3 Proof of Part (b) in the Case α>1
Tải bản đầy đủ - 0trang
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 ˛/ Ä˛; .x/ 1 c˛;
.y t/ '.t/dt
O
:
1
The result follows from the hat version of Lemma 2 and the expression of Ä˛; .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; 1c 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˛
yˇ
1
ˇ
ˇ1
ˇx
Z
0
ˇ
xˇ
yˇ
t˛
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; 1c 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; 1c 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 < 1ı 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
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interval. Bull. Lond. Math. Soc. 28(5), 514–520 (1996)
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Analysis of Stable Processes and Its Extensions. Lecture Notes in Mathematics, vol. 1980
(Springer, Berlin, 2009)
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333–343 (2014)
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395 (1967)
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(1938)
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Math. Szeged. 9, 116–118 (1938)
17. B.A. Rogozin, The distribution of the first hit for stable and asymptotically stable walks on an
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18. M.L. Silverstein, Classification of coharmonic and coinvariant functions for a Lévy process.
Ann. Probab. 8(3), 539–575 (1980)
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(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