The Markov chain asymptotics of random mapping graphs

The Markov chain asymptotics of random mapping graphs

Xinxing Chen, Jiangang Ying

Department of Mathematics, Fudan University, Shanghai, China Received 16 November 2005; accepted 30 May 2006

Available online 15 December 2006

Abstract

In this paper the limit behavior of random mappings withnvertices is investigated. We first compute the asymptotic probability that a fixed class of finite non-intersected subsets of vertices are located in different components and use this result to construct a scheme of allocating particles with a related Markov chain. We then prove that the limit behavior of random mappings is actually embedded in such a scheme in a certain way. As an application, we shall give the asymptotic moments of the size of the largest component.

©2006 Elsevier Masson SAS. All rights reserved.

Résumé

Dans cet article, nous étudions le comportement asymptotique des trasformations aléatoires ànvertex. A titre d’application nous calculons les moments asymptotiques de la taille de la plus grande composante.

©2006 Elsevier Masson SAS. All rights reserved.

MSC:05C80; 60J10

Keywords:Random mapping graphs; Connection; Component; Scheme of allocating particles; Markov chain; Asymptotic behavior

1. Introduction

LetV be a set withnelements, say,V = {1,2, . . . , n}, andΩ_nthe set of all mappings onV, which includesnⁿ elements. LetPnbe the classical probability onΩ_n. Anyf ∈Ω_ninduces a directed graphG_f with the set of vertices V (G_f)=V and edgesE(G_f)= {(u, f (u)): u∈V}. The space(Ω_n,Pn)is called the space of random mappings or random mapping graphs.

The most interesting problems on random mappings are their various asymptotic behavior, by which we means the limit distribution of various graph structures, for example, number of components, size of component, etc., asngoes to infinity. Most of papers on this field focused on these issues. In this paper we shall start with a direct computation of asymptotic connection probability and use it to construct a scheme of allocating particles and a related Markov chain.

* Corresponding author.

E-mail address:jgying@fudan.edu.cn (J. Ying).

1 The research of this author is supported in part by NSFC No. 10671036.

0246-0203/$ – see front matter ©2006 Elsevier Masson SAS. All rights reserved.

doi:10.1016/j.anihpb.2006.05.004

We prove that asymptotic behaviors of random mappings may be represented in the Markov chain in certain sense.

Moreover we may use the limit theorems in theory of probability and powerful tools developed for martingales and Markov chains. As a main result we shall give an explicit answer for the asymptotic moments and distribution of the largest component size.

We should mention that in 1994, Aldous and Pitman [2], also see [6], proved that the uniform random mapping is asymptotically the Brownian bridge in some sense, by which, various limit distributions were obtained. Comparing to that, our approach is more direct and elementary, and involves less machinery.

Throughout the paperNdenotes the set of natural numbers. For anyn∈N,[n] = {1,2, . . . , n}. For any setA, the notation|A|denotes the cardinality ofA. A partition{A1, . . . , Am}of a finite setU⊂Nis said to be ordered if minA1<minA2<· · ·<minAm. The partition involved in this paper will always be ordered. The probability and expectation are taken in the probability space of random mappings withnelements are written asPn,En. The symbol

‘:=’ should be read as ‘is defined to be’. We shall briefly introduce our strategy as follows. The key is to relate this asymptotics with that of a particular Chinese restaurant process. More precisely let(Ωn,Pn)be the probability space of random mappings on[n]andGna sample. ForNn, letA1, . . . , Ambe an ordered partition of[N]. Denote by JA₁⊕ · · · ⊕JA_m the event thati, j ∈ [N]are connected if and only if they are in the same setAl. In other words, JA₁⊕ · · · ⊕JA_m is the event that the trace of the connected components ofGn on[N]is the partition{A1, . . . , Am} (let us call this theN-trace ofGn). Now let(Zk)k1be the(0,1/2)-Chinese restaurant process andPits probability (Zk is the label number of the table where thek-th customer is seated). Set

φ_A1,...,Am(·):=

m l=1

l·1Al.

The asymptotic connection of two models is the following (see Theorem 3.1, though it is actually proved in Section 2) limn Pn(JA₁⊕ · · · ⊕JA_m)=P(Zk=φA₁,...,A_m(k), k=1, . . . , N ).

(Roughly speaking, the asymptotic distribution of theN-trace ofGnis given by the distribution ofZ1, . . . , ZN.) We then show in Theorem 3.2 that the asymptotic(n→ ∞)“difference” between the cardinalities of theN-trace and those of the connected components ofGnvanishes asN goes to infinity. By this way we obtain a connection between the asymptotic of the cardinalities of the connected components ofGnand the process(Zk). As a consequence, we give examples in Section 3 to show how to derive from this approach some known results as in Pittel [7], Stepanov [11], Aldous and Pitman [2], etc. However the main application is Theorem 4.1 in which the asymptotic of all moments of ther-th largest connected component ofGnis given.

2. The limit probability of connection

Two verticesiandj are connected if there is a path of edges, ignoring direction, connecting them. This naturally induces a classification for each graph. LetU be a fixed subset of[n]and{A₁, A₂, . . . , A_m}a fixed partition ofU.

LetJ_A1⊕J_A2⊕ · · · ⊕J_Ambe the event that for anyi, j∈U withi=j,iandj are connected if and only ifi, j∈A_l for some 1lm. SetJ_m^d:=J_{1}⊕J_{2}⊕ · · · ⊕J_{m}. LetY_i=Y_i(G_n)be the number of the vertices connected with vertexi for anyi∈V, and 1α be the indicator ofα for any setα. In order to write with more convenience, we also use the notationa_l := |A_l| −10 for eachl,1lmand set a:=a₁+a₂+ · · · +a_mn−m; and (x)_r:=x(x−1)· · ·(x−r+1).

Lemma 2.1.

Pn(J_A1⊕J_A2⊕ · · · ⊕J_Am)= 1 (n−m)a

En

_m

l=1

(Y_l)_al;J_m^d

.

Proof. LetS= {JS₁ ⊕JS₂⊕ · · · ⊕JS_m: S1, S2, . . . , Smare mutually disjoint subsets ofV and for 1lm,l∈Sl

and|Sl| = |Al|}, a set of events. For any eventα∈S,

En(1α)=Pn(1α=1)=Pn{α} =Pn(J_A1⊕J_A2⊕ · · · ⊕J_Am).

Hence

Pn(JA₁⊕JA₂⊕ · · · ⊕JA_m)=

α∈SEn(1α)

|S| =En(

α∈S1α)

|S| . On the other hand, a direct calculation gives that

|S| = (n−m)_a a1!a2! · · ·am! and

α∈S

1α= ^ml=1

(Y_l)_al

al! , ifJ_m^doccurs,

0, otherwise.

Replacing them into the formula forPn{JA₁⊕JA₂⊕ · · · ⊕JA_m}, the conclusion follows. 2 Refer to pages 129–137, [9] for the following lemma.

Lemma 2.2.

Pn{G_nis connected} =(n−1)! nⁿ

n−1

j=0

n^j j! ∼ π

2n.

Lemma 2.3.For each1lm, letk_l0, withk:=k₁+k₂+ · · · +k_mn−m. Then Pn

Y₁=k₁, . . . , Y_m=k_m;J_m^d

=(n−m)! nⁿ

(n−k−m)^n−k−m (n−k−m)!

m l=1

k_l

j=0

(k_l+1)^j j! .

Proof. SetS := {J_S1 ⊕J_S2 ⊕ · · · ⊕J_Sm: S₁, . . . , S_m are mutually disjoint and l∈S_l ⊂V,|S_l| =k_l+1, for any 1lm}, and forα=JS₁⊕JS₂⊕ · · · ⊕JS_m∈S,1lmwe setl∈αl=Sl. And thatα=Jα₁⊕Jα₂⊕ · · · ⊕Jα_m

forα∈S. ForΛ⊂V we setDΛ be the event that none of the vertices inΛis connected with any vertex inV\Λ.

Then

Pn(Y1=k1, . . . , Y_m=k_m;J_m^d)=

α∈S

Pn(α;Y1=k1, . . . , Y_m=k_m)

=

α∈S

Pn(J_α1⊕J_α2⊕ · · · ⊕J_αm;Y₁=k₁, . . . , Y_m=k_m)

=

α∈S

Pn(J_α1⊕J_α2⊕ · · · ⊕J_αm;D_α1, D_α2, . . . , D_αm)

=

α∈S

Pn(Dα₁Dα₂· · ·Dα_m)Pn(Jα₁⊕Jα₂⊕ · · · ⊕Jα_m|Dα₁, Dα₂, . . . , Dα_m)

=

α∈S

Pn(D_α1D_α2· · ·D_αm) m l=1

Pn(J_αl|D_αl).

By the definition ofSandα, we know thatk_l= |α_l| −1 for eachl,1lm. It is easy to see that

|S| = (n−m)!

k₁! · · ·k_m!(n−m−k)!, Pn(D_α1D_α2· · ·D_αm)=(n−k−m)^n−k−mm

l=1(kl+1)^kl+1

nⁿ .

Employing Lemma 2.2 we know that for eachl,1lmit holds that Pn(J_αl|D_αl)= k_l!

(k_l+1)^kl+1

k_l

j=0

(k_l+1)^j j! .

Substituting them+2 equations intoPn(Y₁=k₁, . . . , Y_m=k_m;J_m^d), we shall have the conclusion directly. 2

Corollary 2.1.Letkl0,1lm. Ifk:=k1+k2+ · · · +km< n−m, then Pn(Y1=k1, . . . , Ym=km;J_m^d)= 1

2^mn^m−1/2(n−k−m)^1/2

1−k₁,...,k_m+²_n,k

1,...,k_m

;

and ifk=n−m, then Pn

Y₁=k₁, . . . , Y_m=k_m;J_m^d

=

√2π 2^mn^m−1/2

1−_k1,...,km+²_n,k

1,...,km

, where

k₁,...,k_m:=

m l=1

2 3√

2π(kl+1) and ²_n,k

1,...,k_m:= C₋₁

n−m−k+C₀ n +

m l=1

C_i k_l+1,

withC₋₁, C₀, C_i, . . . , C_mbeing bounded functions which depend onn, mandk₁, . . . , k_m. Note that we set _n−^C_m⁻₋¹_k =0 ifk=n−m.

Proof. We prove for the casek < n−monly. Employing the Stirling’s formular! =(r/e)^r√

2π re^θr, where_12r¹₊₁<

θr <_12r¹ , we get that (n−m)!

nⁿ

(n−k−m)^n−k−m

(n−k−m)! = n^1/2e^−k−m

n^m(n−k−m)^1/2e^{θn−θn−k−m}

m−1 i=0

1− i

n ₋1

. Furthermore, we have the Ramanujan sequence [8]

1

2e^r =1+ r 1!+r²

2! + · · · + r^r−1 (r−1)!+γr

r^r r!

whereγ_r is decreasing inrand of the formγ_r=¹₃+_135(r⁴_+ηr) with₂₁² η_r45⁸. Then m

l=1 k_l

j=0

(k_l+1)^j j! =

m l=1

e^kl+1

2 −γ_kl+1(k_l+1)^kl+1 (k_l+1)!

=e^k+m 2^m

m l=1

1− 2γkl+1

√2π(kl+1)e^θkl+1

. After some simple but lengthy calculation, it follows that

e^{θn−θn−k−m}

m−1 i=0

1−i

n ₋1m

l=1

1− 2γk_l+1

√2π(kl+1)e^θkl+1

=1−_k1,...,km+²_n,k

1,...,km

where_k1,...,kmand²_n,k

1,...,k_mare defined as above. Therefore, substituting the three equations above into Lemma 2.3, we shall have

Pn

Y₁=k₁, . . . , Y_m=k_m;J_m^d

= 1

2^mn^m−1/2(n−k−m)^1/2

1−_k1,...,km+²_n,k

1,...,km

. 2

The following two theorems give an explicit expression for connection probability and its exact asymptotic behav- ior.

Theorem 2.1.LetA1, A2, . . . , Ambe a partition of a fixed setU⊂V. Then Pn(JA₁⊕ · · · ⊕JA_m)=(n−m−a)!

nⁿ

n−m k=0

kl+···+km=k

(n−k−m)^n−k−m (n−k−m)!

m l=1

(kl)a_l kl

j=0

(k_l+1)^j j! . Proof. It is a direct consequence of Lemmas 2.1 and 2.3. 2

Theorem 2.2.LetA1, A2, . . . , Ambe a partition of a fixed setU⊂V. Then Pn(J_A1⊕ · · · ⊕J_Am)

=(2a1)!! · · ·(2am)!!

(2m+2a−1)!! +(2a1)!! · · ·(2am)!!

(2m+2a−2)!!

1−

m i=1

(2ai−1)!!

(2ai)!!

√ 2π

6 n^−1/2+o n^−1/2

, and

Pn(JA₁⊕ · · · ⊕JA_m)−Pn+1(JA₁⊕ · · · ⊕JA_m)

=(2a1)!! · · ·(2am)!!

(2m+2a−2)!!

1−

m i=1

(2ai−1)!!

(2ai)!!

√ 2π

12 n^−3/2+o n^−3/2

. asngoes to infinity. Note that we set(−1)!! =0!! =1.

Proof. The casem=1, a1=0 is trivial. We shall prove for case I:m=1,a₁=1 and case II:m=2,a₁=a₂=0. The result for the rest cases can be proved by induction tomand the size ofA₁, A₂, . . . , A_m, and we omit the procedure which is something like the procedure that we shall do for case I and case II.

Case I:m=1, a1=1. Without loss of generality, we setA₁= {1,2}. What we want to prove is that Pn{J_{1,2}} =2

3+

√2π

12 n^−1/2+o n^−1/2

, Pn{J_{1,2}} −Pn+1{J_{1,2}} =

√2π

24 n^−3/2+o n^−3/2

.

Case II:m=2,a1=a2=0. Without lost of generality, we setA1= {1}andA2= {2}. What we want to prove is that

Pn{J_{1}⊕J_{2}} =1 3−

√2π

12 n^−1/2+o n^−1/2

, and

Pn{J_{1}⊕J_{2}} −Pn+1

J_{1}⊕J_{2}

= −

√2π

24 n^−3/2+o n^−3/2

.

We prove for case I in three steps. At first we point out that Pn{J_{1,2}}converges to ²₃. Secondly we prove the existence of limitT =limn→∞√

n (Pn{J_{1,2}} −2/3)by induction on the base of limn→∞√

n (Pn{J_{1}} −1)=0.

At last we find outT, where we use the fact limn→∞√

n (Pn{J_{1}} −1)=0 again, and prove the monotonicity of Pn{J_{1,2}}at the same time. We prove for case II straightforward by using the previous results above forPn{J_{1,2}}. (It may be seen as being proved by induction, too.)

Step 1. Using Theorem 2.1, we get Pn{J_{1,2}} =En(Y₁)

n−1 =(n−2)! nⁿ

n−1

k=1

k(n−k−1)^n−k−1 (n−k−1)!

k j=0

(k+1)^j j! . SetnΛ_k, for eachn, k,n=2,3, . . .andk=0,1,2, . . . , n−1,

nΛk= k

n−1Pn(Y1=k)=(n−2)! nⁿ

k(n−k−1)^n−k−1 (n−k−1)!

k j=0

(k+1)^j j! . ThenPn{J_{1,2}} =_n−1

k=1nΛ_k. Employing Corollary 2.1, we get, for 0kn−2

nΛk= 1 2√

n(n−1)

√ k

n−k−1

1− 2

3√

2π(k+1)+ C₋₁

n−k−1 +C₀ n + C₁

k+1

. Therefore,

Pn{J_{1,2}} =P (Y₁=n−1)+

n−2

k=0 nΛ_k∼

n−2

k=0

1 2√

n(n−1)

√ k

n−k−1 ∼1 2

1

0

√ x

1−xdx=2 3. Step 2. As it is known thatPn{J_{1}} =1, we get

(n−1)

n−1

k=1 nΛ_k

k =

n−1

k=1

Pn(Y1=k)=Pn{J_{1}} −Pn(Y1=0)=1+O n⁻¹

. We shall separatePn{J_{1,2}}into several parts.

Pn{J_{1,2}} =

n−1

k=1 nΛ_k=

n−1

k=1

n−1 k ⁿΛk−

n−1

k=1

n−k−1 k ⁿΛk

=1−

n−1

k=1

√n−k−1 2√

n(n−1)

1− 2

3√

2π(k+1)+ C₋1

n−k−1+C0

n + C1

k+1

+O n⁻¹

=1−

n−1

k=1

√n−k−1 2√

n(n−1)+

n−1

k=1

√n−k−1 2√

n(n−1) 2 3√

2π(k+1)+o n^−1/2

=1−1 2

1

0

√xdx+ 1 3√

2π 1

0

1−x

x dx·n^−1/2+o n^−1/2

=2 3 +

√2π

12 n^−1/2+o n^−1/2

. HenceT =limn→∞√

n (Pn{J_{1,2}} −2/3)exists andT =^√₁₂^2π. However, it is not necessary to know the value ofT. What we need only at present is to know its existence, and we will calculate its value by solving out a equation. Such method will do help when we prove the rest cases.

We comparen+1Λk+1withnΛk, and get for each 1kn−1,

n+1Λ_k+1

nΛ_k =k+1 k

n−1 n

e·nⁿ⁺¹ (n+1)ⁿ⁺¹

_k+1 j=0

(k+2)^j j!

e_k

j=0 (k+1)^j

j!

=

1+1 k

1− 3

2n+O 1

n²

1+ 1

3(2π )^1/2k^3/2+C_k k²

=1+1 k− 3

2n+ 1

3(2π )^1/2k^3/2+C_k k²,

whereCk, which may vary from place to place, is a bounded function which depends onk. But it can be shown that n^3/2

n−1

k=1 nΛk

k^3/2 ∼

n−2

k=1

1 2√

k(n−k−1)∼1 2 1

0

√ 1

x(1−x)dx=π 2 and_n−1

k=1ⁿ Λ_k

k² =o(n^−3/2). Therefore, n^3/2

n−1

k=1

(n+1Λk+1−nΛk)=n^3/2

n−1

k=1 nΛ_k

k −3n^1/2 2

n−1

k=1

nΛ_k+ n^3/2 3(2π )^1/2

n−1

k=1 nΛ_k k^3/2+o(1)

=n^1/2−3n^1/2 2

2/3+T ·n^−1/2 +

√2π 12 +o(1)

= −3 2T +

√2π

12 +o(1).

It follows that n^3/2

Pn+1{J_{1,2}} −Pn{J_{1,2}}

=n^3/2·n+1Λ1+n^3/2

n−1

k=1

(n+1Λk+1−nΛk)= −3 2T +

√2π

12 +o(1).

On the other hand, we have

nlim→∞

√n

Pn{J_{1,2}} −2/3

=2 lim

n→∞n√ n

Pn{J_{1,2}} −Pn+1{J_{1,2}} , since the existence of limn→∞n√

n(Pn{J_{1,2}} −Pn+1{J_{1,2}}). Therefore,

−T 2 = −3

2T +

√2π 12 and we getT =√

2π /12.For case II, it holds thatPn(J_{1}⊕J_{2})=Pn(J_{1})−Pn(J_{1,2}).Substituting the result of case I into this, we draw out the conclusion for case II. 2

The following two results are the direct consequences of Theorem 2.2.

Corollary 2.2.

(1) The probability that vertices in[m]are totally disconnected Pn{J_m^d} =Pn{J_{1}⊕J_{2}⊕ · · · ⊕J_{m}} ∼ 1

(2m−1)!!; (2) The probability that vertices in[m]are connected

Pn{J_{1,2,...,m}} ∼(2m−2)!!

(2m−1)!!.

Remark.At the end of this section, we would like to present a table of Monte Carlo simulation. In Table 1 we list some numerical values concerning probabilityPn{J_A1⊕J_A2 ⊕ · · · ⊕J_Am}in some simple cases for comparing. ‘precise’

means ‘precise value’, ‘revised’ means ‘revised value’, ‘test’ means ‘test value’ and ‘limit’ means ‘limit value’. The precise value is given by Theorem 2.1. The revised value is given by approximation

Table 1

Calculation ofPn{JA₁⊕JA₂⊕ · · · ⊕JA_m}

P precise

n=10

revised n=10

test n=10

precise n=50

revised n=50

precise n=100

revised n=100

limit n→ ∞

J_{1,2} .7159 .7327 .7175 .6923 .6962 .6855 .6876 .6667

J_{1,2,3} .5966 .6159 .5990 .5659 .5703 .5572 .5594 .5333

J_{1,2,3,4} .5272 .5480 .5315 .4931 .4978 .4834 .4859 .4571

J{1,2,3,4,5} .4805 .5023 .4830 .4443 .4493 .4341 .4367 .4063

J_{1}⊕J_{2} .2841 .2673 .2825 .3077 .3038 .3145 .3124 .3333

J_{1,2}⊕J_{3} .1193 .1168 .1185 .1264 .1259 .1284 .1281 .1333

J_{1,2}⊕J_{3,4} .0365 – .0430 .0376 – .0378 – .0381

J_{1,2,3}⊕J_{4} .0694 .0679 .0675 .0728 .0725 .0737 .0736 .0762

J_{1,2,3}⊕J_{4,5} .0166 .0176 .0145 .0170 .0172 .0170 .0171 .0169

J_{1,2,3,4}⊕J_{5} .0467 .0456 .0485 .0488 .0485 .0493 .0492 .0508

J_{1}⊕J_{2}⊕J_{3} .0454 .0336 .0445 .0548 .0519 .0578 .0562 .0667

J_{1,2}⊕J_{3}⊕J_{4} .0135 .0108 .0120 .0160 .0154 .0168 .0164 .0190

J_{1,2}⊕J_{3,4}⊕J_{5} .0032 .0029 .0045 .0037 .0036 .0038 .0038 .0042

J_{1,2,3}⊕J_{4}⊕J_{5} .0060 .0047 .0055 .0071 .0068 .0074 .0073 .0085

J_{1}⊕J_{2}⊕J_{3}⊕J_{4} .0050 .0013 .0060 .0068 .0058 .0074 .0069 .0095 J_{1,2}⊕J_{3}⊕J_{4}⊕J_{5} .0011 .0004 .0000 .0015 .0013 .0017 .0016 .0021 J_{1}⊕J_{2}⊕J_{3}⊕J_{4}⊕J_{5} .0004 −.0003 .0010 .0006 .0004 .0007 .0006 .0011

Pn{JA₁⊕JA₂⊕ · · · ⊕JA_m} ≈(2a1)!! · · ·(2am)!!

(2m+2a−1)!! +(2a1)!! · · ·(2am)!!

(2m+2a−2)!!

1−

m i=1

(2ai−1)!!

(2ai)!!

√ 2π 6 n^−1/2. The test value is given by the Monte Carlo simulation

Pn{JA₁⊕JA₂⊕ · · · ⊕JA_m} ≈times of the occurrence ofJ_A1⊕J_A2⊕ · · · ⊕J_AminN times N

withN=2000. The limit value is given by

Pn{J_A1⊕J_A2⊕ · · · ⊕J_Am} →(2a1)!! · · ·(2am)!!

(2m+2a−1)!! . 3. A related Markov chain and asymptotics of components

We shall present the essential relation between large components of a random mapping and large boxes of the related scheme of allocation in this section. We first introduce a probability space which describes a scheme of allo- cating particles and discuss interesting properties of some related random sequences. By Theorem 2.2 the following lemma is direct.

Lemma 3.1.Assume thatA1, . . . , Amis a fixed partition of[N]. Then

nlim→∞Pn{J_A1⊕ · · · ⊕J_Am⊕J_{N+1}|J_A1⊕ · · · ⊕J_Am} = 1 2N+1 and forl=1, . . . , m

nlim→∞Pn{J_A1⊕ · · · ⊕J_Al−1⊕J_Al∪{N+1}⊕J_Al+1⊕ · · · ⊕J_Am|J_A1⊕ · · · ⊕J_Am} = 2|A_l| 2N+1.

The lemma inspires us to consider a scheme of random allocating particles. There are different particles called P₁, P₂, . . .and different boxes calledB₁, B₂, . . .. TheP_nandB_n are also called then-th particle andn-th box. We define the allocation by induction. First we placeP₁intoB₁, and next we placeP₂with probability ²₃ intoB₁ and with probability ¹₃ intoB₂. More generally, suppose that the firstN particles have been placed. Let us place the next particle. Let the firstmboxes be non-empty and haveq₁, . . . , q_m particles in boxes from 1 tomrespectively. Then N=q₁+ · · · +q_m. At the next time we placeP_N+1into theB_lwith probability_2N^2q₊^l₁forl=1, . . . , mand intoB_m+1 with probability _2N¹₊₁. More precisely we have a probability space(Ω,F,P)and a sequence of random variables {Z_n}, which is defined by induction as follows

(1) Z1=1;

(2) Suppose that Z1, . . . , ZN is defined forN 1. Let m:=max{Z1, . . . , ZN}andqi :=N

k=11_{Z_k=i}.ZN+1 is equal toiwith probability_2N^2q₊ⁱ₁for 1imand tom+1 with probability _2N¹₊₁, or in the form of conditional expectation

P(ZN+1=i|Z₁, . . . , Z_N)= 2qi

2N+11_{1im}+ 1

2N+11_{i=m+1}.

The random variableZ_Nrecords the number labelled on the box where theN-th particle is placed. The property (2) actually gives the conditional distribution ofZ_N+1 given {Z₁, . . . , Z_N}which determines the law P. Let HN:=

σ (Z₁, . . . , Z_N)the filtration of{Z_N}andH_∞=σ (HN: N1). Assume again thatA₁, . . . , A_m is a fixed ordered partition of[N]. We denote byA₁⊕A₂⊕ · · · ⊕A_m the event that for anyi, j ∈ [N],P_i andP_j are contained in the same box if and only ifi, j∈A_l for some 1lm. Actually the eventA₁⊕ · · · ⊕A_mdetermines the value of Z1, . . . , ZNand vice versa. Indeed ifZ1, . . . , Znare given, then the partition is natural, and conversely if an ordered partition is given as above,Zk=lifk∈Al. Then

H_N=σ

A₁⊕ · · · ⊕A_m: {A₁, . . . , A_m}is a partition of[N] andA₁⊕ · · · ⊕A_mis an atom ofHN.