n n n−1 1 1
≤ ≤ ≤···≤ ≤
When x Rn and y Rn+1 we add the relation y x . We denote by x the
n+1 n
vector of∈Rn whose c∈omponents are the absolute values≤of those of x. | |
Definition 2.1. Let k be a positive integer.
1
2 MANONDEFOSSEUX
(1) We denote by GT the set of Gelfand-Tsetlin patterns defined by
k
GT = (x1, ,xk):xi Nj−1 Z when i=2j 1,
k
{ ··· ∈ × −
xi Nj when i=2j, and xi−1 xi ,1 i k .
∈ | |(cid:22)| | ≤ ≤ }
(2) If x = (x1,...,xk) is a Gelfand-Tsetlin pattern, xi is called the ith row of
x for i 1,...,k .
(3) Forλ ∈Z{[k+21] the}subsetofGelfand-Tsetlin patternshavingakth rowequal
∈
to λ is denoted by GT (λ) and its cardinality is denoted by s (λ).
k k
Usually, a Gelfand Tsetlin pattern is represented by a triangular array as indi-
cated at figure 1 for k=2r.
x1
1
x2 x2
− 1 1
x3 x3 x3
− 1 2 1
x4 x4 x4 x4
− 1 − 2 2 1
··· ···
x2r−1 x2r−1 x2r−1 x2r−1 x2r−1
− 1 ··· − r−1 r r−1 ··· 1
x2r x2r x2r x2r
− 1 ··· − r r ··· 1
Figure 1. A Gelfand–Tsetlin pattern of GT
2r
3. An interacting particle model with exponential jumps
In this section we construct a random process (X(t)) evolving on the subset
t≥0
of GT of Gelfand-Tsetlin patterns with non negative valued components. This
k
process can be viewed as an interacting particle model. For this, we associate
to a Gelfand-Tsetlin pattern x = (x1,...,xk), a configuration of particles on the
integer lattice Z2 putting one particle labeled by (i,j) at point (k i,xi) of Z2
− j
for i 1,...,k , j 1,...,[i+1] . Several particles can be located at the same
∈ { } ∈ { 2 }
point. In the sequel we will say ”particle xi” instead of saying ”particle labeled by
j
(i,j)locatedatpoint(xi,k i)”. Letq (0,1). Considertwoindependentfamilies
j − ∈
1
(ξji(n+ 2))i=1,...,k,j=1,...,[i+21];n≥0, and (ξji(n))i=1,...k,j=1,...,[i+21];n≥1,
of identically distributed independent random variables such that
1
P(ξ1( )=x)=P(ξ1(1)=x)=qx(1 q), x N,
1 2 1 − ∈
and the Markov kernel R on N defined by
1−q(q|x−y|+qx+y) if y N∗
1+q ∈
R(x,y)=
 1−qqx otherwise,
1+q
for x N. Actually the probability measure R(x,.) on N is the law of the random
∈
variable x+ξ1(1) ξ1(1).
| 1 − 1 2 |
Particles evolveas follows. At time 0 all particles are at zero,i.e. X(0)=0. All
particles,exceptthoselabeledby(2l 1,l)forl 1,...,[k+1] (i.e. particlesnear
− ∈{ 2 }
AN INTERACTING PARTICLE MODEL AND A PIERI-TYPE FORMULA FOR SO(d) 3
the wall),try to jump to the left attimes n+1 andto the rightattimes n, n N.
2 ∈
Particleslabeled by (2l 1,l), for l 1,...,[k+1] , jump on their own volition at
− ∈{ 2 }
integer times only. Notice that these particles can eventually move at half-integer
times if they are pushed by another particle. Suppose that at time n there is one
particle at point (k i,Xi(n)) of Z2, for i = 1,...,k, j = 1,...,[i+1]. Positions
− j 2
of particles are updated downward as follows. Figure 2 gives an example of an
evolution of a pattern between times n and n+1. In that example, the particles
(3,2) and (4,2) are respectively pushed by the particles (2,1) and (3,1) and the
particles (3,1) and (4,2) are respectively blocked by the particles (2,1) and (3,2)
between times n and n+ 1. The particle (3,1) is pushed by the particle (2,1) and
2
the particles (3,2) and (4,2) are respectively blocked by the particles (2,1) and
(3,1) between times n+ 1 and n+1.
2
At time n+1/2: All particles exceptparticles X2l−1(n) for l 1,...,[k+1] , try
l ∈{ 2 }
tojumptotheleftoneafteranotherinthelexicographicorderpushingtheparticles
in order to stay in the set of Gelfand-Tsetlin patterns and being blocked by the
initial configurationX(n) of the particles. Let us indicate how the first three rows
of a pattern are updated at time n+ 1.
2
Particle X1(n) doesn’t move. We let
• 1
1
X1(n+ )=X1(n).
1 2 1
Particle X2(n) tries to jump to the left according to a geometric jump. It
• 1
is blocked by X1(n). If it is necessary it pushes X3(n) to an intermediate
1 2
position denoted by X˜3(n), i.e.
2
1 1
X2(n+ )=max X1(n),X2(n) ξ2(n+ ) ,
1 2 1 1 − 1 2
(cid:0) (cid:1)
1
X˜3(n)=min X3(n),X2(n+ ) .
2 2 1 2
(cid:0) (cid:1)
ParticleX3(n)triestomovetotheleftaccordingtoageometricjumpbeing
• 1
blocked by X2(n) :
1
1 1
X3(n+ )=max X2(n),X3(n) ξ3(n+ ) .
1 2 1 1 − 1 2
(cid:0) (cid:1)
Particle X˜3(n) doesn’t move. We let
2
1
X3(n+ )=X˜3(n).
2 2 2
Suppose now that rows 1 through l 1 have been updated for some l > 1. Then
−
the particles Xl(n),...,Xl (n) of row l are pushed to intermediate positions
1 [l+1]
2
1 l+1
X˜l(n)=min Xl(n),Xl−1(n+ ) , i 1,...,[ ] ,
i i i−1 2 ∈{ 2 }
(cid:0) (cid:1)
whit the convention Xl−1(n+ 1)=+ . Then particles X˜l(n),...,X˜l (n) try to
0 2 ∞ 1 [2l]
jumptotheleftaccordingtogeometricjumpbeingblockedasfollowsbytheinitial
position X(n) of the particles. For i=1,...,[l],
2
1 1
Xl(n+ )=max Xl−1(n),X˜l(n) ξl(n+ ) .
i 2 i i − i 2
(cid:0) (cid:1)
4 MANONDEFOSSEUX
When l is odd, particle X˜l (n) doesn’t move and we let
l+1
2
1
Xl (n+ )=X˜l (n).
l+21 2 l+21
At time n+1: AllparticlesexceptparticlesX2l−1(n+1)forl 1,...,[k+1] ,try
l 2 ∈{ 2 }
to jump to the right one after another in the lexicographic order pushing particles
in order to stay in the set of Gelfand-Tsetlin patterns and being blocked by the
initial configuration X(n + 1) of the particles. Particles X2l−1(n + 1), for l
2 l 2 ∈
1,...,[k+1] ,trytomoveontheirownvolitionaccordingto thelawR(X2l−1(n+
{ 2 } l
1),.). The first three rows are updated as follows.
2
Particle X1(n+ 1) moves according to the law R(X1(n + 1),.) pushing
• 1 2 1 2
X2(n+ 1) to an intermediate position X˜2(n+ 1) :
1 2 1 2
1 1
X1(n+1)= X1(n+ )+ξ1(n+1) ξ1(n+ ) ,
1 1 2 1 − 1 2
(cid:12) (cid:12)
X˜2(n+ 1)=m(cid:12) ax X2(n+ 1),X1(n+1) . (cid:12)
1 2 1 2 1
(cid:0) (cid:1)
ParticleX˜2(n+1)jumpstotherightaccordingtoageometricjumppushing
• 1 2
X3(n+ 1) to an intermediate position X˜3(n+ 1), i.e.
1 2 1 2
1
X2(n+1)=X˜2(n+ )+ξ2(n+1),
1 1 2 1
1 1
X˜3(n+ )=max X3(n+ ),X2(n+1) .
1 2 1 2 1
(cid:0) (cid:1)
Particle X3(n+ 1) tries to move according to the law R(X1(n+ 1),.). It
• 2 2 1 2
is blocked by X2(n+ 1). Particle X˜3(n+ 1) moves to the right according
1 2 1 2
to a geometric jump. That is
1 1 1
X3(n+1)=max( X3(n+ )+ξ3(n+1) ξ3(n+ ) ,X2(n+ )),
2 2 2 2 − 2 2 1 2
(cid:12) (cid:12)
X3(n+1)=X˜3(n(cid:12)+ 1)+ξ3(n+1). (cid:12)
1 1 2 1
Suppose rows 1 through l 1 have been updated for some l >1. Then particles of
−
row l are pushed to intermediate positions
1 1 l+1
X˜l(n+ )=max Xl−1(n+1),Xl(n+ ) , i 1,...,[ ] ,
i 2 i i 2 ∈{ 2 }
(cid:0) (cid:1)
with the convention Xl−1(n + 1) = 0 when l is odd. Then particles X˜l(n +
l+1 1
2
1),...,X˜l (n+ 1) try to jump to the right according to geometric jump being
2 [l] 2
2
blocked by the initial position of the particles as follows. For i=1,...,[l],
2
1 1
Xl(n+1)=min Xl−1(n+ ),X˜l(n+ )+ξl(n+1) .
i i−1 2 i 2 i
(cid:0) (cid:1)
When l is odd, particle Xl (n+ 1) is updated as follows.
l+1 2
2
1 1 1
Xl (n+1)=min( Xl (n+ )+ξl (n+1) ξl (n+ ) ,Xl−1(n+ )).
l+21 (cid:12) l+21 2 l+21 − l+21 2 (cid:12) l−21 2
(cid:12) (cid:12)
AN INTERACTING PARTICLE MODEL AND A PIERI-TYPE FORMULA FOR SO(d) 5
4. An interacting particle model with exponential waiting times
In this section we describe an interacting particle model on Z2 where particles
try to jump by one rightwards or leftwards after exponentially distributed waiting
times. Theevolutionoftheparticlesisdescribedbyarandomprocess(Y(t)) on
t≥0
thesubsetofGT ofGelfand-Tsetlinpatternswithnonnegativevaluedcomponents.
k
Asinthepreviousmodel,attimet 0thereisoneparticlelabeledby(i,j)atpoint
≥
(k i,Yi(t)) of the integer lattice, for i=1,...,k, j =1,...,[i+1]. Every particle
− j 2
tries to jump to the left or to the right by one after independent exponentially
distributed waiting time with mean 1. Particles are pushed and blocked according
to the same rules as previously. That is when particle labeled by (i,j) wants to
jump to the right at time t 0 then
≥
(1) ifi,j 2andYi(t−)=Yi−1(t−)thentheparticlesdon’tmoveandY(t)=
≥ j j−1
Y(t−),
(2) else particles (i,j),(i+1,j),...,(i+l,j)jump to the rightby one for l the
largest integer such that Yi+l(t−)=Yi(t−) i.e.
j j
Yi(t)=Yi(t−)+1,...,Yi+l(t)=Yi+l(t−)+1.
j j j j
When particle labeled by (i,j) wants to jump to the left at time t 0 then
≥
(1) if i is odd, j = (i+1)/2 and Yi(t−) = 0 then particle labeled by (i,j) is
j
reflectedby0andeverythinghappensasdescribedabovewhenthisparticle
try to jump to the right by one,
(2) if i is odd, j =(i+1)/2 and Yi(t−) 1 then Yi(t)=Yi(t−) 1,
j ≥ j j −
(3) if i is even or j = (i+1)/2, and Yi(t−) = Yi−1(t−) then particles don’t
6 j j
move,
(4) ifiisevenorj =(i+1)/2,andYi(t−)>Yi−1(t−)thenparticles(i,j),(i+
6 j j
1,j +1),...,(i+l,j +l) jump to the left by one for l the largest integer
such that Yi+l(t−)=Yi(t−). Thus
j+l j
Yi(t)=Yi(t−) 1,...,Yi+l(t)=Yi+l(t−) 1.
j j − j+l j+l −
This random particle model is equivalent to a random tiling model with a wall,
as it is explained in detail in [1].
5. Markov Kernel on the set of irreducible representations of the
orthogonal group
When a finite dimensionnal representation V of a group G is completely re-
ducible, there is a natural way that we’ll recall later in our particular case to
associateto this decompositiona probability measureonthe set ofirreducible rep-
resentationsofG. Thetransitionprobabilitiesoftherandomprocess(Xk(t),t 0)
≥
which will be proved to be Markovian are obtained in that manner. Actually
we recover them considering decomposition into irreducible components of tensor
products of particular irreducible representations of the special orthogonalgroup.
Let d be an integer greater than 2. Let us recall some usual properties of the
finite dimensional representationsof the compact group SO(d) of d d orthogonal
×
matrices with determinant equal to 1 (see for instance [5] for more details). The
set of finite dimensional representations of SO(d) is indexed by the set
λ Rr :2λ N,λ λ N,i=1,...,r 1 ,
r i i+1
{ ∈ ∈ − ∈ − }
6 MANONDEFOSSEUX
when d=2r+1 and by the set
λ Rr :λ +λ N, λ λ N, i=1,...,r 1 ,
r−1 r i i+1
{ ∈ ∈ − ∈ − }
when d = 2r. Actually we are only interested with representations indexed by a
subset of these sets defined by
d
W
= λ Rr :λ N,λ λ N, i=1,...,r 1 ,
d r i i+1
W { ∈ ∈ − ∈ − }
when d=2r+1 and
= λ Rr :λ Z,λ +λ N, λ λ N, i=1,...,r 1 ,
d r r−1 r i i+1
W { ∈ ∈ ∈ − ∈ − }
when d=2r. For λ , using standardnotations,we denote by V the so called
d λ
∈W
irreducible representation with highest weight λ of SO(d). The subset of of
d
elements having non-negative components is denoted by +. W
Wd
Let m be an integer and λ an element of . Consider the irreducible repre-
d
W
sentations V and V of SO(d), with γ = (m,0, ,0). The decomposition of
λ γm m ···
the tensor product V V into irreducible components is given by a Pieri-type
λ⊗ γm
formulafortheorthogonalgroup. Asithasbeenexplainedin[3],itcanbededuced
from [6]. We have
(1) V V = M (β)V ,
λ⊗ γm ⊕β λ,γm β
where the direct sum is over all β such that
d
∈W
whend=2r+1,thereexists anintegers 0,1 andc Nr whichsatisfy
• ∈{ } ∈
c λ, c β
(cid:22) (cid:22)

 r (λ c +β c )+s=m,
i=1 i− i i− i
s being equal to 0Pif c = 0. In addition, the multiplicity M (β) of the
r λ,γm
irreducible representation with highest weight β is the number of (c,s)
Nr 0,1 satisfying these relations. ∈
whe×n{d=}2r, there exists c Nr−1 which verifies
• ∈
c λ, c β
(cid:22)| | (cid:22)| |

 r−1(λ c +β c )+ λ µ =m.
k=1 k− k k− k | r− r|
Inaddition,tPhemultiplicityM (β)oftheirreduciblerepresentationwith
λ,γm
highest weight β is the number of c Nr−1 satisfying these relations.
∈
Let us consider a family (µ ) of Markov kernels on defined by
m m≥0 d
W
dim(V )
β
µ (λ,β)= M (β),
m dim(V )dim(V ) λ,γm
λ γm
forλ,β andm 0. Itis knownthat for λ the dimensionofV is given
d d λ
∈W ≥ ∈W
by the number s (λ) defined in definition 2.1. Thus
d−1
s (β)
d−1
µ (λ,β)= M (β).
m s (λ)s (γ ) λ,γm
d−1 d−1 m
Let ξ ,...,ξ be independent geometric random variables with parameter q and ǫ
1 d
a Bernoulli random variable such that
q
P(ǫ=1)=1 P(ǫ=0)= .
− 1+q
Consider a random variable T on N defined by
AN INTERACTING PARTICLE MODEL AND A PIERI-TYPE FORMULA FOR SO(d) 7
d−1
T = ξ +ǫ,
i
Xi=1
when d=2r+1 and
d
T = ξ ξ + ξ ,
1 2 i
| − |
Xi=3
when d=2r.
Lemma 5.1. The law of T is a measure ν on N defined by
1
ν(m)= (1 q)d−1qms (γ ), m N.
d−1 m
1+q − ∈
Proof. When d=2r+1, for m=0 the property is true. For m 1
≥
d−1 d−1
q 1
P(T =m)= P( ξ =m 1)+ P( ξ =m)
i i
1+q − 1+q
Xi=1 Xi=1
d−1
1
= (1 q)d−1qmCard (k ,...,k ) Nd−1 : k m 1,m
1 d−1 i
1+q − { ∈ ∈{ − }}
Xi=1
1
= (1 q)d−1qm (21 +1 )
1+q − k1≥1 k1=0
(k1,...,kd−1)∈NXd−1:Pdi=−11ki=m
1
= (1 q)d−1qms (γ ).
d−1 m
1+q −
So the lemma is proved in the odd case. Moreover
21−qqk if k 1,
1+q ≥
P(ξ ξ =k)=
1 2
| − |  1−q otherwise.
1+q

Thus when d=2r,
1
P(T =m)= (1 q)d−1qm (21 +1 )
1+q − k1≥1 k1=0
(k1,...,kd−1)∈NXd−1:Pdi=−11ki=m
1
= (1 q)d−1qms (γ ).
d−1 m
1+q −
(cid:3)
Lemma 5.1 implies in particular that the measure ν is a probability measure.
Thus one defines a Markov kernel P on by letting
d d
W
+∞
(2) P (λ,β)= µ (λ,β)ν(m),
d m
mX=0
for λ,β . We’ll see that the kernel P describes the evolution of the (d 1)th
d d
∈W −
rowoftherandomprocessonthesetofGelfand-Tsetlinpatternsobservedatinteger
times.
8 MANONDEFOSSEUX
Proposition 5.2. For λ,β ,
d
∈W
Pd(λ,β)= (1−q)d−1ssd−1((βλ))qPri=1(λi+βi−2ci)(1cr>0+ 11c+r=q0)
c∈NXr:c(cid:22)λ,β d−1
when d=2r+1 and
Pd(λ,β)= (1−q)d−1sd−1(β)qPir=−11(λi+βi−2ci)+|λr−βr|
q+1 s (λ)
c∈Nr−X1:c(cid:22)|λ|,|β| d−1
when d=2r.
Proof. The proposition follows immediately from the tensor product rules recalled
for the decomposition (1). (cid:3)
6. Random matrices
Let us denote by d,d′ the set of d d′ real matrices. A standard Gaussian
M ×
variableon d,d′ isarandomvariablehavingadensitywithrespecttotheLebesgue
M
measure on d,d′ equal to
M
1 1
M ∈Md,d′ 7→ dd√′ 2π exp(−2tr(MM∗)).
We write for the set M : M +M∗ = 0 of antisymmetric d d real
d d,d
A { ∈ M } ×
matrices, and i for the set iM :M . Since a matrix in i is Hermitian,
d d d
A { ∈A } A
it has real eigenvalues λ λ λ . Morever, antisymmetry implies that
1 2 d
≥ ≥ ··· ≥
λ = λ , for i = 1, ,[d/2]+1, in particular λ = 0 when d is odd.
d−i+1 i [d/2]+1
− ···
Consider the subset of R[d2] defined by
Cd +
Cd ={x∈R[d2] :x1 >···>x[d2] >0},
and its closure
C¯d ={x∈R[d2] :x1 ≥···≥x[d2] ≥0}.
Definition 6.1. We define the function h on by
d d
C
h (λ)=c (λ)−1V (λ), λ ,
d d d d
∈C
where the functions V and c are given by :
d d
V (λ)= (λ λ ) (λ +λ ) λε,
n i− j i j i
1≤i<Yj≤[d] 1≤i<Yj≤[d] 1≤Yi≤[d]
2 2 2
d 1
c (λ)= (j i) (d j i) ([ ]+ i)ε,
n
− − − 2 2 −
1≤i<Yj≤[d] 1≤i<Yj≤[d] 1≤Yi≤[d]
2 2 2
whit ε equal to 1 when d / 2N and 0 otherwise.
∈
The next proposition is a consequence of Propositions 4.8 and 5.1 of [3].
Proposition 6.2. Let (M(n),n 0), be a random process on i defined by
d
≥ A
n
0 i
M(n)= Y Y∗,
l(cid:18) i 0(cid:19) l
Xl=1 −
where the Y ’s are independent standard Gaussian variables on . If Λ(n) is
l d,2
M
the vector of ¯ whose components are the [d] largest eigenvalues of M(n), n N,
Cd 2 ∈
AN INTERACTING PARTICLE MODEL AND A PIERI-TYPE FORMULA FOR SO(d) 9
then the random process (Λ(n),n 0) is a Markov chain on ¯ with transition
d
≥ C
probabilities
h (y)
d
p (x,dy)= m (x,y)dy,
d d
h (x)
d
for x,y , where dy is the Lebesgue measure on R[d2] and
∈Cd +
md(x,y)= 1{z(cid:22)x,y}e−Pmi=1(yi+xi−2zi)dz
ZRr
+
when d=2r+1 and
md(x,y)= 1{z(cid:22)|x|,|y|}e−Pir=−11(xi+yi−2zi)(e−|xr−yr|+e−(xr+yr))dz
ZRr−1
+
when d=2r.
7. Results
The main result of our paper states in particular that if only one row of the
patterns (X(t),t 0) is considered by itself, it found to be a Markov process too.
≥
Actually we state the result for the process observed at integer times, even if the
process observed at the whole time is also Markovianas we’ll see in section 8.
Theorem 7.1. The random process (Xk(n)) is a Markov process on + . If
n≥0 Wk+1
we denote by R its transition kernel then
k
R =R.
1
•
when k is even R =P ,
k k+1
•
when k is an odd integer greater than 2
•
Pk+1(x,y)+Pk+1(x,y˜) if yk+1 =0
2 6
R (x,y)=
k

P (x,y) otherwise,
k+1

for x,y ∈Wk++1, where y˜=(y1,...,yk−21,−yk+21).
Corollary 7.2. Let (e1,...,e[k+1]) be the canonical basis of R[k+21]. The random
2
process (Yk(t),t 0) is a Markov process with infinitesimal generator defined by
≥
A (λ,β)= 2sskk((βλ))1β∈Wk if k is odd, λk+21 =0 and βk+21 =1
k  sk(β)1 otherwise,
sk(λ) β∈Wk

for λ∈Wk, and β ∈{λ+e1,...,λ+e[k+21],λ−e1,...,λ−e[k+21]}.
If (Λ(n),n 0) is the process of eigenvalues considered in Proposition 6.2 with
≥
d=k+1 then the following theorem holds.
Theorem 7.3. Letting q = 1 1, the process (Xk(n),n 1) converges in distri-
− N N ≥
bution towards the process of eigenvalues (Λ(n),n 1) as N goes to infinity.
≥
10 MANONDEFOSSEUX
8. proofs
8.1. Proof of Theorem 7.1. For k = 1, Theorem 7.1 is clearly true. The proof
of the theorem for k 2 rests on an intertwining property and an application of a
≥
Pitman and Rogers criterion given in [7].
Notation 8.1. Let ξ and ξ be two independent geometric random variables. For
1 2
x,a N such that x a, the law of the random variable
∈ ≥
max(a,x ξ ),
1
−
a← →b
is denoted by P (x,.). For x,b N such that x b we denote by P (x,.) and
∈ ≤
→b
R(x,.) the laws of the random variables
min(b,x+ξ ) and min(b, x+ξ ξ ).
1 1 2
| − |
For x,y R2 such that x y we let
∈ ≤
P(x,y)=(1 q)qy−x.
−
The two following lemmas are proved by straightforwardcomputations.
Lemma 8.2. For a,x,y N such that a y x
∈ ≤ ≤
a← (1 q)qx−y if a+1 y
P (x,y)= − ≤
(cid:26) qx−a if y =a.
For b,x,y N such that b y x
∈ ≥ ≥
→b (1 q)qy−x if y b 1
P (x,y)= − ≤ −
(cid:26) qb−x if y =b.
For b,x,y N such that b y,x
∈ ≥
1−q(q|y−x|+qx+y) if y b 1,y >0
1+q ≤ −

→Rb(x,y)= 11+−1qqqqxb(q−x+qx) iiff yy ≤=bb,−y1>,y0=0
1+q
 1 if y =b,y =0.
Lemma 8.3. For (x,y,z) N3 such that 0<z y
∈ ≤
z
u←
(3) (1 +21 )R(u,x)P (y,z)=(1 q)(1 +21 )qx∨z+y−2z.
u=0 u>0 x=0 x>0
−
uX=0
For (x,y,a) N3 such that a y and y x
∈ ≤ ≤
y
u←
(4) quP (x,y)=qx−yqa.
uX=a
For (x,y,a) N3 such that y a and x y
∈ ≤ ≤
a
→v
(5) q−vP (x,y)=qy−xq−a.
vX=y