Even Walks and Estimates of High Moments of Large Wigner Random Matrices
aa r X i v : . [ m a t h - ph ] D ec EVEN WALKS AND ESTIMATES OF HIGH MOMENTSOF LARGE WIGNER RANDOM MATRICES
O. Khorunzhiy ∗ and V. Vengerovsky † November 8, 2018
Abstract
We revisit the problem of estimates of the moments m ( n )2 s = E { Tr A sn } ofrandom n × n matrices of the Wigner ensemble by using the approach elaborated byYa. Sinai and A. Soshnikov and further developed by A. Ruzmaikina. We continueto investigate the structure of closed even walks w s and their graphs g ( w s ) thatarise in these studies. One of the key problems here is related with the graphs g ( w s ) that have at least one vertex β that is the tail of a large number of edges.This situation occurs when the corresponding Dyck path (or equivalently, the planerooted tree) has a vertex of large degree; in the opposite case this can happenwhen the self-intersection degree of β is large. We show that there exists one morepossibility; it is given by the case when w s has a large number of open instants ofself-intersections, or more precisely, a large total number of the instants of brokentree structure. Basing on this observation, we modify the technique mentionedabove and prove the estimates of the moments m ( n )2 s n in the limit s n , n → ∞ when s n = O ( n / ). Random matrices of infinite dimensions represent a very rich and interesting subject ofstudies that relates various branches of mathematical physics, analysis, combinatoricsand many others.The spectral theory of large random matrices started half a century ago by E. Wigneris still a source of interesting and challenging problems. An important part of theseproblems concerns the universality conjecture for the local spectral properties of en-sembles of large real symmetric (or hermitian) matrices (see e.g. the monograph [8]). ∗ Universit´e de Versailles - Saint-Quentin, Versailles, FRANCE; e-mail: [email protected] † Mathematical Division, B. Verkin Institute for Low Temperature Physics, Kharkov, UKRAINE n × n realsymmetric random matrices of the form( A n ) ij = 1 √ n a ij , (1 . { a ij , ≤ i ≤ j ≤ n } are jointly independent random variables such that thefollowing conditions are verified E { a ij } = 0 , and E { a ij } = 1 , (1 . E {·} denotes the mathematical expectation. E. Wigner proved that the normalizedmoments of A n converge in the limit n → ∞ lim n →∞ n E { Tr A pn } = ( (2 s )! s ! ( s +1)! , if p = 2 s ,0 , if p = 2 s + 1 (1 . a ij exist and the probability distribution ofrandom variables a ij is symmetric [14, 15].To study the moments (1.3), E. Wigner has interpreted the trace of the product E { Tr A pn } = n X i ,i ,...,i p =1 E (cid:8) A i ,i · · · A i p − ,i (cid:9) (1 . I p = ( i , i , . . . , i p − , i ) [14]. Theset of these sequences can be separated into classes of equivalence that in the case ofeven p = 2 s can be labelled by simple non-negative walks of 2 s steps θ s that startand end at zero. These walks are known as the Dyck paths and the Catalan number C ( s ) = (2 s )! /s !( s + 1)! standing in (1.3) represents the number of all Dyck paths with2 s steps. Obviously, E { Tr A s +1 n } = 0 due to the symmetric distribution of randomvariables a ij and it is sufficient to consider the even moments m ( n )2 s = E { Tr A sn } only.The method proposed by E. Wigner has been used as the starting point in thestudies of the asymptotic behavior of variables m ( n )2 s in the limit when s goes to infinityat the same time as n does [1, 3, 4]. In particular, it was shown in the beginning of80-s that the estimate m ( n )2 s ≤ n (2 s )! s ! ( s + 1)! (1 + o (1)) as s, n → ∞ (1 . s = s n = o ( n / ) as n → ∞ provided random variables a ij are boundedwith probability 1 and their probability distribution is symmetric [3]. The method of[3] uses certain encoding of the paths I s additionally to the Dyck paths representation.2ore precisely, the graph theory approach has been developed in [3], where, in par-ticular, the Dyck paths θ s are considered as the canonical runs over the plane rootedtrees T s = T ( θ s ) of s + 1 vertices.Regarding the problem of estimates of high moments of large random matrices, it isimportant to determine the maximally possible rate of s n such that (1.5) is still valid.Here a breakthrough step was made in paper [10], where (1.5) was shown to be truefor all s n = o ( n / ), n → ∞ in the case when the moments of random variables a ij areof the sub-Gaussian form and the probability distribution of a ij is symmetric. Also theCentral Limit Theorem for the random variable 4 − s (cid:16) m ( n )2 s − E { m ( n )2 s } (cid:17) was proved inthis limit. It was shown that the limiting expressions for the corresponding correlationfunctions do not depend on the particular values of the moments of a ij . This was afirst step toward the proof of the universality conjecture for the Wigner ensembles. Itshould be stressed that in these studies the estimate (1.5) represents a key result thatis crucial for the proof of the Central Limit Theorem.The proof of [10] is based on the Wigner’s approach added by an important notionof the self-intersection of the sequence I s ; in [10] these sequences are called the paths of2 s steps. In certain sense, the self-intersections of the path from the class θ s correspondto gluing of the vertices of the corresponding tree T ( θ s ). Then the self-intersectiondegree of a vertex can be determined as a number of arrivals to this vertex by theedges of this tree. In [10], the set of all paths has been separated into the classesof equivalence according to the number of vertices of self-intersections and the self-intersection degrees of these vertices. This gives the tools to control the number ofpaths in the limit s, n → ∞ .This method was modified in the subsequent paper [11] to prove that (1.5) is validin the limit s n , n → ∞ , s n = o ( n / ). Here the family of self-intersections has beenfurther classified and the notion of the open vertex of two-fold self-intersection hasbeen considered for the first time. The next in turn step has been made in [12] to showthat an estimate of the form (1.5) is true in the limit n → ∞ , s n = O ( n / ) under thesame conditions on the probability distribution of a ij as in [10].The technique of [10, 11, 12] was further developed in paper [9], where the caseof arbitrary distributed random variables a ij with symmetric law of polynomial decaywas considered. It was indicated in [9] that certain estimate of [12] was not establishedin the full extent and a way was proposed to complete the proof. However, the wayproposed in [9] is partly correct and the proof of corresponding estimate suffers in itsown turn from a serious gap.Let us briefly explain the problem. Following [11, 12], it was assumed in [9] thatfor the paths with typical θ s the presence of vertices with large number of steps ”out”is possible only when there is a sufficiently large number of steps ”in”; here the steps”in” and ”out” are meant to be the steps that correspond to the ascending steps of θ s .3owever, this is not the case. The walk can arrive at a given vertex β by the steps thatcorrespond to the descend parts of the Dyck path θ s and bring to β the edges that donot belong the ascending steps ”in”. We call these edges the imported ones and referto the corresponding arriving steps as to the imported cells. At the end of this paper,we present examples of paths with a number of imported cells and edges.We see that the description of the family of paths { I s } of (1.4) should be improved.This is the main subject of the present work. We will see that the studies require anumber of generalizations of the notions introduced in [11]. In particular, the notionof the open vertex of two-fold self-intersection should be generalized for the case ofself-intersections of any degree. Also the further analysis of the open instants of self-intersections leads us to the notion of the instant of broken tree structure. It playseven more important role with respect to the estimates of the moments m ( n )2 s thanthat played by the open vertices of self-intersections. Clearly, these new circumstancesrequire essential modifications of the technique proposed by [9, 12, 11].The paper is organized as follows. In Section 2 we repeat some of the definitions ofnotions introduced [10, 11, 12] and [9] and formulate generalizations of them we need.Then we present our main result about the primary and imported cells. Namely, weprove that the number of imported cells at given vertex β is determined by the self-intersection degree of β and by the total number of broken tree instants of the path.We show that the number of the instants of broken tree structure is bounded by thenumber of open arrival instants and this finally gives us a tool to control the numberof paths that have vertices of large degree.The descriptive part given by Section 2 is followed by Section 3, where the principlesof construction of the set of walks and corresponding estimates are described. Here wemainly follow the lines of [11, 12] with necessary modifications and specifications. InSection 4 we prove estimates of the form (1.5) of the averaged traces E { Tr ( A n ) s n } .Our main result concern the case when the matrix entries are given by bounded randomvariables a ij .The reason for this restriction is two-fold. First, this helps us not to overload thepaper and to present clearer the key points of the technique used. From another side,the description of classes of paths and walks in terms of vertices of self-intersectionsdoes not fit very well the problem of counting the number of multiple edges in thegraphs of these paths and walks. As a result, the technique described can be applied towider classes of random variables a ij while the optimal conditions are far to be reached.At the end of the paper we prove one of the possible results in this direction, where werequire that a finite number of moments exist, E | a ij | q < ∞ , q ≤ q .Finally, in Section 5 we consider several examples of the walks with primary andimported cells and show why the estimates of [9, 12] do not work in the correspondingcases. 4 Even closed walks
As we mentioned above, it is natural to consider (1.4) as the sum of weights E n Tr A sn o = 1 n s n X i ,i ,...,i s − =1 E (cid:8) a i ,i · · · a i s − ,i (cid:9) = 1 n s X I s ∈I s ( n ) Q ( I s ) , (2 . I s can be regarded a trajectory of 2 s steps I s = ( i , i , . . . , i s − , i ) , i l ∈ { , . . . , n } (2 . I s ( n ) denotes the set of all such trajectories; the weight Q ( I s ) is given by theaverage of the product of corresponding random variables a ij . Here and below we omitsubscripts in s n when they are not necessary. In papers [10, 11, 12] the sequences I s are referred as to the paths , so we keep this terminology in the present paper.In the present section we study the paths I s and their graphs and describe partitionsof the set I s ( n ) into the classes of equivalence according to the self-intersection prop-erties of I s . To do this, we give necessary definitions based on those of [9, 10, 11, 12]and consider their generalizations. Then we prove our main technical result about thepaths with primary and imported cells. In (2.2), it is convenient to consider the subscripts of i l as the instants of the discretetime. Introducing variable 0 ≤ t ≤ s , we write that I s ( t ) = i t . We will also say thatthe couple ( t − , t ) with 1 ≤ t ≤ s represents the step number t of the path I s .We determine the set of vertices visited by the path I s up to the instant t U ( I s ; t ) = (cid:8) I s ( t ′ ) , ≤ t ′ ≤ t (cid:9) and denote by |U ( I s ; t ) | its cardinality.Regarding a particular path I s , one can introduce corresponding closed walk w ( I s ; t ),0 ≤ t ≤ s that is given by a sequence of 2 s labels (say, numbers from (1 , . . . , n ) orletters). Also we can determine the minimal closed walk w min ( t ) = w min ( I s ; t ) con-structed from I s by the following recurrence rules:1) at the origin of time, w min (0) = 1;2) if I s ( t + 1) / ∈ U ( I s ; t ), then w min ( t + 1) = |U ( I s ; t ) | + 1;if there exists such t ′ ≤ t that I s ( t + 1) = I s ( t ′ ), then w min ( t + 1) = w min ( t ′ ).One can interpret w min ( t ) as a path of 2 s steps, where the number of each new labelis given by the number of different labels used before increased by one. The followingsequences I = (5 , , , , , , , , w min ( I ) = (1 , , , , , , , . I s with 2 s = 8 and of the corresponding minimalwalk w min ( I ). The set of all possible paths I ( n )2 s can be separated into the classesof equivalence C ( w min ) labelled by the minimal walks w min . We say that I s and I ′ s are equivalent, I s ∼ I ′ s when w min ( I s ) = w min ( I ′ s ). In what follows, we considerthe minimal walks only, so we omit the subscript min in w min and write simply that w s = w ( I s ).Regarding a minimal closed walk of 2 s steps w s , one can consider the graph of thewalk g s = g ( w s ) with the set of vertices V ( g s ) and the set of oriented edges E ( g s );sometimes we will write that g s = ( V , E ).It is natural to accept equality V ( g s ) = U ( w s ; 2 s ). We denote the vertices g ( w s )with the help of the Greek letters α, β, ... . Given a walk w s , one can reconstruct thepaths from C ( w s ) by assigning to the vertices of g ( w s ) different values from the set { , , . . . , n } . This procedure will be considered in more details in Section 3.Let us determine E ( g s ). The edge e = ( α, β ) is present in E ( g s ) if and only ifthere exists an instant t , 0 < t ≤ s such that w s ( t −
1) = α and w s ( t ) = β . In thiscase we will write that e ( t ′ + 1) = ( α, β ). If t ′ is the instant of the first arrival to β ,we say that the vertex β and the edge ( α, β ) are created at the instant t ′ . In general,the graph g ( w s ) is a multigraph because the couple α, β can be connected by severaledges of E ( g s ); these could be oriented as ( α, β ) or ( β, α ). We will denote by | α, β | corresponding non-oriented edges. We can easily pass to the graph ˆ g = ( V , ˆ E ), wherethe set of non-oriented edges ˆ E contains the couples { a, β } such that α and β are joinedby elements of E . In this case we denote the corresponding element of ˆ E by [ α, β ].Obviously, |E ( g s ) | = 2 s . We will say that the number of non-oriented edges | α, β | determines the number of times that the walk w s passes the edge [ α, β ].We denote by m w ( α, β ; t ) the multiplicity of the non-oriented edge [ α, β ], or in otherwords, the number of times that the walk w passes the edge [ α, β ] up to the instant t ,1 ≤ t ≤ s : m w ( α, β ; t ) = (cid:8) t ′ ∈ [1 , t ] : ( w ( t ′ − , w ( t ′ )) = ( α, β ) or ( w ( t ′ − , w ( t ′ )) = ( β, α ) (cid:9) . Certainly, this number depends on the walk w s but we will omit the subscripts w .As we have seen from (2.1), the paths and the walks we consider are closed bydefinition, that is w s (2 s ) = w s (0). There is another important restriction for thepaths and walks we consider. It follows from the fact that the probability distributionof a ij is symmetric: • the weight Q ( I s ) is non-zero if and only if each edge from ˆ E ( w s ) is passed by w s an even number of times.In this case we will say that the path I s and the corresponding walk w ( I s ) are even .We see that our studies concern the even closed paths and even closed walks only. Then6 s is always a multigraph and the following equality holds m w ( α, β ; 2 s ) = 0(mod 2) . (2 . A n are real symmetric. In the case of hermitian matrices corresponding conditionis more restrictive and requires that each edge is passed an even number of times in theway that the numbers of ”there” and ”back” steps are equal. We will call such walksas the double-even walks. It is easy to see that all definitions and statements of thepresent section do not change when switching from the even to the double-even walks.In the present paper, we do not consider the case of hermitian matrices in details.We denote by W s the set of all possible minimal even closed walks of 2 s steps. Thenext subsections are devoted to the further classification of its elements. Given w s ∈ W s , we say that the instant of time t with w ( t ) = β is marked if the walkhas passed the edge [ α, β ] with α = w ( t −
1) an odd number of times during the timeinterval [0 , t ], t ≤ s ; m w ( α, β ; t ) = 1(mod 2) , α = w ( t − , β = w ( t ) . ε α β γ ρ δ Figure 1:
A graph g ( w ) of the walk w with two open and one closed instants of self-intersections In this case we will also say that the step ( t − , t ) and the corresponding orientededge ( α, β ) = ( w ( t − , w ( t )) ∈ E are marked. Other instants of time are referred toas the non-marked ones. 7n Figure 1 we present an example of a minimal walk where the marked instantsand corresponding edges are given in boldface.Given a vertex β of the graph g s , we determine the set of all marked steps( t i , t i + 1) such that w s ( t i ) = β . The set of corresponding edges of the form ( β, γ i ) =( w s ( t i ) , w s ( t i + 1)) ∈ E is called the exit cluster of β ; we denote it by D e ( β, w s ) = D e ( β ). The cardinality |D e ( β ) | = deg e ( β ) is called the exit degree of β . If |D e ( β ) | = 0,then we will say that the exit cluster of β is empty.Each even closed walk w s generates a binary sequence θ s = θ ( w s ) of 2 s elements0 and 1 that correspond to non-marked and marked instants, respectively. It is clearthat θ s represents path of 2 s steps known as the Dyck path [13]. We denote by Θ s the set of all Dyck paths of 2 s steps.Given β ∈ V ( g s ), let us denote by 1 ≤ t ( β )1 < . . . < t ( β ) N ≤ s − w s ( t ( β ) j ) = β . We call t ( β ) j , ≤ j ≤ N the marked arrival instants at β . The non-marked arrival instants at β are defined in obvious manner. We will alsosay that the step ( t ( β ) i − , t ( β ) i ) and the corresponding edge e ( t ( β ) i ) ∈ E are the arrivalstep at β and the arrival edge at β , respectively. If N = 2, then the correspondingvertex is called the vertex of simple self-intersection [10]. If N = k , then we say that β is the vertex of k -fold self-intersection and that the self-intersection degree of β is equalto k ; we denote the self-intersection degree of β by κ ( β ) = κ w s ( β ).As it is mentioned in [12], one has to consider the origin of time t = 0 as the markedinstant of time. This is needed to include the walks of the form (1 , , , , , ,
1) with β = { } and only one marked arrival t (1)1 = 3 into the family of walks with self-intersections. Let us note that such ”hidden” marked instants of time can differ from t = 0 and can be numerous. For example, this happens each time when the walkreturns at its origin with all of the existing edges closed. Summing up, we accept that κ ( α ) ≥ α ∈ V ( g s ).The following definition generalizes the notion of the open vertex of (simple) self-intersection introduced in [11] and used in [9] and [12]. Definition 2.1.
The instant t is called the non-closed (or open) arrival instant at thevertex β ∈ V ( g ( w s )) , if the step ( t − , t ) with β = w s ( t ) is marked and if there existsat least one non-oriented edge [ β, γ ] ∈ ˆ E attached to β that is passed an odd number oftimes during the time interval [0 , t − ; m w ( β, γ ; t −
1) = 1( mod . In this case we say that the edge [ β, γ ] of the graph ˆ g ( w s ) is open up to the arrivalinstant t = t ( β ) , or more briefly that this edge is t -open. The instant t can be also calledthe open instant of self-intersection. Correspondingly, one can define the t -open vertex β of self-intersection of the walk w s . emarks
1. Definition 2.1 remains valid in the case when γ coincides with β = w s ( t ),more precisely in the case when there exists another marked instant t ′ < t such that β = w s ( t ′ −
1) = w s ( t ′ ) and the graph g ( w s ) has a loop at the vertex β . Definition2.1 is also valid in the case when β = w s (0) = w s ( t ).2. Both of the definitions of the open vertex of self-intersection [11] and the openarrival instant are based on the following property: the walk, when arrived at β atsuch a marked instant, has more than one possibility to continue its way with the non-marked step. In the opposite case, when the only one continuation with the non-markedstep is possible, we say that the arrival instant is closed. A vertex of a walk can changethe property to be closed or open several times during the run of the walk. On Figure1 we present an example of the walk where two instants of self-intersection are open(these are t = 4 and t = 8) and one instant of self-intersection t = 15 is closed.Regarding the simple self-intersections only, we see that the definition 2.1 coincideswith the definition of the open vertex of simple self-intersection formulated first in[11, 12] and then used in [9]. However, the definition of the open vertex of simpleself-intersection presented in [9] slightly differs from that of [11]. In [9], the vertex β of the simple self-intersection at the marked instant t is called the open one when theedge ( α, β ) of the first arrival to β is not used again up to the instant t , while in [11]only returns in the direction ( β, α ) are prohibited. Looking at the Figure 1, we see that β = w (2) is the open vertex of the self-intersection at the instant t = 8 accordingto the definition of [11], and is not the open vertex according to the definition of [9].However, as we will see later, this slight divergence in definitions does not alter muchthe estimates one obtains (see subsection 3.4). Given a walk w s , let us consider the following procedure of reduction that we denoteby P : find an instant of time 1 ≤ t < s such that the step ( t − , t ) is marked and w s ( t −
1) = w s ( t + 1); if it exists, consider a new walk w ′ s − = P ( w s ) determinedby a sequence w ′ s = ( w s (0) , w s (1) , . . . , w s ( t − , w s ( t + 1) , . . . , w s (2 s )) . Performing this procedure once more, we get w ′′ s = P ( w ′ s ). We repeat this procedureas many times as it is possible and denote by ¯ W ( w s ) the walk obtained as a resultwhen all of the reductions are performed. Let us note that ¯ W ( w s ) is again a walk thatcan be transformed into the minimal one by renumbering the values of ¯ W ( t ), t ≥ g ( w s ) = g ( ¯ W ( w s )) as it is done in subsection 2.1.9e accept the point of view when the graph ¯ g ( w s ) is considered as a sub-graph of g ( w s ) V (¯ g ( w s )) ⊆ V ( g ( w s )) , E (¯ g ( w s )) ⊆ E ( g ( w s ))and assume that the edges of ¯ g ( w s ) are ordered according to the order of the edges of g ( w s ). Definition 2.2 . Given a walk w s , we consider a vertex of its graph β ∈ V ( g ( w s )) and refer to the marked arrival edges ( α, β ) ∈ E ( g ( w s )) as to the primary cells of w s at β . If β ∈ V ( g ( ¯ W )) with ¯ W = ¯ W ( w s ) , then we call the non-marked arrival edges ( a ′ , β ) ∈ E ( g ( ¯ W )) the imported cells of w s at β . As we will see later, in order to control the exit degree of a vertex of a walk withtypical θ , one needs to take into account the number of imported cells at this vertex.The main observation here is that the presence of the imported cells is closely relatedwith the breaks of the tree structure performed by the walk. To formulate the rigorousstatement, we need to introduce the notion of the instant of broken tree structure thatwill play the crucial role in our studies. Definition 2.3.
Any walk ¯ W = ¯ W ( w s ) contains at least one instant ¯ η such thatthe step (¯ η − , ¯ η ) is marked and the step (¯ η, ¯ η + 1) is not. We call such an instant ¯ η theinstant of broken tree structure (or the BTS-instant of time) of the walk ¯ W . Passingback to the non-reduced walk w s , we consider the edge e ( η ) that corresponds to the edge e (¯ η ) ∈ E ( ¯ W ) and refer to the instant η as the BTS-instant of the walk w s . α γρ β Figure 2:
The graph g ( ¯ W ) of the reduced walk ¯ W = ¯ W ( w ) Examples . Let us consider the Figure 2, where we present the graph of the walk¯ W = ¯ W ( w ) obtained from the walk w given on Figure 1 as a result of four reduc-10ions P . Then each non-marked edge of the graph g ( ¯ W ) represents an imported cellwith respect to the vertices of g ( w s ). One can make the following observations. • The vertex β ∈ V (¯ g ) has one non-marked edge attached to it, so there is oneimported cell at β of g ( w ); this is given by the edge e (5). There are two primarycells at β ∈ V ( g ) given by the edges e (2) and e (8). The root vertex ̺ ∈ V ( g ) has oneimported cell e (14) and the vertex ε ∈ V ( g ) has two primary cells, e (7) and e (15), andno imported cells. • The walk ¯ W has only one BTS-instant, ¯ η = 4 because the edge e (4) is marked andthe edge e (5) is not. The same is true for the non-reduced walk w where η = ¯ η = 4.However, it can happen that in the non-reduced walk w s the marked BTS-instant η isseparated from the corresponding non-marked instant by a tree-type sub-walk that isto be removed during the reduction procedure. We will not give the rigorous definitionof the tree-type sub-walk because we do not use it here in the full extent. • Returning to the graph of w depicted on Figure 1, we can explain the term”imported cell” on the example of the vertex β = w (2). The exit cluster of thisvertex consists of two edges, e (3) and e (6). Regarding the tree T ( w ), we see thatthese edges have different parents in T ( w ); the edge of the tree that corresponds to e (3) has the origin determined by the end of the edge corresponding to e (2) while theedge e (6) is imported at β by the edge e (5) from another part of the tree T ( w ). Wegive more examples of the walks with primary and imported cells at the end of thispaper.Now we will present a proposition that trivially follows from the Definitions 2.1and 2.3. However, it plays so important role in our studies that we formulate it as theseparate statement. Lemma 2.1.
If the arrival instant τ is the BTS-instant of the walk w s , then τ isthe open instant of self-intersection of w s . Given a vertex β ∈ V ( w s ), we refer to the BTS-instants η i such that w s ( η i ) = β as to the β -local BTS-instants. All other BTS-instants are referred to as the β -remoteBTS-instants. Now we can formulate the main result of this section. Lemma 2.2.
Let K ( \ β ) w s be the number of all β -remote BTS-instants of the walk w s .Then the number of all imported cells at β denoted by J w s ( β ) is bounded as follows J w s ( β ) ≤ K ( \ β ) w s + κ w s ( β ) . (2 . Proof.
Let us consider the reduced walk ¯ W = ¯ W ( w s ) and a vertex β ∈ V ( g ( ¯ W )).We introduce the function Λ β ( t ; ¯ W ) determined as the number of t -open edges attachedto β ; Λ β ( t ; ¯ W ) = { i : m ( α i , β ; t ) = 1(mod 2) } W arrivesat β . The following considerations show that this value can be changed by 0 , +2 and − β stays unchanged is possible in two situations.a) The first situation happens when the walk ˆ W leaves β by a non-marked edge andarrives at β by a marked edge. Then the corresponding cell is the primary one and wedo not care about it.b) The second situation occurs when the walk ˆ W leaves β by a marked step ( x, x +1)and arrives at β by a non-marked step ( y − , y ). Then the interval of time [ x + 1 , y − τ when the non-marked step follows immediately after the marked one. It is clear that w s ( τ ) = β andtherefore τ is the β -remote BTS-instant.It should be noted that another such interval [ x ′ + 1 , y ′ −
1] contains another η ′ thatobviously differs from η ; η ′ = η . This is because the any couple of such time intervals[ x + 1 , y − x ′ + 1 , y ′ −
1] has an empty intersection. Then each imported cell ofthe type (Ib) has at least one corresponding β -remote BTS-instant and the sets of theBTS-instants that correspond to different intervals do not intersect.II. Let us consider the arrival instants at β when the value of Λ β is changed.a) The change by +2 takes place when the walk ¯ W leaves β by a marked edge andarrives at β by a marked edge also.b) The change by − W leaves β with the help ofthe non-marked edge and arrives at β by a non-marked edge.During the whole walk, these two different passages occur the same number of times.This is because Λ β = 0 at the end of the even closed walk ¯ W . Taking into accountthat the number of changes by +2 is bounded by the self-intersection degree κ ¯ W ( β ),we conclude that the number of imported cells of this kind is not greater than κ ¯ W ( β ).To complete the proof, we have to pass back from the reduced walk ¯ W ( w s ) to theoriginal w s . Since the number of imported cells of w s and the number of BTS-instantsof w s are uniquely determined by ¯ W ( w s ), and κ ¯ W ( β ) ≤ κ w s ( β ) , then (2.4) follows provided β belongs to g ( ¯ W ) as well as to g ( w s ). If β / ∈ V ( g ( ¯ W )),then J w s ( β ) = 0 and (2.4) obviously holds. Lemma 2.2 is proved. ⋄ Corollary of Lemma 2.2.
Given a vertex β of the graph of w s , the number ofprimary and imported cells L ( β ) at β is bounded L ( β ) ≤ κ ( β ) + K, (2 . where K is the total number of the BTS-instants performed by the walk w s . roof. The number of the primary cells at β is given by the κ ( β ). The number ofthe imported cells J ( β ) is bounded by the sum κ ( β ) + K ( \ β ) , where K ( \ β ) ≤ K . Then(2.5) follows. ⋄ Let us note that relations (2.4) and (2.5) reflect that non-local character of theimported cells that can arise at β due to the BTS-instants that can happen rather”far” from β at any remote part of the walk. In Section 4 we show that these cells can”bring” to β the edges that originally do not belong to the corresponding vertex in theinitial tree T s . This explains the use of the term imported cells for the correspondingedges (or instants of time).The results we have formulated and proved in this section are sufficient to obtainthe estimates we need for the high moments of large random matrices (see Section 4).So, at this stage we terminate the study of the fairly rich and interesting subject givenby the family of even closed walks. In the previous section, we have described the properties of the even closed paths I s based on the notion of the vertices and the instants of self-intersections. In thepresent section we describe the procedure of construction of the set of paths and presentcorresponding estimates. Here we mostly follow the scheme of [10] and [11] added byconsiderations of the vertices of open self-intersections that are not necessary simple. To describe the constructions procedure, we first complete and summarize the descrip-tion of the even walk paths I s started in the Section 2. There a way to separate theset of all paths I s ( n ) into classes of equivalence C ( w s ) was considered.Each class of equivalence C ( w s ) is uniquely determined by an element w s ∈ W s .It is clear that the weights of the equivalent paths are also equal; if I s ∼ I ′ s , then Q ( I s ) = Q ( I ′ s ) = Q ( w s ). Therefore we can rewrite (2.1) in the form E n Tr A sn o = X w s ∈W s |C ( w s ) | · Q ( w s ) (3 . C ( w s ) is determined by the number of vertices ofthe graph g ( w s ); |C ( w s ) | = n ( n − · · · ( n − |V ( g s ) | + 1).Regarding g ( w s ), we determine the partition of V ( g ) into subsets N , . . . , N s ; if κ ( α ) = k , then α ∈ N k . We denote by ν k the cardinality of N k , ν k = |N k | , ν k ≥ . N the subset of of vertices α of g ( w s ) with κ ( a ) = 1, we get obviousequality s = P sk =1 kν k , where ν = |N | .Given a walk w s that has ν k vertices of k -fold self-intersections, 2 ≤ k ≤ s , wesay that it is of the type ¯ ν s = ¯ ν ( w s ) = ( ν , ν , . . . , ν s ). Then we can separate the setof all walks W s into classes of equivalence. We say that two walks w s and w ′ s areequivalent, w s ∼ w ′ s if their types of self-intersections coincide, ¯ ν ( w s ) = ¯ ν ( w ′ s ). Thenumber | ¯ ν s | = s X k =2 ( k − ν k (3 . V ( w s ); namely, |V ( w s ) | = s + 1 − | ¯ ν s | . We consider therearrangement of (3.1) according to the classes of equivalence described above in thenext subsection.Finally, let us recall that the walk w s generates a Dyck path θ s = θ ( w s ) that isin one-by-one correspondence with a sequence of s marked instants Ξ s = ( ξ , . . . , ξ s )such that ξ < ξ < . . . < ξ s and ξ j ∈ { , s − } . As before, here it is convenient toconsider the subscript of ξ j as a sort of the discrete ”time” we denote by τ , 1 ≤ τ ≤ s .Sometimes we will refer to τ as to the instants of τ -time, or more simple as to the τ -instants. Given a value of τ , we say that ξ τ = Ξ s ( τ ).Remembering the definition of the vertex α of k -fold self-intersection of the walk w s , we see that it is determined by an ordered k -plet of variables τ (1) < . . . < τ ( k ) such that the marked instants ξ τ (1) < . . . < ξ τ ( k ) indicate the marked arrival instantsat the vertex α determined by the first arrival α = w s ( ξ τ (1) ). We also observe that theelements α j ∈ N k , 1 ≤ j ≤ ν k are naturally ordered in the chronological way. Thereforethe walk w s generates an ordered set of ordered k -plets that we denote by T ( k ) ( w s ).More generally, we denote the ordered set of ν k ordered k -plets by T ( k ) s ( ν k ) = { ( τ (1) j ( k ) , . . . , τ ( k ) j ( k )) , ≤ j ≤ ν k } , where all values τ ( l ) j ( k ) are distinct. In what follows, we omit variable k in τ ( l ) j ( k ) whenregarding such k -plets. Also, we will use denotations ( τ ′ , τ ′′ ) for the τ -instants of simpleself-intersection.The last observation concerns the fact that each instant of self-intersection can bycharacterized by one more property - is the corresponding arrival instant open or not.So, the variable τ ( l ) j , l ≥ ξ τ ( l ) j . An obvious but important remark is that theopenness of ξ τ ( l ) j depends on the whole pre-history of walk w s ; that is on its sub-walkdetermined by the time interval [1 , ξ τ ( l ) j − .2 Cover the set of walks Let us recall that the walk w s of 2 s steps is determined as a sequence of 2 s + 1 symbols(labels or numbers). To determine a particular walk, one starts with the initial (root)label and then indicates the symbols that appear at the subsequent instants of time t ,1 ≤ t ≤ s .In the case of closed even walk one needs less information. The first conditionimplies relation w s (2 s ) = w s (0) and the second says that, for instance, if w s has noself-intersections, then it is sufficient to indicate the values of w s ( t ) at s instants oftime. Indeed, elementary reasoning shows that this choice of values corresponds to thechoice of one of the Dyck paths from the set Θ s . Then the choice of labels at s instantsof time corresponds to the choice of the marked instants. The values of w s ( t ) at eachnon-marked instant of time are uniquely determined by the rule that the non-markedstep ( β, α ) closes the last marked arrival ( α, β ). This arrival is unique in the case when w s has no self-intersections.The rule described above uniquely determines the even walk by the knowledge of theDyck path θ s and the partition ¯ N s = ( N , N , . . . , N s ). We call this rule the canonicalrun and refer to such a walk as to the tree-like walk.The situation becomes more complicated in the case when we want to determinethe set of all even walks with a number of self-intersections. The simplest example isgiven by the following two walks of 2 s = 8 steps˜ w = (1 , , , , , , , ,
1) and ˘ w = (1 , , , , , , , , θ = (1 , , , , , , ,
0) and the same partitions with N = { , } and N = { } . The walk ˜ w is the tree-like walk but ˘ w is not of thetree-like structure.We denote the set of minimal even walks that have the same θ s and ¯ N s by W ( θ s , ¯ N s ) and denote by ¯ N ( r ) s with r , 0 ≤ r ≤ ν open vertices of simple self-intersections.It is argued in [11] that given θ s , the cardinality of W ( θ s , ¯ N s ) is bounded asfollows; |W ( θ s , ¯ N ( r ) s ) | ≤ W (¯ ν s , r ) = 3 r s Y k =3 (2 k ) kν k . (3 . Lemma 3.1.
Consider a vertex β with the self-intersection degree κ ( β ) = k . Thetotal number of non-marked edges of the form ( β, α i ) is equal to k and at any instantof time ≤ t ≤ s , the number of t -open marked edges attached to β is bounded by k . roof. To prove the first part of this lemma, we denote by In m and In nm thenumbers of marked and non-marked enters at β and by Out m and Out nm the numbersof marked and non-marked exits from β , respectively.The walk is closed and therefore number of enters at β is equal to the number ofexits from β . Since the walk is even, then the number of the marked edges attached to β is equal to the number of non-marked edges attached to β . Corresponding equalities,In m + In nm = Out m + Out nm , and In m + Out m = In nm + Out nm result in 2( In m − Out nm ) = 0. Then Out nm = k .Given t , we consider the numbers In m ( t ) , In nm ( t ) , Out m ( t ), and Out nm ( t ) thatcount the corresponding numbers of edges during the time interval [0 , t − t is theinstant of the exit from β , thenIn m ( t ) + I nm ( t ) = Out m ( t ) + Out nm ( t ) + ϕ, (3 . ϕ = 0 in the cases when β is the root vertex and ϕ = 1 otherwise. The choice ofedges to close by the non-marked exit edge is bounded by the number of marked edgesthat enter and leave β , that is by the number In m + Out m − In nm − Out nm . Takinginto account (3.4), we get inequalityIn m + Out m − In nm − Out nm ≤ m − Out nm ) − ϕ ≤ m − ϕ ≤ k. Lemma 3.1 is proved. ⋄ Lemma 3.1 says that the multiplicative contribution of each vertex β to the estimate(3.3) is bounded by (2 κ ( β ) − ϕ )!!. Let us recall that in the case when the root vertex ̺ has l marked edges ( γ i , ρ ) attached, then we accept that κ ( ̺ ) = l + 1 because of thefirst marked arrival instant at ̺ that is hidden.In the case of κ ( β ) = 2 the bound 4 − ϕ can be immediately improved. If the secondarrival instant at β is closed, then there is only one possibility to continue the walk atthe non-marked instant. If the second arrival instant in open, then we have not morethan three possibility to continue the run at the non-marked departure from β in thecase when β = ̺ and not more than two possibilities in the case when β = ̺ . Thisterminates the proof of (3.3).In fact, the estimate (3.3) can be further improved by the analysis of the numberof BTS-instants at the vertices of high self-intersection degree. This goes out of theframeworks of the present paper. In this subsection we estimate the number of possibilities to choose ν k instants of k -foldself-intersections that is obviously bounded by the number of possibilities to choose the16et of k -plets T ( k ) s ( ν k ). We denote the latter number by | T ( k ) s ( ν k ) | . In this subsectionwe do not do the distinction between the open and the closed arrival instants. Lemma 3.2.
Given any θ s , the number of possibilities the number of possibilitiesto choose ν instants of simple self-intersections is bounded by | T (2) s ( ν ) | ≤ ν ! · s ! ν . (3 . If k ≥ , then the number of possibilities the number of possibilities to choose ν k instants of k -fold self-intersections is bounded by | T ( k ) s ( ν k ) | ≤ ν k ! · s k ( k − ! ν k . (3 . Proof.
Let us start with the proof of (3.5). Here we mostly repeat the computationsof [11]. Regarding the second arrival instant at the vertex β j of simple self-intersection,we have to point out its position τ (2) j among the s marked instants, 1 ≤ j ≤ ν .Denoting for simplicity τ (2) j = l j , we can write that | T (2) s ( ν ) | ≤ X ≤ l 3, we use the estimate of T ( k ) s (1) with k replaced by k − 1. Namely, pointing outthe last arrival τ -instant τ ( k ) , we can write that | T ( k ) s (1) | = s X τ ( k ) = k | T ( k − τ ( k ) − (1) | ≤ s X τ ( k ) = k τ ( k ) − k − ! ≤ s X τ ( k ) = k ( τ ( k ) − · · · ( τ ( k ) − k + 1)( k − ≤ s k − ( k − s X τ ( k ) = k ≤ s k ( k − . (3 . ν k ≥ ν k vertices τ ( k )1 < . . . < τ ( k ) ν k . Using therepresentation described in (3.8) and ignoring the fact that some of the marked instantsare already in use, we get inequality | T ( k ) s ( ν k ) | ≤ X <τ (1) k <...<τ ( k ) νk 0) with two self-intersections determined by the couplesof instants t of time (1 , 4) and (2 , t ′′ = 8 is the open or theclosed instant of self-intersection in dependence on the value of w at the non-markedinstant t = 5.However, the number of choices of the edge e ′ does not depend on the particular19un of the walk and is bounded by the number of ( t ′′ − θ s ( t ′′ ).Taking into account the previous reasoning, we describe the following procedure ofconstruction of the set of walks with closed and open simple self-intersections.1. Given θ s , we choose among s τ -instants the positions of ν second arrivals τ ′′ <. . . < τ ′′ ν at vertices of simple self-intersections. Among them, we choose ν − r E − r S instants that will be the instants of closed self-intersections, and r E and r S instants ofE-open and S-open self-intersections, respectively. We denote r = r E + r S . With thisinformation in hands, we start the run of the walk.2. At the instant τ ′′ we have choose a vertex of the first self-intersection. Then thefollowing estimates are valid: • if ( τ ′ , τ ′′ ) is prescribed to be the closed self-intersection, then there is not morethat τ ′′ − τ ′ ; • if ( τ ′ , τ ′′ ) is prescribed to be the E-open self-intersection, we choose the vertexamong the heads of the edges that are ( ξ τ ′′ − θ s ( ξ τ ′′ ); • if ( τ ′ , τ ′′ ) is prescribed to be the S-open self-intersection, we choose the vertexamong the tails of the ( ξ τ ′′ − θ s ( ξ τ ′′ )of such edges.3. To continue the run if the walk at the instant ξ τ ′′ + 1, we look whether it ismarked or not. If it is marked and is not the instant of self-intersection, we produce anew vertex; if it is the instant of self-intersection, we return to the paragraph 2 givenabove.4. If the instant ξ τ ′′ + 1 is non-marked, and the self-intersection ( τ ′ , τ ′′ ) is theclosed one, then vertex w s ( ξ τ ′′ + 1) is uniquely determined. If ( τ ′ , τ ′′ ) is open, then weconsider all possibilities to choose the vertex w s ( ξ τ ′′ + 1); this produces several sub-walks that correspond to different runs. In the previous subsection we have seen thatthere are not more than three possible runs in the case of simple open self-intersection.Then we proceed with the next step ξ τ ′′ + 2 till we meet the second arrival instant ξ τ ′′ of the second self-intersection. At this stage we redirect ourselves to the paragraph 2.We continue the procedure described until the last vertex of the self-intersection( τ ′ ν , τ ′′ ν ) is determined. In this construction, we take into account that the choice ofvertices of j -th closed self-intersection is bounded by l j − j − 1, where l j is the positionof τ ′′ j in Ξ( θ s ). 20aving all the steps of the construction performed, we obtain not more than 3 r E + r S different runs, and whatever run is chosen, there is not more than( l − · · · ( l ν − ν − · M rθ , M θ = max t θ s ( t )possibilities to choose the instants and vertices of the self-intersections. We see that inthe case of walks with simple self-intersections, we have not more than X ≤ l <... 1. There are two different cases.Let us assume that β is such that the ( ξ τ (3) − τ (2) is possible from the set of cardinality less or equal to θ ( ξ τ (3) ). Since thearrival instant ξ ( τ (2) ) is open, the same concerns the choice of the instant τ (1) . Thenthe number of possibilities to created such a vertex is bounded by M θ .Now let us consider the second possibility when the ( ξ τ (3) − w s ( ξ τ (2) ) but to w s ( ξ τ (1) ). In this case we have no restrictions for the choice of τ (2) but the choice of τ (1) is restricted to the set of cardinality less than θ ( ξ τ (2) ). Thisgives the estimate of the number of choices by τ (3) M θ .Regarding the cases (i)-(iii) and summing over all possible values of τ (3) , we concludethat in the case of triple self-intersection the number of choices to construct a vertexwith one or two open arrival instants is bounded by max (cid:8) sM θ ; s M θ / (cid:9) ≤ s M θ .Now it is not difficult to estimate the number of possibilities to choose the valuesof T s ( ν ; r ) with given number r of open arrival instants at the vertices of tripleself-intersections. Lemma 3.4 Given θ s , the number of possibilities | T (3) s ( ν ; r ) | to construct ν vertices of triple self-intersections with the total number of r open arrival instants isbounded by the following expression; | T (3) s ( ν ; r ) | ≤ ν ! ν r ! · (cid:16) s M θ (cid:17) h r / i s ! ν −h r / i , ≤ r ≤ ν , (3 . here we denoted h r / i = (cid:26) l, if r = 2 l , l + 1 , if r = 2 l + 1 . Proof. First we point out r arrival instants among 2 ν ones; this produces thefactor (cid:0) ν r (cid:1) . Then we set the values of the last arrival τ -instants τ (3)1 < . . . < τ (3) ν .The vertex α ′ j that has at least one open arrival instant can be constructed by notmore than 4 sM θ choices of the values of the arrival τ -instants τ (1) j and τ (2) j , where thefactor 4 estimates the choice of the head or the tail of the open edges. The number ofsuch vertices α ′ j is not less than h r / i .The two first arrival instants τ (1) i and τ (2) i at the vertices that have no open arrivalinstants can be chosen in not more than s / τ -instants,we repeat the computations of (3.9) and obtain (3.12). Lemma is proved. ⋄ In the previous subsections we have described the construction of all walks with thesame θ s that have closed and open instants of simple and triple self-intersections. Thisis important for the estimate of number of walks because the instants of the open self-intersections leave a certain freedom for the walk to continue its run at the non-markedinstants of time. This run can differ from the canonical run of the walk accordingto the tree structure dictated by the corresponding θ s . So, the open self-intersectionrepresent a potential possibility to choose the canonical run or the non-canonical runat the non-marked instants of time. The arrival instants followed by the non-canonicalcontinuation are the BTS-instants considered in Section 2.The BTS-instants are important when counting the number of imported cells atone or another vertex of the walk. So, let us inverse somehow the point of view ofthe previous subsections and look at the set of walks that have a certain number R of BTS-instants. We assume that R = ρ + ρ + . . . + ρ s , where ρ k is the number ofBTS-instants that happen in the vertices β such that κ ( β ) = k .As usual, let us consider first the vertices of simple self-intersections. Supposingthat the walk w s has ρ BTS-instants, we conclude that there are at least ρ openinstants of self-intersections in w s . Then this walk belongs to the class ( ν , r ) with r ≥ ρ .If w s is such that ρ > 0, then there is a number of vertices of triple self-intersectionsthat contain at least one open instant of self-intersection. Clearly, the number of suchvertices is greater than h ρ / i . 23n what follows, we will need the estimates of choices of vertices with κ = 2 and κ = 3 that have at least one open instant of self-intersection. In contrast, for thevertices with κ ≥ w s is such that ρ k > k ≥ 4, we observe that this walk contains at least one vertex of k -foldself-intersection.At this point we terminate the study of such a rich and interesting subject of evenwalks and pass to the estimates of high moments of large random matrices. In this section we use the results of our studies of even walks and prove the estimatesof the moments (1.2) in the asymptotic regime n → ∞ , s = O ( n ). We are restrictedto the simplest case when the random matrix entries are given by bounded randomvariables. Therefore our results represent a particular case of statements formulatedfor the sub-gaussian random variables [12] and random variables having a number ofmoments finite [9]. However, it should be stressed that the arguments presented in [9]and [12] are not sufficient to get the full proof of the estimates even in the simplest caseconsidered here. In the next subsection we address this question in more details.Another important thing to say is that the results of [9, 11, 12] rely strongly on theassumption that the following property of the Dyck paths is true B ( λ ) = lim s →∞ C ( s ) X θ s ∈ Θ s exp ( λM ( s ) θ √ s ) < + ∞ , λ > , (4 . s is the set of all Dyck paths of 2 s steps, C ( s ) = (2 s )! s ! ( s + 1)! , and M ( s ) θ =max ≤ t ≤ s θ s ( t ). This estimate is closely related with the corresponding property ofthe normalized Brownian excursion that is proved to be true (see [2] and referencestherein). This is because the limiting distribution of the random variable M ( s ) θ / √ s considered on the probability space generated by Θ s coincides with that given by thishalf-plane Brownian excursion. We did not find any explicit reference where (4.1) wouldbe proved, but it is widely believed to be true. So, we also prove our statements underthe hypothesis (4.1). The main result of the present section is given by the following statement.24 heorem 4.1 Consider the ensemble of real symmetric matrices A n (1.1) whoseelements ( A n ) ij = 1 √ n a ij are given by a family A n = { a i,j , ≤ i ≤ j ≤ n } of jointly independent identicallydistributed random variables that have symmetric distribution. We assume that thereexists such a constant U ≥ that a i,j are bounded with probability 1, sup ≤ i ≤ j ≤ n | a ij | ≤ U (4 . and we denote the moments of a by V m with V = 1 ; E { a ij } = 1 , E { a mij } = V m . (4 . If s n = µn with µ > , then in the n → ∞ the estimate √ πµ s n · E n Tr ( A n ) ⌊ s n ⌋ o ≤ B (6 µ / ) exp { Cµ } (4 . is true with a constant C that does not depend on n and on V m , m ≥ . Here we denoted by ⌊ x ⌋ , x > x . Not tooverload the formulas, we will omit this sign when no confusion can arise. Then onecan rewrite (4.4) in the following form similar to (1.5) s n ! ( s n + 1)! n (2 s n )! · E n Tr ( A n ) s n o ≤ B (6 µ / ) exp { Cµ } (4 . ′ )that is asymptotically equivalent to (4.4) due to the Stirling formula. Remarks. 1. We denote the limiting transition s n , n → ∞ such that s n = µn by ( s n , n ) µ → ∞ or simply by ( s, n ) µ → ∞ . It can be easily seen from the proof of Theorem 4.1 that theestimate (4.4) is also true in the limit s n , n → ∞ such that s n /n → µ . More generally,one can prove (4.4) in the limit when s n = O ( n ) with µ = lim sup n →∞ s n /n ≥ E { Tr A sn } are tobe replaced by the closed double-even walks. This could lower the estimate (4.4); inparticular, one could replace the coefficient 6 in B (6 µ / ) of (4.4).3. One can compare these results with the non-asymptotic estimates of the momentsof real symmetric (or hermitian) random matrices of the form (4.2) whose elements havejoint Gaussian distribution. Corresponding ensembles are represented by the GaussianOrthogonal Ensemble (GOE) and the Gaussian Unitary Ensemble (GUE) (see e.g. [8]).25t is proved in [6] that the moments of GUE m ( n )2 k are bounded by (cf. (1.5)) n (2 s )! s ! ( s + 1)! (cid:18) α k ( k − k + 1) n (cid:19) for all values of k, n such that k /n ≤ χ , where α > (12 − χ ) − with χ < 12. Then inthis case the estimate (4.4 ′ ) can be replaced by more explicit inequality, say s ! ( s + 1)! n (2 s )! m ( n )2 s ≤ (cid:18) µ − µ (cid:19) , µ < . For more result on the non-asymptotic estimates of the moments of random matrices,see e.g. [7]. Scheme of the proof of Theorem 4.1. We mainly follow the scheme proposed inpaper [9] that slightly modifies the approach of [10, 11, 12] and gives more detailedaccount on the computations involved.The strategy is to consider the natural representation of Tr A sn as the sum overthe set I s ( n ) of all possible paths I s (cf. 2.1) and split this sum into four sub-sumsaccording to the properties of the graphs of the paths I s ∈ I s ( n ).These properties are related with the value of | ¯ ν s | (3.2) and the maximal exit degreeof the graph g ( I s ) = g s ∆( I s ) = max α ∈V ( g s ) deg e ( α ) , where deg e ( α ) = |D e ( α ) | . Namely, we use the same partition of Tr A sn as in [9, 12]given by relation E n Tr A sn o = X l =1 Z ( l )2 s , (4 . • Z (1)2 s is the sum over the set I (1)2 s ⊂ I s ( n ) of all possible paths I s such that | ¯ ν ( I s ) | ≤ C s /n and there is no edges in E ( I s ) passed by I s more than twotimes; • Z (2)2 s is the sum over the set I (2)2 s of all the paths I s such that | ¯ ν ( I s ) | ≤ C s /n and ∆( I s ) ≤ s / − ǫ , ǫ > E ( I s ) passed by I s more than two times; • Z (3)2 s is the sum over the set I (3)2 s of all the paths I s such that | ¯ ν ( I s ) | ≤ C s /n and ∆( I s ) ≥ s / − ǫ ; • Z (4)2 s is a sum over the set I (4)2 s of all the paths I s such that | ¯ ν ( I s ) | ≥ C s /n .26he constant C of (4.4) is chosen according the condition C > C + 36 where C isdetermined in the proof of the estimate of Z (4)2 s ; it is sufficient to take C > eC U ,where C = sup k ≥ k (( k − /k . The appropriate value of ǫ will be determined in theestimate of Z (3)2 s ; it is sufficient to choose 0 < ǫ < / s, n ) µ → ∞ , the sub-sum Z (1)2 s contributes to (4.5) as a non-vanishingterm, while other three sub-sums are of the order o (1). In the following four subsectionswe consider these sub-sums one by one. Let us explain the use of the results of Sections2 and 3 in the proof of the estimates of Z ( i )2 s .In Section 3 we presented the three types of estimates; first we estimatedA) the number of walks with given θ s and ¯ ν s ;then we specified these estimates and studiedB) the number of walks with a number of simple open self-intersections;finally, we estimatedC) the number of walks that have a number of triple self-intersections with openarrival instants.In the present section we use these estimates completed in some cases by the informationaboutD) walks with self-intersections that produce factors V m , m ≥ • to estimate Z (1)2 s , we use mostly parts (A) and (B) described above; here ourcomputations repeat almost word-by-word those of [11, 12]; • to estimate Z (2)2 s , we use the parts (A), (B), and (D); the structure of simple andtriple self-intersection that produce the factors V m is studied in more details thanit is done in [9, 12]; • to estimate Z (3)2 s , we use the items (A), (B), (C), and (D); this is the most compli-cated part of the present section that involves the results of Section 2; here we usethe new ingredient of the structure of even closed walk we called the primary andimported cells; this makes our arguments and computations essentially differentfrom those of [12] and [9]; • to estimate Z (4)2 s , the part (A) is sufficient; here we slightly modify the reasoningof [9] by adding some missing elements of the proof.Let us start to perform the program presented.27 .2 Estimate of Z (1)2 s Taking into account observations and results of Section 3, we can write that Z (1)2 s ≤ X θ ∈ Θ s C s /n X σ =0 X ¯ ν : | ~ν | = σ n ( n − · · · ( n − |V ( g s ) | + 1) n s × ν X r =0 | T (2) s ( ν ; r ; θ s ) | · s Y k =3 | T ( k ) ( ν k ; θ s ) | · W s (¯ ν s ; r ) , (4 . | T (2) s ( ν ; r ) | represents an estimate of the number of possibilities to point out ν vertices of simple self-intersections such that r self-intersections are non-closed (3.11),the variables | T ( k ) ( ν k ; θ s ) | with k ≥ W s (¯ ν s ; r ) is given by (3.3).Remembering that |V ( g s ) | = s + 1 − | ¯ ν | and that σ = P sk =2 ( k − ν k , we can writethe following inequality n ( n − · · · ( n − s + σ ) n s − σ · n σ ≤ n exp ( − s n + sσn ) · s Y k =2 n ( k − ν k . (4 . Lemma 4.1 ([10]). If s < n , then for any positive natural σ the following estimateholds s − σ Y k =1 (cid:18) − kn (cid:19) ≤ exp ( − s n ) exp (cid:26) sσn (cid:27) . (4 . Proof. The proof mainly repeats the one of [10]. We present it for completeness.Elementary computations show that s − σ Y k =1 (cid:18) − kn (cid:19) = exp ( s − σ X k =1 log (cid:18) − kn (cid:19)) = exp − s − σ X k =1 ∞ X j =1 k j jn j ≤ exp ( − s − σ X k =1 kn ) ≤ exp ( − ( s − σ ) n ) ≤ exp ( − s n + sσn ) . Lemma is proved. ⋄ Taking into account results (3.6) and (3.11) of Lemmas 3.2 and Lemma 3.3 andusing the estimate of W n (3.3), we derive from (4.6) with the help of (4.7) the following28nequality Z (1)2 s ≤ n exp ( C s n − s n ) X θ ∈ Θ s C s /n X σ =0 X ¯ ν : | ¯ ν | = σ ν ! s n + 6 sM θ n ! ν × s Y k =3 ν k ! (2 k ) k s k ( k − n k − ! ν k . Passing to the sum over ν i ≥ , i ≥ Z (1)2 s ≤ n exp { C µ } X θ ∈ Θ s exp M θ √ µ √ s + 36 µ + X k ≥ ( C s ) k n k − , (4 . C = sup k ≥ k/ (( k − /k . It is easy to see that the Stirling formula impliesrelation n | Θ s | = nC ( s ) = n (2 s )! s ! ( s + 1)! = 4 s √ πµ (1 + o (1)) , ( s, n ) µ → ∞ . Multiplying and dividing the right-hand side of (4.10) by C ( s ) and using (4.1), weconclude that √ πµ s · Z (1)2 s ≤ B (6 µ / ) · exp { Cµ } , ( s, n ) µ → ∞ , (4 . C > C + 36. This is the estimate for Z (1)2 s we need. Z (2)2 s In the present subsection we study the walks whose weight contains at least one factor V m with m ≥ 2. The first important observation here is that it is sufficient to considerin details the multiple edges attached to the vertices of the self-intersection degree κ ≤ U with the appropriate degree.Also we have to say that here one meets the following general inconvenience of themethod: the classes of equivalence of the paths and the walks are characterized by theset ¯ ν that determines the multiplicities of the vertices of self-intersections. However,this description does not take explicitly into account the multiplicities of the edges.Sometimes this makes the study of the classes of walks whose weight contains thefactors V m , m ≥ .3.1 Simple self-intersections of the first kind Let us consider first the vertices attached to the edges that produce the factors V .Assume that among ν vertices of simple self-intersections there are r open vertices ofself-intersection. We assume also that among ν − r vertices of simple self-intersectionsthere are q vertices β such the following condition is verified: the second arrival to β at the marked instant is performed along the edge oriented in the same direction as theedge corresponding to the first arrival to β . We denote the edge of this first arrival by( α, β ). δ α β γ Figure 3: The marked part of g ( w ) with the simple self-intersections of different types We say that these q simple self-intersections are of the type one or I-type simple self-intersections. On Figure 3 we present the marked edges of the graph of the walk thathas one simple self-intersection ( t ′ , t ′′ ) = (2 , 16) of the I-type.Let us formulate the following elementary proposition. Lemma 4.2. ([10]) Given the position of the second arrival instant τ ′′ of simpleself-intersection of the first kind, there exist not more than ∆ ( α ) possibilities to choosethe first arrival τ -instant τ ′ , where ∆ ( α ) = deg e ( α ) , α = w s ( ξ τ ′′ − . Proof. The proof immediately follows from the observation that the vertex of thesimple I-type self-intersection ( τ ′ , τ ′′ ) is such that the edge of first arrival belongs tothe exit cluster β ∈ D e ( α ) of the sub-walk w [1 ,ξ τ ′′ ] of w s . ⋄ orollary of Lemma 4.2. If one is restricted with the class of walks with given ¯ ν and bounded maximal exit degree ∆( I s ) = max α ∈V ( g s ) deg e ( α ) ≤ d, then the number of possibilities to construct such a walk of r open self-intersectionswith q of the I-type self-intersections is bounded by | T ( ν , r , q ) | · s Y k =3 | T ( k ) s ( ν k ) | · W s (¯ ν ; r ) , (4 . where | T ( ν ; r ; q ) | ≤ (2 sM θ ) r ( sd ) q r ! q ! ( ν − r − q )! · s ! ν − r − q . (4 . Proof. Relation (4.13) corresponds to the choice of the vertices of simple self-intersections. Then (4.12) obviously follows after the reasoning similar to that used inSection 3. The case of simple self-intersections of the first kind is described in [9, 10]. Now let usconsider other possibilities to construct the walks whose weight contains the factors V .First let us assume that among remaining ν − r − q vertices there are p vertices γ that verify the following property: the second arrival at γ at the marked instant isperformed along the edge ˜ e = ( δ, γ ) such that the marked edge ( γ, δ ) = ˆ e already existsin g ( w s ). We say that these simple self-intersections are of the type two, or II-typesimple self-intersections. On Figure 4 we give an example of the walk w s with p = 2and q = 1. We show there the marked edges of the graph g ( w s ).To construct the self-intersection of II-type, the walk has to close the edge ˆ e . The di-rection of this closure indicates two different ways to create the II-type self-intersection.In the first case the closure of ˆ e = ( γ, δ ) is performed in the inverse direction ( δ, γ ).Then the marked edge ˜ e can be created after the marked arrival at δ . We see that thevertex δ is by itself the vertex of a self-intersection. When arrived at δ , the walk has todecide about the choice of the next vertex; there is not more than three marked edgesthat arrive at γ and the number of choices of the vertex γ is bounded by 3. Let usrecall that we consider the edges attached to vertices with κ ≤ Const · s n · n = O ( s /n ).Another possibility is given by the closure of the edge ˆ e = ( γ, δ ) in the direction( γ, δ ). In this case the walk has to perform an open self-intersection before this closure31nd such a II-type self-intersection contributes by the factor of the order M θ /n , where M θ estimates the number of choices of τ ′′ . We do not give the rigorous proofs of thestatements presented above.Now let us consider the situation when p vertices of simple self-intersections of thesecond type are organized into several chains of neighboring self-intersections as it isshown on Figure 4. Regarding one group only that contains l elements in the chain,we see that there is less than 3 s possibilities to produce the last self-intersection at thevertex γ (1) where w s arrives at the second time at the instant t . To produce the secondelement of this chain, the next in turn instant of self-intersection is to be chosen fromthe exit cluster of w ( t ). Therefore the number of possibilities to perform this is notgreater than sup t deg e ( w ( t )) = ∆( I s ) = ∆. Then the total number of possibilities toproduce such a chain is bounded by 3 s ∆ l − ≤ s ∆ l , where we assumed for simplicitythat ∆ ≥ v chains of l i elements with l + . . . + l v = p , then the number ofpossibilities is bounded by s v ∆ p and the number of vertices in the correspondinggraph g ( w s ) is bounded by s − | ~ν | − v . This means that such a configuration entersinto the sum with the factor1 v ! s n ! v X l + ... + l v = p − v (cid:18) ∆ n (cid:19) l · · · (cid:18) ∆ n (cid:19) l ≤ v ! s n ! v X l ≥ (cid:18) ∆ n (cid:19) l v ≤ v ! s ∆ n ! v . Here we have taken into account that ∆ /n ≤ s / /n = o (1) as n → ∞ . V j , j ≥ V . Regarding the simple self-intersections of the typeone, we see that one can add an arrival edge at β and keep the factor V with nochanges. If there are q vertices β with κ ( β ) = 3 of this type, then there is not morethan ( s ∆) q /q ! possibilities to choose the instants to create such a group. This groupenters Z (2)2 s with the factor n − q . Some of the vertices of the chains described abovecan be also the vertices of triple self-intersection. Then each of the factors (∆ /n ) l i given above should be replaced by (cid:18) ∆ n (cid:19) l i l i X u i =0 l i + 1 u i ! (cid:18) sn (cid:19) u i = (cid:18) ∆ n (cid:19) l i (cid:18) sn (cid:19) l i = (cid:18) ∆ n (1 + o (1)) (cid:19) l i . The same concerns each of the v factors s /n .Let us pass to factors V . To consider these, we have to study the vertices of tripleself-intersections or the mixed cases given by the vertices of simple self-intersections of32ype two and/or the vertices of triple self-intersections. Slightly modifying the previousreasonings, it is easy to see that ( V ) Q enters with the factor bounded in these two casesby 1 Q ! s ∆ n + s ∆ n (1 + o (1)) ! Q . The first factor takes into account either the edges whose ends are the vertices of simpleself-intersections of the types I and II or the triple self-intersections of the type one;the second one corresponds to the triple self-intersection of the type two and the chainsof them. Both of these types an be determined by straight analogy with the types ofsimple self-intersections. We do not present the details here.Taking into account q vertices of triple self-intersections described above and as-suming that v chains of edges are constructed with the help of p vertices of triple self-intersections, we can write that the contribution of the vertices of triple self-intersectionsthat produce moments V j with j ≥ q + p + Q )! s ∆ n V + s ∆ n V (1 + o (1)) + s ∆ n V ! q + p + Q . Here the factor 1 + o (1) corresponds to the chains of vertices of triple self-intersectionsof the type two.The factors V can arise due to the presence of triple self-intersections of the specialtype similar to the type two of the simple and triple self-intersections. It is not hardto see that the factor ( V ) P enters together with the number estimated by expression P ! ( s ∆ /n ) P .Gathering the observations of this subsection and using the computations of theprevious subsection, we conclude that Z (2)2 s ≤ n X θ ∈ Θ s C s /n X σ =0 X ¯ ν : | ¯ ν | = σ ν X r =0 ν − r X p + q =0 X Q + P + ν ′ = ν exp ( − s n + C µ ) × ν − p − q − r )! s n ! ν − p − q − r · r ! (cid:18) sM θ n (cid:19) r · q ! (cid:18) s ∆ n V (cid:19) q × p ! p X v =1 v ! s ∆ n V ! v · Q ! s ∆ n V + 2 s ∆ n V + s ∆ n V ! Q × P ! s n ! P V P · I [1 ,s ] ( p + q + P + Q ) · ν ′ ! s n ! ν ′ · ν k ! s Y k =4 C k U k s k n k − ! ν k , (4 . o (1) by 2 and have denoted by I B ( · ) the indicatorfunction I B ( x ) = (cid:26) , if x ∈ B ,0 , if x / ∈ B and by ν ′ the number of vertices from N that are not included into the subsets con-sidered above. Although we could use inequality V m ≤ U m in (4.14), we prefer tokeep the factors V m to indicate clearly the origin of the corresponding factors.Remembering that ∆ ≤ s / − ǫ with ǫ > 0, we repeat the computations of theprevious subsection that lead to (4.10) and deduce from (4.14) inequality √ πµ s Z (2)2 s ≤ exp { ( C + 36) µ } · B (6 µ / ) · (cid:18) exp (cid:26) √ µs ǫ V (1 + o (1)) (cid:27) − (cid:19) . Then √ πµ s Z (2)2 s = o (1) , as ( s, n ) µ → ∞ . (4 . I (2)2 s do not contribute to m ( n )2 s in the limit weconsider. Z (3) We have seen in the previous subsection that the presence of V m in the weight is relatedmainly with the I-type simple self-intersections and therefore with the exit degree of avertex. The exit degree of β is determined as the cardinality of the exit cluster D e ( β )defined in subsection 2.2. In the present subsection we concentrate on the classes ofwalks such that their maximal exit degree is large; ∆ = max β |D e ( β ) | ≥ s / − ǫ .Given θ s ∈ Θ s , we determine the canonical walk w (0)2 s = w ( θ s ) as the walk withoutself-intersections constructed with the help of θ s . Clearly, the graph g ( w (0)2 s ) representsa rooted half-plane tree of s edges T s = T ( w (0)2 s ) = T ( θ s ) introduced in Section 1. Wedetermine the vertices and the exit clusters of the tree T s in the obvious way. L -property of the Dyck paths Given a walk w s , we consider a vertex β of g ( w s ) and denote by 0 < ζ < . . . < ζ L < s the arrival instants at β that represent either primary or imported cells. We determinea partition of the exit cluster D e ( β ) into subsets D ( l ) ( β ) , ≤ l ≤ L , where the elementsof D ( l ) β are given by the marked edges created during the time interval ( ζ l , ζ l +1 ) with ζ L +1 ≡ s . If there is no such marked edges, we say that the corresponding subset D ( l ) ( β ) is empty. The following statement is a simple consequence of the definition ofthe primary and imported cells. 34 emma 4.3 Consider a walk w s and its Dyck path θ ( w s ) with the correspondingtree T ( θ ) . Then the edges of the same subset D ( l ) ( β ) correspond the edges of the tree T ( θ ) that belong to the same exit cluster of T ( θ ) . Denoting by L ′ the number of allsuch exit clusters of T ( θ ) that correspond to D ( l ) ( β ) , ≤ l ≤ L , we have L ′ ≤ L. (4 . Proof. Let us denote by t ( l )1 = t and t ( l )2 = t the instants when the first and thelast elements of D ( l ) ( β ) are created. According to the definition of ζ i , the time interval[ ζ l + 1 , ζ l +1 − 1] contains the non-marked arrival instants at β only, if they exist, andthese arrivals do not represent the imported cells. Then the sub-walk ˜ W = w [ t ,t − starts and ends at β and is of the tree-type structure. This means that after a series ofreductions described in subsection 2.3 this sub-walk can be reduced to the empty sub-walk. Not to overload this paper, we do not present corresponding rigorous definitionsand proofs.Since ˜ W is of the tree-type structure, then the edges of D ( l ) β correspond to the chil-dren of the same parent in T ( θ ). It can happen that the edges of T ( θ ) that correspondto different clusters D ( l ) β and D ( l ′ ) β have the same parent. Lemma 4.3 is proved. ⋄ Now let us introduce an important characterization of the Dyck paths that representa simplified version of the property proposed in [12] and used in [9]. Definition 4.1 . We say that the Dyck path θ ∈ Θ s verifies the L ( m ) -property, ifthere exists α ∈ V ( T ( θ s )) such that deg e ( α ) ≥ m . We denote by Θ ( m )2 s the subset ofDyck paths that verify this property. Let us explain the use of the L -property in the estimate of Z (3)2 s . The set of walksinvolved in Z (3)2 s is characterized by the fact that the graph of each of these walkscontains at least one vertex β such that its exit degree D e ( β ) is greater than d ≥ s / − ǫ .We assume β to be the first vertex of this kind in the chronological order. Let us denoteby N the self-intersections degree of β and by K the total number of the BTS-instantsperformed by the walk. Then one can observe that some of the Dyck paths θ cannotbe used to construct the walks of the type ( d, N, K ) determined.Indeed, it follows from the corollary of Lemma 2.1 that the number L of primary andimported cells at β is bounded by 2 N + K . According to (4.16), the number of parents L ′ = L ′ β in the corresponding tree T is also bounded by 2 N + K . If θ ′ / ∈ Θ ( d/ (2 N + K ))2 s ,then any tree T ( θ ′ ) has no vertices with the exit degree greater than or equal to d/ (2 N + K ). Then obviously deg e ( β ) is strictly less than d .35n [12] it is argued that the subset of Dyck paths Θ ( m )2 s has an exponentially boundedcardinality with respect to C ( s ) = | Θ s | ; | Θ ( m )2 s | ≤ as b C ( s ) exp {− C m } , (4 . a = 1 , b = 2 and C is a constant. In [5] (4.17) is proved with a = 2 , b = 1 and C = log(4 / Z (3)2 s To estimate the sum Z (3) , we determine the values of variables d, N, K and considerthe paths such there exists a vertex β of the corresponding graph of the exit degreedeg e ( β ) = d with the self-intersection degree κ ( β ) = N ; the sum over θ is restrictedto the subset Θ ( d/ (2 N + K ))2 s and according to Lemma 2.1, the structure of the path issuch that the corresponding walk has R open instants of self-intersection with R ≥ K and R = ρ + . . . + ρ s , where ρ k is the number of open arrival instants at the verticesof self-intersection degree k .Repeating the reasoning of the subsection 4.3 leading to the estimate (4.16) andtaking into account the arguments of the previous subsection, we conclude that Z (3)2 s isbounded by the following expression Z (3)2 s ≤ n exp ( C s n − s n ) X θ ∈ Θ s X d ≥ s / − ǫ s X N =1 X K ≥ I Θ ( d/ (2 N + K ))2 s ( θ ) × C s /n X σ =0 ( N ) X ¯ ν : | ¯ ν | = σ σ X R = K X r ... + rs = R r i ≥ r ! · (cid:18) sM θ n (cid:19) r × ν − r )! s n + sdV n (1 + o (1)) ! ν − r · ν − h r / i )! V s n ! ν −h r / i × h r / i ! V s M θ n ! h r / i · s Y k =4 ( k − ν k r k ! · ν k ! C k U k s k n k − ! ν k . (4 . P ( N )¯ ν is taken over the sets ¯ ν such that ν N ≥ 1. The term (cid:0) ( k − ν k r k (cid:1) stands for the choice of r k BTS-instants among ( k − ν k arrival instants atthe corresponding vertices; here we assume that (cid:0) l (cid:1) = δ l, for any integer l ≥ 0. Thefactor 72 = 2 · 36 gives the estimate of the corresponding choices for the vertices oftriple self-intersections multiplied by the estimate that comes from the correspondingpart of W n . 36et us explain the presence of the factors V in (4.18). The last product correspondsto the vertices with κ ≥ 4. Regarding the vertex β of the triple self-intersection, wedenote three marked arrival edges at β by ( α i , β ), i = 1 , , 3. The vertices α i cancoincide between them. However, we see that the vertex β produces that maximalweight in the case when all α i are distinct and each of α i is the head of two markededges ( β, α i ). This gives the factor V ≤ V . Here we have taken into account theagreement that κ ( α ) ≥ α ∈ V ( g s ).Remembering that s = µ / n / , we can write equality U k C k s k − sn k − = U k C k µ k/ n / n ( k − / = δ n H k − n · n ( k − / , where δ n = U C µ / n − / and H n = U C µ / n − / . (4 . a )Summing a part of (4.18) over all possible values of r i , i ≥ 4, we obtain with the helpof multinomial theorem that X r + ... + r s = ρ s Y k =4 r k ! (( k − ν k ) r k · ν k ! C k U k s k n k − ! ν k = σ ρ ρ ! · n σ / s Y k =4 δ ν k n ν k ! · H ( k − ν k n , (4 . b )where we denoted σ = P k ≥ ( k − ν k . Here we have used the obvious estimate ( k − ν k r k ! ≤ r k ! (( k − ν k ) r k . Now we are ready to estimate the right-hand side of (4.18). To better explain theprinciple and the estimates, we split our considerations into three parts. In the firstone we consider sub-sum given by the right-hand side of (4.18) with σ = 0 and r = 0.We denote this sub-sum by ˇ Z (3)2 s . When estimating ˇ Z (3)2 s , we illustrate the main toolsof the present subsection.Then we denote the sub-sum of (4.18) with σ = 0 and r ≥ Z (3)2 s and thesub-sum of (4.18) with σ ≥ Z (3)2 s . Obviously, Z (3)2 s ≤ ˜ Z (3)2 s + ˘ Z (3)2 s . We show that ˜ Z (3)2 s = o (1) and ˘ Z (3)2 s = o (1) as ( s, n ) µ → ∞ in the second and the thirdparts of the proof. This implies the conclusion that Z (3)2 s = o (1) as ( s, n ) µ → ∞ .37 .4.3 The basic case of σ = 0 and r = 0Relation σ = 0 implies equality ρ = 0. Also we observe that the sum over N runsfrom 1 to 3. Remembering that the corresponding sum is denoted by ˇ Z (3) and takinginto account that r = R , we can write the following estimateˇ Z (3) ≤ X d ≥ s / − ǫ C s /n X σ =0 X ν +2 ν = σ σ X K =0 ν X r = K X θ ∈ Θ s r ! · (cid:18) sM θ n (cid:19) r · ν − r )! s n ! ν − r × ν ! V s n ! ν · s · exp ( C s n − s n ) · exp (cid:26) sdn V − C d K (cid:27) · I Θ ( d/ K )2 s ( θ ) . (4 . X = s / (2 n ) and Φ = 6 sM θ /n and consider the sum over r ; S ( K ) ν ( X, Φ) = ν X r = K ν − r )! X ν − r · r ! Φ r = 1 ν ! ν X r = K ν r ! X ν − r Φ r . Multiplying and dividing by h K , we conclude that if h > 1, then S ( K ) ν ( X, Φ) = 1 h K ν ! Φ ν h K + . . . + ν K ! X ν − K Φ K h K ! ≤ h K · ( X + h Φ) ν ν ! . (4 . Z (3) . Regarding the lastline of (4.20), we see the factor exp {− s / (2 n ) } that normalises the sum of the powersof s /n diverging in the limit ( s, n ) µ → ∞ . This sum is in certain sense not completebecause of the presence of powers of asymptotically bounded factors 6 sM θ /n , and thismakes possible to use the corresponding exponentially decaying factor h − K .Regarding the normalized sum over Θ s as a kind of the the mathematical expec-tation E {·} , we use elementary inequality E { f I A } ≤ P ( A )( Ef ) / and deduce from(4.20) with the help of (4.21) the following estimate;ˇ Z (3) ≤ C s · exp { C µ + 36 V µ } · (cid:16) B (12 hµ / ) (cid:17) / · (2 s )! s ! ( s + 1)! × X d ≥ s / − ǫ X K ≥ h K · exp (cid:26) − C d K (cid:27) · exp (cid:26) sdn V (cid:27) . (4 . Z (3)2 s is reduced to the question aboutthe maximum value of the expression F ( K ) = 2 sdn V − C d K − K log h 38s a function of variable K . Function f ( h ) ( x ) = 2 µ / dV n / − C d x − x log h, x ≥ x = q C d log h − 6. This gives the estimate f ( h ) ( x ) ≤ √ d µ / d / V n / − p C log h ! + 6 log h. Remembering that s / − ǫ ≤ d ≤ s , we see that if h = h = exp n µV C o , then F ( K ) ≤ f ( h )max = − (cid:16) n / µ / (cid:17) / − ǫ/ + 6 log h . Returning to (4.22), we conclude that the choice of sufficiently small positive ǫ , say ǫ < / 6, leads to the estimateˇ Z (3)2 s ≤ C s · (2 s )! s ! ( s + 1)! · exp {− µ / n / +6 log h +( C +36 V ) µ }· (cid:16) B (12 h µ / ) + 1 (cid:17) . Then obviously ˇ Z (3)2 s = o (1) in the limit ( s, n ) µ → ∞ . ˜ Z (3)2 s In the present subsection we consider (4.18) with σ = 0 and r + r = R denoted by˜ Z (3)2 s . To estimate this sub-sum, we use the same principle of the estimates as in theprevious subsection: in a part of terms the infinitely increasing factor X is replacedeither by Φ = 6 sM θ /n or by Ψ = 72 V s M θ /n that are asymptotically bounded. Allthat we need here is the following elementary computation, where we use (4.22); X r r R r ≤ ν , r ≤ ν ˜ X ν − r ( ν − r )! · Φ r r ! · Ψ h r / i h r / i ! · Y ν −h r / i ( ν − h r / i )! ≤ (cid:16) ˜ X + h Φ (cid:17) ν ν ! ν X r =0 h R − r · Ψ h r / i h r / i ! · Y ν −h r / i ( ν − h r / i )! ≤ h R · ( X + h Φ) ν ν ! · (cid:0) Y + h Ψ (cid:1) ν ν ! . (4 . Y = 36 V s /n , we get the following esti-mate;˜ Z (3)2 s ≤ n (2 s )! s ! ( s + 1)! exp ( C s n − s n ) X θ ∈ Θ s X d ≥ s / − ǫ s X N =1 X K ≥ I Θ ( d/ (2 N + K ))2 s ( θ ) × C s /n X σ =0 X ν +2 ν = σ X R ≥ K X r + r = R r ! · Φ r × ν − r )! (cid:18) X + 2 sdV n (cid:19) ν − r · Y ν −h r / i ( ν − h r / i )! · Ψ h r / i h r / i ! . (4 . X = X + 2 sdV /n to the last two sums of (4.24) and repeatingcomputations of the previous subsection, we can write that˜ Z (3)2 s ≤ C s · exp {− µ / n / + 6 log h + ( C + 36 V ) µ } · (cid:16) B (24 h µ / ) + 1 (cid:17) . (4 . o (Φ) as ( s, n ) µ → ∞ . Clearly, ˜ Z (3)2 s = o (1) in thelimit ( s, n ) µ → ∞ . ˘ Z (3)2 s In this subsection we obtain the estimate of the right-hand side of (4.18) in the case of σ ≥ 1. It differs from the previous case of σ = 0 by the factor (( k − ν k ) ρ k /ρ k ! thatestimates the number of choices of ρ instants among σ ones. Relation (4.19) showsthat this factor is compensated because of the presence of the vertices with κ ≥ n − σ / of (4.19b).Repeating the arguments and the computations of the previous subsection, we getthe estimate˘ Z (3)2 s ≤ n s X d = s / − ǫ C s /n X σ =0 s X N =1 σ X K =0 X R ≥ K X r + r + ρ = R ( N ) X ¯ ν : σ ≥ (2 s )! s !( s + 1)! × ν − r )! (cid:18) X + 2 sdV n (cid:19) ν − r · Φ r r ! · Y ν −h r / i ( ν − h r / i )! · Ψ h r / i h r / i ! × n σ / · σ ρ ρ ! s Y k =4 δ ν k n ν k ! · H ( k − ν k n × exp ( C µ − s n ) · exp (cid:26) − C d N ′ + K (cid:27) · U N C N s N n N − ! I ( N ) , (4 . N ′ = max { , N } and I ( N ) = I [4 , + ∞ ) ( N ).Let us describe the operations we perform to estimate the right-hand side of (4.24).First we estimate the sum over all possible sets ( ν , ν , . . . , ν s ) as follows; X ν + ... + ν s = σ ≥ s Y k =4 δ ν k n ν k ! · H ( k − ν k n ≤ exp δ n X k ≥ H k − n , (4 . k is obviously convergent. The last expression tends to 1 as( s, n ) µ → ∞ .Next, using an analog of (4.23), we can write that X r + r = R − ρ Φ r r ! · ˜ X ν − r ( ν − r )! · Y ν −h r / i ( ν − h r / i )! · Ψ h r / i h r / i ! ≤ h R − ρ · ν ! ( ˜ X + h Φ) ν · ν ! ( Y + h Ψ) ν . (4 . X R ≥ K h R R X ρ =0 ( hσ ) ρ ρ ! · n σ / ≤ hh K ( h − · e hσ n σ / . (4 . Z (3)2 s ≤ s X d = s / − ǫ n (2 s )! s ! ( s + 1)! exp { C µ + 36 V µ } · (cid:16) B (24 hµ / ) + 1 (cid:17) × C hs h − X N ≥ X K ≥ h K exp (cid:26) sdV n − C d N ′ + K (cid:27) · U N C N s N n N − ! I ( N ) (4 . n such that for n ≥ exp { h } .Similarly to the situation encountered in (4.22), we consider the following functionof two variables F ( N, K ) = 2 sdV n − C d , N ) + K − K log h − N − I ( N ) log n. Denoting N ′′ = ( N − I [4 , + ∞ ) ( N ), we see that the problem of estimate of ˘ Z (3)2 s for largeenough values of n such that log n / ≥ log h is reduced to the study of the maximalvalue of the function˘ F ( N ′′ , K ) = 2 sdV n − C d K + 2 N ′′ − ( K + 2 N ′′ ) log h. f ( h )max determined in the previous subsectionsand we can write that F ( N, K ) ≤ ˘ F ( N ′′ , K ) ≤ − (cid:16) n / µ / (cid:17) / − ǫ/ + 6 log h provided log n ≥ h .Then we deduce from (4.30) inequality˘ Z (3)2 s ≤ n (2 s )! s ! ( s + 1)! · exp { C µ + 36 V µ } · (cid:16) B (24 hµ / ) + 1 (cid:17) × C h s h − · exp n − (cid:16) n / − ǫ/ µ / − ǫ/ (cid:17) + 6 log h o (4 . n such that n ≥ exp { h } . We see that the choice of 0 < ǫ < / s, n ) µ → ∞ .Then ˘ Z (3)2 s = o (1) as ( s, n ) µ → ∞ provided (4.1) holds.This result together with (4.25) shows that the estimate of Z (3)2 s is completed. Z (4)2 s In this part, we follow the general description of the walks, and give the estimate of Z (4)2 s in the way that slightly differs from that presented in [9].Remembering (3.10) and W n , we can write that Z (4)2 s ≤ (2 s )! s ! ( s + 1)! X σ ≥ C s /n X ¯ ν : | ¯ ν | = σ n ( n − · · · ( n − s + σ ) n s s Y k =2 (2 kU s ) k k ! ! ν k . The main difference between this sub-sum and the previous ones is that in (4.8) thefactor exp { sσ } is not bounded for σ ≥ C s /n and exp {− s /n } cannot be used as thenormalizing factor for the terms ( s /n ) σ /σ !. In this case the last expression is boundedby itself. This is the main idea of the proof of the estimate of Z (4)2 s proposed in [9]. Wereconstruct this proof with slight modifications and corrections.Denoting | ¯ ν | = X k ≥ ( k − ν k , we can write that s Y k =2 s kν k = s σ − χ . Z (4)2 s ≤ n (2 s )! σ ! ( s + 1)! X σ ≥ C s /n σ X χ =0 X ¯ ν : | ¯ ν | = σ, | ¯ ν | = χ ( n − · · · ( n − s + σ ) n s − σ · n σ × s σ s χ · ν ! ν ! · · · ν s ! s Y k =2 (cid:16) C U (cid:17) kν k . Multiplying and dividing by σ ! and by ( σ − χ )!, we obtain inequality Z (4)2 s ≤ n (2 s )! σ ! ( s + 1)! X σ ≥ C s /n s σ σ ! n σ σ X χ =0 σ ! s χ ( σ − χ )! × X ¯ ν : | ¯ ν | = σ, | ¯ ν | = χ ( σ − χ )! ν ! ν ! · · · ν s ! s Y k =2 (cid:16) C U (cid:17) kν k . Using the definition of | ¯ ν | , we can rewrite the previous estimate in the form Z (4)2 s ≤ n (2 s )! s ! ( s + 1)! X σ ≥ C s /n σ X χ =0 σ ! s n ! σ × X ¯ ν : | ¯ ν | = σ, | ¯ ν | = χ ( σ − χ )! ν ! ν ! · · · ν s ! s Y k =2 C U σs ! ( k − ν k (cid:16) C U (cid:17) ν k . (4 . σ into two parts: the first sub-sum we denoteby ˙ Z (4)2 s corresponds to the interval C s n ≤ σ ≤ s C U , the second denoted by ¨ Z (4)2 s corresponds to the remaining part σ ≥ s C U .Aiming the estimate of ˙ Z (4)2 s , we use the multinomial theorem in the form X ν + ... + ν s = σ − χ ( σ − χ )! ν ! · · · ν s ! s Y k =2 ( k − ν k ≤ (cid:18) . . . + 12 s − (cid:19) σ − χ ≤ σ − χ . Since 2 C U > 1, we deduce from (4.32) with the help of the Stirling formula that43 Z (4)2 s ≤ n (2 s )! s ! ( s + 1)! X C s /n ≤ σ ≤ s/ (2 C U ) σ ! s n ! σ σ X χ =0 (2 C U ) σ − χ ≤ n (2 s )! s ! ( s + 1)! X σ ≥ C s /n (2 C U ) σ +1 C U − · es nσ ! σ · o (1) √ πσ ≤ n (2 s )! s ! ( s + 1)! · C U C U − X σ ≥ C s /n C U eC ! σ · √ πσ ≤ n (2 s )! s ! ( s + 1)! · C U C U − · p πC s /n · C U eC ! C s /n . (4 . s, n ) µ → ∞ in the case when C ≥ eC U . Now it remains to consider the part that is complementary to ˙ Z (4)2 s ; it is estimatedby the following expression¨ Z (4)2 s ≤ n (2 s )! s ! ( s + 1)! X σ ≥ s/ (2 C U ) σ X χ =0 s n ! σ s χ × X ¯ ν : | ¯ ν | = σ, | ¯ ν | = χ ν ! · · · ν s ! s Y k =2 ( C U ) kν k . (4 . Q k ≥ (2 C U ) ( k − ν k = (2 C U ) σ , we can replace the product of thelast four factors of (4.34) by1 σ ! · C U s n ! σ · σ χ s χ · σ !( σ − χ )! σ χ · ( σ − χ )! ν ! · · · ν s ! s Y k =2 C U k − ! ν k . Using again the Stirling formula, remembering that σ ≤ s , and applying the multino-mial theorem to the sum over ¯ ν , we get that¨ Z (4)2 s ≤ n (2 s )! s ! ( s + 1)! X σ ≥ s/ (2 C U ) r σ π · C U µ / en / ! σ . The last series is obviously o (1) as ( s, n ) µ → ∞ . The estimate of Z (4)2 s is completed.44 More examples of the walks Let us consider a walk W (0)18 that has a number of open simple self-intersections. βα γ γδ βα 47 10 1314 15161718 γ γ δγ γ Figure 4: The marked edges of the graph g ( W (0)18 ) and the full graph g ( W (0)18 ) Basing on the example given on Figure 4, one can easily construct a sequenceof walks of 2 s steps such that their graphs contain a vertex with the exit degree thatinfinitely increases as s → ∞ ; the self-intersection degree of this vertex remains boundedand the corresponding Dyck path θ s is such that its tree T ( θ s ) has no vertices withlarge degree. Below we present one of the possible examples.Regarding the walk W (0)2 s with 2 s = 18 steps depicted on Figure 4, we see that themarked instants are given by the instants of time 1 , , , , , , , , 12 and the vertex β is the vertex of the self-intersection degree 1. This walk contains four BTS-instantsgiven by 3 , , , 11. There are four imported cells at the vertex β determined by theinstants of time 4 , , 10 and 13.Let us modify this walk W (0) to W (1) by adding some ”there-and-back” steps aftereach non-marked arrival ( γ i , β ), i = 1 , , 3; let them be three each time. Then the graphof the walk is added by nine edges and we get the walk W (1) = ( α, β, γ , α, β, γ , γ , β, ε , β, η , β, µ , β | {z } three edges , γ , γ , β, ε , β, η , β, µ , β | {z } three edges , δ, γ , ... } The number of marked steps that leave β denoted by ν n in papers [12] and [9] and bydeg e ( β ) in the present paper is increased by 9 but the self-intersection degree of β isstill equal to one; κ ( β ) = N = 1. 45ertainly, one can consider analogs of W (1) with more vertices of the type γ i , sayten: γ , ..., γ Q , Q = 10. If we add ten triplets of ”there-and-back” edges with vertices ε i , η i , µ i and pass them after each arrival to β by non-marked steps ( γ i , γ ), respectively,then we get a walk W (2) ( Q ) with deg e ( β ) = ν n = 3 Q + Q + 1 = 41 and still κ ( β ) = 1.If Q → ∞ , then deg e ( β ) infinitely increases.Regarding the walks W (2) ( Q ) with arbitrarily large Q , we see that the expressionexp { c deg e ( β ) s n /n } of (4.14 [9]) goes out of the control in the limit of large s becauseit cannot be suppressed by the factor of the form s N /n N − with N = κ ( β ), κ ( β ) = 1.From another hand, the trees T ( W (2) ( Q )) are such that for any value of Q there isno vertices in T ( W (2) ( Q )) with the exit degree greater than 5. This situation is possiblebecause the groups of three edges are imported at β from different parts of the tree T ; freely expressed, these edges grow in g ( W (2) ( Q )) from imported cells. As we haveseen in Section 2, the presence of the imported cells is possible due to the presence ofBTS-instants in the walk. Thus we conclude that the trees under consideration do notverify the L (6)-property (see subsection 4.1) and therefore there is no factors with theexponential estimates of the form (4.17) that would suppress the growth of (4.14 [9]).Therefore the proof of the estimate of Z (3)2 s presented in [9] is not correct. The same istrue with respect to the proof of the Lemma 3 of [11]. The main aim of the paper [9] was to extend the universality results of the papers[11, 12] to the more general case when the entries of the Wigner random matrices a ij (1.1), (1.2) are given by random variables with polynomially decaying probabilitydistribution. We are going to prove the statement that shows that this is indeed thecase. To do this, we need just a slight modification of the proof of Theorem 4.1.In paper [9] the Wigner ensemble is considered, where the random variables a ij verify condition P {| a ij | > x } ≤ x − . However, to make inequality (4.7 [9]) true, onehas to require more restrictive conditions, say with exponent 18 replaced by 36. In thepresent paper, we do not aim the optimal conditions for a ij and prove our statement inthe frameworks of [9] and [12]; modifications of the proof of Theorem 4.1 necessitate alsomore conditions on a ij than those of [9] in the case of polynomially bounded randomvariables. Theorem 6.1 Let us consider the random matrix ensemble described in Theorem4.1 with the bound (4.2) replaced by the following condition; E | a ij | q < ∞ for all q ≤ q = 76 . Then the estimate (4.4) is true in the limit ( s, n ) µ → ∞ , wherethe constants do not depend on particular values of the moments E a kij , k ≥ . roof. By the standard approach of the probability theory, we introduce the trun-cated random variables ˆ a ij = (cid:26) a ij , if | a ij | ≤ U n ;0 , if | a ij | > U n , (6 . U n = n α with α = 1 / 25. Then we consider the random matrices ˆ A ij = ˆ a ij /n / and write down equality E n Tr ˆ A sn o = X l =1 ˆ Z ( l )2 s , where ˆ Z ( l )2 s are determined exactly as it is done in (4.5). We are going to show thatthese sub-sums admit the same asymptotic estimates as Z ( l )2 s of (4.5).The first sub-sum ˆ Z (1)2 s is estimated as Z (1)2 s (4.11) with no changes.To estimate Z (2)2 s , we repeat the reasonings of subsection 4.3.1 that lead to inequality(4.14) and (4.15). The estimate Z (2)2 s = o (1) as ( s, n ) µ → ∞ is valid due to relations C k U kn s k n k − = C k µ k/ n k/ k/ n k − = C k µ k/ n − k/ → , n → ∞ (6 . k ≥ Z (3)2 s , we introduce a slight modification of the computations used toestimate Z (3)2 s . All that we need here is to redefine the variables δ n and H n of (4.19).We rewrite (4.19a) in the form C k U kn n / n ( k − / = C µ / n α +2 / n − β C µ / n α n (1 − β ) / ! k − n ( k − β/ = ˆ δ n ˆ H k − n n ( k − β/ . (6 . α + ≤ − β , then ˆ δ n = O (1) and ˆ H n → 0. The choice of β = 1 / 75 leads to thevalue α = 1 / 25 imposed in (6.2). We see that the value α = 1 / 24 represents the lowerbound for α in the approach developed.Then we can use the analogue of (4.19b) with n σ / replaced by n σ / . All othercomputations that lead to the estimate of Z (3)2 s can be repeated as they are.The estimate of ˆ Z (4)2 s requires somehow more work. To estimate this sub-sum, letus prove the following auxiliary statement. Lemma 6.1 Given any walk w s of the type ¯ ν , the weight Q ( w s ) (2.1) is boundedas follows; Q ( w s ) ≤ s Y k =2 (cid:16) V U k − n (cid:17) ν k . (6 . roof. Let us consider a vertex γ with κ ( γ ) ≥ g ( w s ) = ( V , E )and color in certain color the first two marked arrival edges at γ and their non-markedclosures. Passing to another vertex with κ ≥ 2, we repeat the same procedure and finallyget 4 P sk =2 ν k colored edges. Clearly, it remains 2 ν +2 P sk =2 ( k − ν k non-colored (grey)edges in E ( g ). Let us remove from the graph ( V , E ) all grey edges excepting the markededges e ′ j whose heads are the vertices of N ; also we do not remove the closures of theseedges e ′ j . We denote the remaining graph by g ◦ = ( V , E ◦ ). The number of the edgesremoved from g is greater or equal to 2 P sk =2 ( k − ν k . When estimating the weight of w s , we replace corresponding random variables by non-random bounds U n when theremoved marked edges end at the vertices with κ ≥ 2. Then we can write that Q ( w s ) ≤ U k − ν k n · Q ◦ ( w s ) , where Q ◦ ( w s ) represents the product of the mathematical expectations of the randomvariables associated with the edges of E ◦ . Obviously, we did not replace by U n thoserandom variables that give factors V = 1.In the remaining (possibly non-connected) graph g ◦ the set V contains a subset V ◦ such that if β ∈ V ◦ , then β is the head of of two colored marked edges. Clearly, |V ◦ | = P sk =2 ν k . Regarding a head β ∈ V ◦ of two marked colored edges, we see thattheir tails α ′ and α ′′ can be either equal or distinct.Let us consider first the case when the tails are distinct, α ′ = α ′′ . Then each of themulti-edges ( α ′ , β ) and ( α ′′ , β ) can produce factors V or V in dependence how manyedges arrive at α ′ and α ′′ from β . Then the contribution of the vertex β to Q ◦ ( w s ) isbounded by V ≤ V .Now let us consider the case when the tails are equal, α ′ = α ′′ = α . Then themulti-edge | α, β | produces the factors equal to either V or V or V in dependence ofhow many marked edges of E ◦ arrive at α from β . This can be either one grey edge ortwo colored edges. Since 1 ≤ V ≤ V ≤ V , then the contribution of this vertex β to Q ◦ ( w s ) is bounded by V ≤ V . Here we have used inequality κ ( α ) ≥ , α ∈ V ( g s ).Collecting the contributions of all vertices of V ◦ , we get the estimate Q ◦ ( w s ) ≤ s Y k =2 V ν k . It is clear that we do not need to consider those vertices of V \ V ◦ that are the headsof non-marked edges because the contributions of the corresponding random variablesis already taken into account. Lemma 6.1 is proved. ⋄ Now it is easy to complete the estimate of ˆ Z (4)2 s . Indeed, formula (4.32) can berewritten in the form 48 Z (4)2 s ≤ n (2 s )! s ! ( s + 1)! X σ ≥ C s /n σ X χ =0 σ ! s n ! σ × X ¯ ν : | ¯ ν | = σ, | ¯ ν | = χ ( σ − χ )! ν ! ν ! · · · ν s ! s Y k =2 C U n σs ! ( k − ν k (cid:16) C V (cid:17) ν k . (6 . C s /n ≤ σ ≤ s/ (2 C U n ). Using the multinomialtheorem and the Stirling formula, we get from (6.5) the following analog of the estimate(4.32); ˙ˆ Z (4)2 s ≤ n (2 s )! s ! ( s + 1)! X C s /n ≤ σ ≤ s/ (2 C U n ) σ ! s n ! σ σ X χ =0 (2 C V ) σ − χ ≤ n (2 s )! s ! ( s + 1)! · C V C V − · p πC s /n · eC V C ! C s /n . This expression is o (1) in the limit ( s, n ) µ → ∞ provided C ≥ eC V .To estimate the sub-sum ¨ˆ Z (4)2 s that corresponds to the interval σ ≥ s/ (2 C U n ), werepeat word by word the computations presented at the end of Section 4 (see formulas(4.34) and (4.35)).Summing up the arguments presented above, we see that the following analog of(4.4) √ πµ s E n Tr ˆ A sn o ≤ B (6 µ / ) exp { (36 + C ) µ } (6 . s, n ) µ → ∞ .By the standard arguments of the probability theory, we have P { A n = ˆ A n } ≤ − (cid:0) − n − q α E | a ij | q (cid:1) n ( n +1) / = O ( n − − δ ) , δ > n → ∞ . Then by the Borel-Cantelli lemma, P { A n = ˆ A n infinitely often } = 0 . (6 . 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