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3 Proof of Part (b) in the Case α>1

3 Proof of Part (b) in the Case α>1

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340



C. Profeta and T. Simon



Using Theorem A and the computations of the case ˛ < 1, the expression may be

transformed into

Z

1

y x Á˛ 1 z

˛; .t/ dt

.˛ / .˛ O/

2

1

Â

Ã

Z 1

˛ 1

.1 ˛/ IJ; .x/ 1 c˛;

.y t/ '.t/dt

O

:

1



The result follows from the hat version of Lemma 2 and the expression of IJ; .x/:

t

u



4 Proof of the Corollaries

4.1 Proof of Corollary 1

By duality, it is enough to consider the case x > y: From Part (a) of Theorem B and

a change of variable, we see that g.x; y/ extends by continuity on the diagonal, with

g.y; y/ D







1

1/ .˛ / .˛ O/



Â



y2



1

2



Ã˛



1



:



Moreover, it is clear that g vanishes on the boundary fjxj D 1g [ fjyj D 1g and is

hence bounded on . 1; 1/ . 1; 1/: By Proposition VI.4.11, Exercise VI.4.18 and

Formula V.3.16 in [4], we deduce

Px ŒTy < T D



g.x; y/

g.y; y/

t

u



and the conclusion follows by Theorem B.

Remark 6



(a) In the case ˛ Ä 1; the process L does not hit points, so that the problem is

irrelevant. In general, one can ask for an evaluation of the probability Px ŒTI < T

where I is a closed subinterval of . 1; 1/ not containing x; and TI is its first

hitting time. In the transient case ˛ < 1; this problem is solved theoretically as

a particular instance of the so-called condenser problem—see Formula (3.4) in

[8]. It is an interesting open problem to find out an explicit formula in the real

stable framework.

(b) By the Markov property, we can write down the following expression for the

harmonic measure Hxfyg .dt/ of the set fyg [ Œ 1; 1c W

Hxfyg .dt/ D



.x; y/.ıfyg .dt/



Hy .dt// C Hx .dt/:



(10)



On the Harmonic Measure of Stable Processes



341



In particular, for every x; y 2 . 1; 1/; one has

Px ŒLT 2 dt; T < Ty  D Hx .dt/



.x; y/Hy .dt/:



(c) It is interesting to mention that using the Gauss formula, we can deduce the

asymptotic behaviour of Px ŒTy > T when x ! y; which is fractional. For

instance, if y D 0; one has

Px ŒT0 > T



.2

.1



x!0C



˛/ .˛ /

.2x/˛

˛ O/



1



and Px ŒT0 > T



.2

x!0



˛/ .˛ O/

j2xj˛ 1 :

.1 ˛ /



(d) By (2) and spatial homogeneity, it is easy to deduce from Corollary 1 the

following expression of Q.x; y/ D Px ŒTy <  where D infft > 0; Lt > 1g W

one finds

Q.x; y/ D .˛



ˇ

ˇx

1/ ˇˇ

1



ˇ

y ˇˇ˛





1



ˇ

ˇ1

ˇx



Z

0



ˇ













1



.t C 1/˛ O



1



dt



if x > y; and Q.x; y/ D QO. x; y/ if x < y: When y D 0; this is Theorem 1.5

in [13], correcting a misprint (the 1 1=x in the second integral should be

1=x) therein. Notice that Corollary 1.6 in [13] is also analogously recovered

from (10).



4.2 Proof of Corollary 2

By the general theory of Martin boundary—see e.g. Theorem 1 in [11], we need to

compute the Martin kernels

M1 .x/ D lim



y!1



g.x; y/

g.0; y/



and



M 1 .x/ D lim



y! 1



g.x; y/

g.0; y/



Part (a) of Theorem B and a straightforward asymptotic analysis show that these

Martin kernels exist and equal respectively

M1 .x/ D .1



x/˛



whence the result.



1



.1 C x/˛ O



and



M 1 .x/ D .1 C x/˛ O 1 .1



x/˛ ;

t

u



342



C. Profeta and T. Simon



5 Final Remarks

In this section, we briefly describe the analogues of the above results in the case of

semi-finite intervals and in the spectrally one-sided situation.



5.1 The Case of Semi-finite Intervals

By scaling and spatial homogeneity, one can deduce from Theorem A—either its

Part (a) or its Part (b)—the following expression of the density of L under Px ;

where x < 1 and D infft > 0; Lt > 1g: One finds

fL .y/ D



c˛; .1 x/˛

.y 1/˛ .y x/



This expression has been found by several authors and can be obtained in different

ways (see Exercise VIII.3 in [2] and the references therein). Observe that it serves

as a starting formula in [17] in order to prove Part (a) of Theorem A. Notice last

that the expression extends to the case with no negative jumps, by the Skorokhod

continuity argument. In the relevant case with no positive jumps ˛ > 1; D 1=˛;

the law of L is a Dirac mass at one.

The Green function is

g .x; y/ D



.y x/˛ 1

.˛ / .˛ O/



Z



1 y

y x



0



˛;



.t/ dt



if x < y < 1 and g .x; y/ D gO .y; x/ if y < x < 1: In the case ˛ > 1; the analogue

of Corollary 1 which is already given in Remark 6 (d) above, can then be recovered.

Finally, one finds that the non-negative harmonic functions vanishing on .1; C1/

are of the type

.1

with ;



x/˛ C



.1



x/˛



1



0; in accordance with Theorem 4 in [18] and the paragraph thereafter.



5.2 The Case of Stable Processes with One-Sided Jumps

By duality, it is enough to consider the two cases ˛ < 1; D 1 and ˛ > 1; D 1=˛:



On the Harmonic Measure of Stable Processes



5.2.1 The Case ˛ < 1;



343



D1



It follows readily from the above paragraph that

c˛;1 .1 x/˛

1fy>1g

.y 1/˛ .y x/



h.x; y/ D



for all x 2 . 1; 1/: See also Example 3 in [9] and the references therein for the

expression of the density of .LT ; LT / under Px : Similarly, one has

h .x; y/ D



c˛;1 j1 C xj˛

1fjyj<1g

.1 C y/˛ .y x/



for all x < 1 and h .x; y/ D 0 for all x > 1: In accordance with the fact that L is a

subordinator, the Green function is

g.x; y/ D



.y



x/˛

.˛/



1



1fx


for all x; y 2 . 1; 1/;

g .x; y/ D



.y



x/˛

.˛/



Z



1



1fx


1g



C c˛;1



j1Cxj.y 1/

2



t



!

˛ 1



0



.1 C t/



1



!



dt 1fy>1g



for all x < 1; and g .x; y/ D g.x; y/ for all x > 1: The problem of Corollary 1

is irrelevant. Finally, the non-negative harmonic functions on . 1; 1/ vanishing on

Œ 1; 1c are constant multiples of .1 x/˛ 1 :

5.2.2 The Case ˛ > 1;



D 1=˛



Using Skorokhod continuity in Theorem A (a) and the absence of positive jumps,

one has

Hx .dy/ D c˛;1



1=˛ .1



x/



.1 C x/˛ 1 jy C 1j1

.1 y/.x y/



˛



1fy<



1g dy



C Px ŒT1 < T ı1 .dy/:



Either taking the limit in Remark 6 (d) or integrating the first term, we can compute

the weight of the Dirac mass, and find

Hx .dy/ D c˛;1



1=˛ .1



.1 C x/˛ 1 jy C 1j1

x/

.1 y/.x y/



Â



˛



1fy<



1g dy



C



xC1

2



Ã˛



1



ı1 .dy/:



344



C. Profeta and T. Simon



The corresponding Green function is

g.x; y/ D



Â



1

.˛/



.1



y/.1 C x/

2



Ã˛



!



1



.x



y/



˛ 1



1fx>yg :



The hitting probabilities are

Â

Px ŒTy < T D



1Cx

1Cy



Ã˛



1



for every x Ä y; which is also a consequence of a well-known result on scale

functions—see e.g. Theorem VII.8 in [2], and

Â

Px ŒTy < T D



1Cx

1Cy



Ã˛



1



Â



2.x y/

1 y2



Ã˛



1



for every x > y: Finally, the non-negative harmonic functions on . 1; 1/ which

vanish on Œ 1; 1c are of the type .1 x/˛ 1 .1Cx/˛ 2 C .1Cx/˛ 1 with ;

0:

It is clear that Hx .dy/ D ı 1 .dy/ for all x < 1: To compute Hx .dy/ for x > 1;

let us introduce

D infft > 0; Lt < 1g: The absence of positive jumps and the

formula for semi-finite intervals imply after some computation

Hx .dy/ D 1fjyj<1g Px ŒL 2 dy C Px ŒL < 1ı 1 .dy/

Â

.x 1/˛ 1 .1 y/1 ˛

D c˛;1 1=˛

1fjyj<1g dy

x y

!

!

Z x 1

xC1

˛ 2

1 ˛

C

z .1 z/ dz ı 1 .dy/ ;

0



in accordance with Remark 3 in [14]—see also Proposition 1.3 in [13].

Acknowledgements Nous savons gré à Jean Jacod d’un chat instructif sur la distance de

Skorokhod. C. P. a bénéficié du support de la Chaire Marchés en Mutation, Fộdộration Bancaire

Franỗaise. Travail dộdiộ lassociation Laplace-Gauss.



References

1. G.E. Andrews, R. Askey, R. Roy, Special Functions (Cambridge University Press, Cambridge,

1999)

2. J. Bertoin, Lévy Processes (Cambridge University Press, Cambridge, 1996)

3. J. Bertoin, On the first exit time of a completely asymmetric stable process from a finite

interval. Bull. Lond. Math. Soc. 28(5), 514–520 (1996)



On the Harmonic Measure of Stable Processes



345



4. R.M. Blumenthal, R.K. Getoor, Markov Processes and Potential Theory (Academic, New York,

1968)

5. R.M. Blumenthal, R.K. Getoor, D.B. Ray, On the distribution of first hits for the symmetric

stable processes. Trans. Am. Math. Soc. 99, 540–554 (1961)

6. K. Bogdan, T. Byczkowski, T. Kulczycki, M. Ryznar, R. Song, Z. Vondraˇcek, Potential

Analysis of Stable Processes and Its Extensions. Lecture Notes in Mathematics, vol. 1980

(Springer, Berlin, 2009)

7. T. Carleman, Über die Abelsche Integralgleichung mit konstanten Integrationsgrenzen. Math.

Z. 15, 111–120 (1922)

8. K.L. Chung, R.K. Getoor, The condenser problem. Ann. Probab. 5(1), 82–86 (1977)

9. N. Ikeda, S. Watanabe, On some relations between the harmonic measure and the Lévy measure

for a certain class of Markov processes. J. Math. Kyoto Univ. 2, 79–95 (1962)

10. J. Jacod, A.N. Shiryaev, Limit Theorems for Stochastic Processes (Springer, Berlin, 1987)

11. H. Kunita, T. Watanabe, Markov processes and Martin boundaries. Bull. Am. Math. Soc. 69,

386–391 (1963)

12. A.E. Kyprianou, A.R. Watson, Potential of stable processes. Séminaire de Probabilités XLVI,

333–343 (2014)

13. A.E. Kyprianou, J.-C. Pardo, A.R. Watson, Hitting distributions of ˛-stable processes via pathcensoring and self-similarity. Ann. Probab. 42(1), 398–430 (2014)

14. S.C. Port, Hitting times and potentials for recurrent stable processes. J. Anal. Math. 20, 371–

395 (1967)

15. M. Riesz, Intégrales de Riemann-Liouville et potentiels. Acta Sci. Math. Szeged. 9, 1–42

(1938)

16. M. Riesz, Rectification au travail “Intégrales de Riemann-Liouville et potentiels”. Acta Sci.

Math. Szeged. 9, 116–118 (1938)

17. B.A. Rogozin, The distribution of the first hit for stable and asymptotically stable walks on an

interval. Theory Probab. Appl. 17, 332–338 (1972)

18. M.L. Silverstein, Classification of coharmonic and coinvariant functions for a Lévy process.

Ann. Probab. 8(3), 539–575 (1980)

19. S.J. Taylor, Sample path properties of a transient stable process. J. Math. Mech. 16, 1229–1246

(1967)



On High Moments of Strongly Diluted Large

Wigner Random Matrices

Oleskiy Khorunzhiy



Abstract We consider a dilute version of the Wigner ensemble of n n random

real symmetric matrices H .n; / , where denotes the average number of non-zero

.n; /

elements per row. We study the asymptotic properties of the moments M2s D

.n; / 2s

E Tr.H

/ in the limit when n, s and tend to infinity. Our main result is that the

.n; /

sequence M2sn n with sn D b n c,

> 0 and n D o.n1=5 / is asymptotically

. /



. /



O s gs 0 are determined by an

close to a sequence of numbers nm

O sn n , where fm

explicit recurrence that involves the second and the fourth moments of the random

variables .H .n; / /ij , V2 and V4 , respectively. This recurrent relation generalizes

the one that determines the moments of the Wigner’s semicircle law given by

ms D lim !1 m

O s . /, s 2 N. It shows that the spectral properties of random matrices

at the edge of the limiting spectrum in the asymptotic regime of the strong dilution

essentially differ from those observed in the case of the weak dilution, where the

dependence on the fourth moment V4 does not intervene.



1 Introduction, Main Results and Discussion

Spectral theory of high dimensional random matrices represents an intensively

developing branch of modern mathematical physics that reveals deep links between

probability theory, analysis, combinatorics and other various fields of mathematics

(see monographs [1, 16]). The first studies of spectral properties of random matrices

of infinitely increasing dimensions were started by E. Wigner (see e.g. [25]), where

the ensemble of real symmetric matrices of the form

1

.A.n/ /ij D p aij

n



(1)



was introduced and the limiting eigenvalue distribution of A.n/ , n ! 1 was

determined explicitly. The random matrix entries of A.n/ (1) are given by jointly



O. Khorunzhiy ( )

Université de Versailles - Saint-Quentin, 45, Avenue des Etats-Unis, 78035 Versailles, France

e-mail: oleksiy.khorunzhiy@uvsq.fr

© Springer International Publishing Switzerland 2016

C. Donati-Martin et al. (eds.), Séminaire de Probabilités XLVIII, Lecture Notes

in Mathematics 2168, DOI 10.1007/978-3-319-44465-9_13



347



348



O. Khorunzhiy



independent random variables faij ; i Ä jg that have all moments finite and the odd

moments zero. At present, this ensemble is referred to as the Wigner ensemble

of random matrices. It was proved in [25] that the eigenvalue distribution of A.n/

converges in average as n ! 1 to a non-random limit with the density of the

semi-circle form. At present this convergence is widely known as the semicircle (or

Wigner) law for random matrix ensembles.

The semicircle law was generalized in several directions. One group of generalizations concerns the properties of the probability distributions of elements aij ,

another one is related with the studies of the spectral norm of A.n/ and other local

properties of the eigenvalue distribution at the border of the limiting spectrum or

inside of it.

A large number of works is related with various generalizations of the Wigner

ensemble that involve modifications of the random matrix entries. In the present

paper we study one of such generalizations given by the ensemble of dilute random

matrices. We consider a family of real symmetric random matrices fH .n; / g whose

elements are determined by equality

H .n;



.n; /



/

ij



D aij bij



;



1 Ä i Ä j Ä n;



(2)



where A D faij ; 1 Ä i Ä jg is an infinite family of jointly independent identically

.n; /

distributed random variables and Bn D fbij ; 1 Ä i Ä j Ä ng is a family of jointly

independent between themselves random variables that are also independent from

A. We denote by E D En the mathematical expectation with respect to the measure

P D Pn generated by random variables fA; Bn g. We assume that the probability

distribution of random variables aij is symmetric and denote their even moments by

V2l D E.aij /2l ;

.n; /



Random variables bij

.n; /

bij



l D 1; 2; 3; : : :



are proportional to the Bernoulli ones,



1

Dp



(



1

0;



ıij ;



with probability =n;

with probability 1



=n,



(3)



where ıij D ıi;j is the Kronecker ı-symbol. In the case when the dilution parameter

is equal to n, one gets the Wigner ensemble of real symmetric random matrices

p

An (1). Let us note that the random matrix B.n; / with the entries

bij (3) can

be regarded as the adjacency matrix of the Erd˝os-Rényi random graph [3]. In this

interpretation, the dilution parameter represents the average degree of a given

vertex of the graph.

The initial interest in the dilute versions of Wigner ensemble was motivated by

theoretical physics studies (see for instance, the pioneering works [17, 18] and the

review [14] for more references), where the spectral properties of large systems with

a number of broken interactions were considered. This kind of random matrices is



High Moments of Strongly Diluted Random Matrices



349



also important in the studies of various mathematical models, such as random graphs

[4, 5, 7, 15] and many others.

In the present paper we study the asymptotic behavior of the moments of H .n; /

given by expression

.n; /

M2s



D E



n

X



!

.H .n; / /2s

ii



D E Tr .H .n; / /2s :



iD1



The moment method represents an effective tool of the spectral theory. It is used in

the studies of the spectral properties of large random matrices since the pioneering

works of E. Wigner [25]. In particular, the semicircle law was proved initially by

.n;n/

the convergence of the moments M2s in the limit of infinite n and given s. The

principal idea of the Wigner’s approach is to consider the trace of the product of

random matrices as the sum over the family of trajectories of 2s steps and then to

compute the weights of these trajectories given by the mathematical expectation of

the products of corresponding random variables.

.n;n/

The moments M2s of Wigner random matrices A.n/ (1) in the limit n ! 1

with infinitely increasing s D sn were studied in a long series of papers, where the

eigenvalue distribution at the edge of the limiting spectra was studied in more and

more details [2, 7, 8]. The crucial step has been performed in papers [19, 20], where

the original Wigner’s moment method has got a powerful and deep improvement. In

these studies, the Tracy-Widom law for random matrices A.n/ established in the case

of normally distributed entries aij is shown to be true in the general case of arbitrary

probability distribution of aij [21, 24]. This result is obtained by analysis of the high

.n;n/

moments M2sn in the limit n; sn ! 1 with sn D O.n2=3 /.

The high moments of large dilute random matrices H .n; n / (2) were studied in

[10] in the asymptotic regime when D n D O.n˛ / with 2=3 < ˛ < 1. It

.n; /

was proved that the limiting expression of the moments M2sn n with sn D O.n2=3 /

.n;n/



coincides with that of the moments of the Wigner random matrices M2sn . This fact

can be regarded as an evidence of the universal behavior of the local eigenvalue

statistics for weakly dilute random matrices, i.e. when the dilution parameter is

sufficiently large. In the present paper we study the opposite asymptotic regime of

strongly dilute random matrices, i.e. when the dilution parameter n tends to infinity

as n ! 1 but with much lower range than before, n D O.n˛ / with ˛ < 1=5. We

show that in the limit

n;



n



! 1;



n



D o.n1=5 /; sn D b



n c;



> 0;

.n; /



(4)



where bxc is the integer part of x, the limiting expressions of M2sn n are different

from those obtained for the Wigner random matrix ensemble. This difference is due

.n; /

to the fact that the leading contribution to the moments M2s in the asymptotic

regime (4) is given by the trajectories that generalize in certain sense the Catalan

numbers that describe the moments of the Wigner ensemble. Up to our knowledge,



350



O. Khorunzhiy



these trajectories of the new type were not considered before. Their combinatorial

properties are of their own interest and this fact has strongly motivated the work

presented. In our studies, in particular, we obtain a number of explicit relations that

were not known in the context of random matrices and plane rooted trees (see, for

example, relation (9) below and formulas (106) and (108) of Sect. 5).

To make more compact the formulas we use, everywhere below we refer to the

limiting transition (4) as .n; s; / ! 1. Our main result is given by the following

statement.

Theorem 1 Assume that V2 D 1 and that for all 1 Ä i Ä j the random variables aij

are bounded with probability 1,

jaij j Ä U:



(5)



There exists a constant 0 D 0 .U/ > 0 such that for any given 0 <

following upper bound holds in the limit (4),

1 .n;

M2sn

nt

s

/!1



lim sup

.n;s;



n/



Ä 4e16



V4



<



0



;



the



(6)



where

ts D



.2s/Š

;

sŠ .s C 1/Š



s D 0; 1; 2; : : :

.n; n /



are the Catalan numbers. The moments M2sn

relation,



are given by the following asymptotic



.n; /



M2sn D nm

O s.n n / .1 C o.1//;

. /



where the sequence fm

O s gs

verifies equation



0



(7)



.n; s; / ! 1;



is such that its generating function F .z/ D



F .z/ D 1 C z F .z/



2



C



z2 V4



Â

1



1

zF .z/



(8)

P



. /



s



zs m

Os



Ã4

(9)



. /



with the initial condition m

O 0 D 1.

Remarks

1. We restrict the rate of n by n1=5 (4) not to overload the technical part of

the paper. In fact, it follows from the proof of Theorem 1 that relations (6)

and (8) can be obtained with (4) replaced by the limit n ! 1 such that

1=2

/ (see formula (57) and the discussion below). Moreover, one can

n D o.n

expect that Theorem 1 remains valid in the asymptotic regime when n D n˛

with 0 < ˛ < 2=3. This asymptotic regime is complementary to the one studied



High Moments of Strongly Diluted Random Matrices



351



in [10]. However, in the present paper we are aimed mostly at the lowest rates

of n having a particular interest in the asymptotic regime when n D O.log n/;

n ! 1.

2. In contrast to the technical restriction (4), it is not clear whether condition (5)

can be essentially relaxed, especially in the case of the asymptotic regime when

n D O.log n/; n ! 1 and sn D b

n c. However, a part of the estimates that

concerns the tree-type walks can be proved under considerably less restricted

conditions than (5) (see relation (66) of Sect. 4 below).

. /

3. We will show that the numbers m

O s are uniquely determined and verify the

following upper bound [cf. (6)],

1 ./

m

O Ä 4e3V4 s= :

ts s



(10)



Therefore the generating function F .z/ (9) exists and is bounded in absolute

value for any given > 0. Then it follows from (9) that the limiting function

f .z/ D lim !1 F .z/ exists and verifies the following relation,

f .z/ D 1 C z. f .z//2 :



(11)



This equation has a unique solution

P that determines the generating function of

the Catalan numbers (7), f .z/ D k 0 tk zk .

4. Relation (8) can be rewritten in slightly more precise form. We will show that

there exists a constant C > 0 such that the following relation holds

lim sup

.n;s; /!1



n



nts



.n; n /



M2sn



Á

nm

O s.n n / Ä C e16



V4



(12)



in the limit n; sn ; n ! 1 such that sn D b n c, 0 < Ä 0 and n D o.n1=6 /

(see Sect. 4.2 below). In fact, one can show that the left-hand side of relation (12)

admits the asymptotic expansion in powers of and that the first terms of this

expansion are given by relation

1

1 .n; /

M

D

nts 2s

ts



Â

1

m

O .s / C R.1/

s C o.



1



Ã

/ ;



(13)



where

R.1/

s D



4V42



.s



.2s/Š

V6

C

4/Š .s C 4/Š

.s



.2s/Š

C O.

3/Š .s C 3/Š



1



/



and s; ! 1 are such that s D b c with > 0 (see Sect. 5.2).

. /

5. As we will see, the numbers fm

O s gk 0 of (8) can be regarded as a generalization

of the Catalan numbers tk ; k

0 in the following sense. The Catalan number

tk counts the half-plane rooted trees Tk of k edges. Regarding the chronological



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