Wigner matrices, the moments of roots of Hermite polynomials and the semicircle law
aa r X i v : . [ m a t h . C A ] M a y Wigner matrices, the moments of roots of Hermite polynomialsand the semicircle law
M. Kornyik
E¨otv¨os Lor´and UniversityDepartment of Probability Theory and StatisticsP´azm´any P´eter s´et´any 1/C., H-1117, Budapest, Hungary email :[email protected]
Gy. Michaletzky
E¨otv¨os Lor´and UniversityDepartment of Probability Theory and StatisticsP´azm´any P´eter s´et´any 1/C., H-1117, Budapest, Hungary email :[email protected] 26, 2016
Abstract
In the present paper we give two alternate proofs of the well known theorem that the empiricaldistribution of the appropriately normalized roots of the n th monic Hermite polynomial H n convergesweakly to the semicircle law, which is also the weak limit of the empirical distribution of appropriatelynormalized eigenvalues of a Wigner matrix. In the first proof – based on the recursion satisfied bythe Hermite polynomials – we show that the generating function of the moments of roots of H n isconvergent and it satisfies a fixed point equation, which is also satisfied by c ( z ), where c ( z ) is thegenerating function of the Catalan numbers C k . In the second proof we compute the leading andthe second leading term of the k th moments (as a polynomial in n ) of H n and show that the firstone coincides with C k/ , the ( k/ th Catalan number, where k is even and the second one is givenby − (2 k − − (cid:0) k − k (cid:1) ). We also mention the known result that the expectation of the characteristicpolynomial ( p n ) of a Wigner random matrix is exactly the Hermite polynomial ( H n ), i.e. Ep n ( x ) = H n ( x ), which suggest the presence of a deep connection between the Hermite polynomials and Wignermatrices. Keywords:
Random matrix; characteristic polynomial; semicircle law; moments of roots of Hermitepolynomials
MSC[2010] : 15A52; 60B20; 33C45
In random matrix theory to analyse the behaviour of the eigenvalues of a random matrix one possibilityis to consider the sum of the k th powers of its eigenvalues. This can be done either via analysing thetrace of the k th power of the random matrix, or through the k th moments of the roots of its characteristicpolynomial. One can find many results of the former type (see [2], [11]), but the latter has not yet beenthoroughly investigated ([4], [5], [8]). Since there is an implicit connection between the moments andelementary symmetric polynomials of the roots (Newton’s identities), one can make observations of themoments of roots via examining the coefficients of the characteristic polynomial.Let us introduce the following Definition 1
A random symmetric matrix A = [ a ij ] i,j =1 ,...,n is called a Wigner matrix, if all the elements ( a ij ) ≤ i ≤ j ≤ n are independent with zero mean, the elements on the diagonal are identically distributed,and the off-diagonal elements are identically distributed with finite second moments. Forrester and Gamburd proved in [5] that if A is a Wigner matrix with its off-diagonal elements havingvariance c > E det[ λI − A ] = c n H n ( x/c ) , (1)1here H n ( x ) is the n -th monic Hermite polynomial given by H n ( x ) = ⌊ n/ ⌋ X k =0 ( − k (cid:18) n k (cid:19) (2 k − x n − k . We would like to remark, that in order to get (1) it is sufficient to assume independence of all the freeelements of A , i.e. the independence of a ij for 1 ≤ i ≤ j ≤ n , E [ a ij ] = 0 for 1 ≤ i ≤ j ≤ n and E [ a kl ] = c < ∞ for 1 ≤ k < l ≤ n . Their proof goes per definition, that is computing E det[ λI − A ] = X σ ∈ S n ( − | σ | E n Y i =1 ( λ i δ iσ ( i ) − a iσ ( i ) ) . Note that although the assumptions on the random matrix are not very restrictive, yet the resultingexpectation of the characteristic polynomial is a very specific one, namely the one orthogonal withrespect to the density function of the standard normal distribution. This fact suggests the presence ofan intrinsic connection between Wigner matrices and Hermite polynomials. Hermite polynomials haveanother interesting property; they coincide with the matching polynomial M K n ( x ) of the complete graph K n . The matching polynomial of a graph G = ( V, E ) is defined by M G ( x ) = X k ≥ ( − k m k ( G ) x n − k , where m k ( G ) denotes the number of matchings with exactly k edges and n = | V | .In the first theorem of Section 1 we are going to present a short, direct proof of the well knowntheorem stating that the semicircle law describes the asymptotic distribution of the (normalized) rootsof the Hermite polynomials by showing that the generating function (without computing the actualcoefficients) of the sum of k th power of the roots converges to P ∞ k =0 C k z k = c ( z ), where c ( z ) denotesthe generating function of the Catalan numbers, using a fixed point argument similar to Girko’s idea in[7]. In the second theorem we explicitly compute the leading and the second leading coefficient in n ofthe sum of k th power of the roots and show that the first one is equal to C k/ and the second oneis equal to − (2 k − − (cid:0) k − k (cid:1) ) by using the implicit connection between the moments and elementarysymmetric polynomials of the roots of an arbitrary polynomial (also known as Newton’s identities orVi´eta’s formulae). This result also implies – after proper scaling – the weak convergence of the empiricaldistribution of the scaled roots of H n as n → ∞ to the semicircle law and the convergence rate cannotbe faster than O (1 /n ). Let us now consider the roots of the Hermite polynomials. Denote by ξ ( n )1 , . . . , ξ ( n ) n its zeros and denoteby µ n = n P nj =1 δ λ ( n ) j the empirical distribution determined by the normalized roots, where λ ( n ) j = ξ ( n ) j √ n ,and by M n ( k ) = P nj =1 (cid:16) ξ ( n ) j (cid:17) k the sum of the k th powers. Theorem 1 (See [6])
The limit distribution of the empirical distribution of the roots of the Hermitepolynomial H n , as n → ∞ , is given by the semicircle distribution, that is µ n w −−−−→ n →∞ ρ sc ( x ) dx (2) where ’ w −→ ’ means weak convergence and ρ sc ( x ) = π √ − x · [ − , ( x ) with [ − , ( x ) denoting theindicator function of the set [ − , ⊂ R . Proof:
To prove the weak convergence we apply the methods of moments. First for the sake of thereader we present a direct proof of the following known lemma (see [10]) connecting the moments of theempirical distribution of the roots and the corresponding polynomial.2 emma 1
Let p ( x ) = P nj =0 b j x j be a monic polynomial (i.e. b n = 1 ) with real coefficients, let η , η , . . . , η n denote its roots, let m ( k ) = P nj =1 η kj , and let M ( z ) = P ∞ k =0 m ( k ) z k denote the generating function ofthe sums of powers of the roots. Then M ( z ) = − z b p ′ ( z ) b p ( z ) + n (3) where b p ( z ) = P nj =0 b n − j z j denotes the conjugate polynomial. Proof:
According to the Newton identities one has k X j =0 m ( k − j ) b n − j = ( n − k ) b n − k (4)The following computation is straightforward: M ( z ) b p ( z ) = ∞ X l =0 m ( l ) z l n X j =0 b n − j z j = ∞ X k =0 k X j =0 m ( k − j ) b n − j z k == ∞ X k =0 ( n − k ) b n − k z k = n b p ( z ) − z b p ′ ( z )hence M ( z ) = − z b p ′ ( z ) b p ( z ) + n so the proof is complete. (cid:3) Let us return to the proof of the proposition. Introduce the notation M n ( z ) := P ∞ k =0 M n ( k ) z k . Weare going to show that 1 n M n ( z/ √ n ) → ∞ X k =0 C k z k , for 0 ≤ z ≤ , (5)where C k = k +1 (cid:18) kk (cid:19) is the k th Catalan number.The proof of this claim will be based on the well-known recursive identities of the (probabilists’)Hermite polynomials (similar as in [12]): H ( x ) = 1 H ( x ) = xH n +1 ( x ) = xH n ( x ) − nH n − ( x ) ddx H n ( x ) = nH n − ( x ) . Denoting by b H n ( x ) = x n H n ( x ) the conjugate polynomial it can be easily checked that b H ( x ) = 1 (6) b H ( x ) = 1 (7) b H n +1 ( x ) = b H n ( x ) − nx b H n − ( x ) (8) ddx b H n ( x ) = nx (cid:16) b H n ( x ) − b H n − ( x ) (cid:17) . (9)Since all the roots of H n are no greater in absolute value than 2 q n + (See [13] p. 131. Theorem 6.32.)we obtain that the conjugate polynomials do not vanish in the interval h − √ n , √ n i . Since b H n (0) = 1,they are in fact positive in that interval. This observation combined with equation (8) above implies thatin this interval b H n − ( x ) b H n ( x ) ≤ nx , b H n − ( z/ √ n ) b H n ( z/ √ n ) ≤ z , for | z | ≤ . (10)On the other hand using Lemma 1 equation (9) implies that M n ( z ) = n b H n − ( z ) b H n ( z ) , furthermore M n ( z ) is a monotonically increasing (for z ≥ M n ( k ) = 0 when k is odd).Now b H n − ( x ) b H n ( x ) = 1 + Z x ddy b H n − ( y ) b H n ( y ) dy > x ddy b H n − ( y ) b H n ( y ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) y = x/ , since b H n − ( x ) b H n ( x ) is a positive, convex, monotonically increasing function on R ≥ , hence ddy b H n − ( y ) b H n ( y ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) y = z/ √ n ≤ (cid:18) z − (cid:19) √ nz , for 0 ≤ z ≤ ddy b H n − ( y ) b H n ( y ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) y = z/ √ n = O ( √ n ) . Straightforward computation gives that ddx b H n − b H n ( x ) = nx b H n − ( x ) b H n ( x ) − b H n − ( x ) b H n ( x ) ! + 1 x b H n − ( x ) b H n ( x ) − b H n − ( x ) b H n ( x ) ! hence ddy b H n − ( y ) b H n ( y ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) y = z/ √ n = n √ nz b H n − ( z/ √ n ) b H n ( z/ √ n ) − b H n − ( z/ √ n ) b H n ( z/ √ n ) ! ++ √ nz b H n − ( z/ √ n ) b H n ( z/ √ n ) − b H n − ( z/ √ n ) b H n ( z/ √ n ) ! and so b H n − ( z/ √ n ) b H n ( z/ √ n ) − b H n − ( z/ √ n ) b H n ( z/ √ n ) = O (cid:18) n (cid:19) , for 0 ≤ z ≤ . (11)Now let f n ( z ) := b H n − ( z/ √ n ) b H n ( z/ √ n ) , then equation (8) implies that1 = f n ( z ) − n − n z f n ( z ) f n − r n − n · z ! and from (11) it follows that f n ( z ) − f n ( z ) f n − r n − n · z ! = b H n − ( z/ √ n ) b H n ( z/ √ n ) − b H n − ( z/ √ n ) b H n ( z/ √ n ) == O (cid:18) n (cid:19) . Therefore if for some fixed z in the interval above h ( z ) is a limit point of the sequence f n ( z ) then itsatisfies the following equation: 1 = h ( z ) − z h ( z ) . (12)4n fact 1 = f n ( z ) − n − n z f n ( z ) − n − n z · " f n ( z ) f n − r n − n · z ! − f n ( z ) f n ( z ) − n − n z f n ( z ) + O (cid:18) n (cid:19) . Introducing the notation c ( z ) = P ∞ k =0 C k z k the usual computation gives that1 − zc ( z ) = 1 − ∞ X k =0 z k +1 ( k + 1)! 2 k +1 (2 k − − ∞ X k =0 z k +1 k +2 ( − k (cid:18) k + 1 (cid:19) = √ − z . Thus c ( z ) = −√ − z z which is the smaller solution of the equation 1 = c ( z ) − zc ( z ) , consequently setting h ( z ) = c ( z ) we arrive at (12) (for more details see [9] p. 27-28).Since acording to (10) the sequence f n ( z ) is uniformly bounded, in order to prove the convergence ofthe whole sequence it is enough to prove that f n ( z ) ≤ c ( z ) for n ≥ ≤ z ≤ , implying that c ( z ) is the only limit point of this sequence.For n = 1 one has f ( z ) = 1 ≤ c ( z ). Equation (8) implies that1 = b H n − b H n (cid:18) z √ n − (cid:19) − z b H n − b H n (cid:18) z √ n − (cid:19) == f n (cid:18)r nn − · z (cid:19) − z f n (cid:18)r nn − · z (cid:19) f n − ( z ) . Let us look at the following map ξ η ( ξ ), where1 = η ( ξ ) − z η ( ξ ) ξ, and z ∈ [0 , /
3] is arbitrary, but fixed. Note that the fixed points of η ( ξ ) are the same as the solutions to(12). Expressing η in term of ξ we get η ( ξ ) = 11 − z ξ . Observe that η is a strictly monotonically increasing function on the set ξ < z with η (0) = 1 ≤ c ( z ),and so η ( ξ ) ≤ c ( z ) if ξ ≤ c ( z ). Now put ξ = f n − ( z ), then η = f n ( p n/ ( n − · z ) ≤ c ( z ). Since f n ismonotonically increasing we have f n ( z ) ≤ f n (cid:18)r nn − · z (cid:19) ≤ c ( z )as well. Thus the only accumulation point of ( f n ( z )) n ∈ N is c ( z ). Remember that n M n ( z/ √ n ) = f n ( z ),hence the proof of (5) is complete.Now the convergence of the power series on the interval [0 , ] implies the convergence of the coefficients,so 12 k n k +1 M n ( k ) → ( C k/ k k is even , . Shortly, they tend to the corresponding moments of the semicircle distribution since Z R x k ρ sc ( x ) dx = ( C k/ k k is even , . This concludes the proof of the theorem. (cid:3) The sum of the k th power of the roots of Hermite polynomials In this section we are going to prove a stronger statement than the one in the previous section, namely:
Theorem 2 M n ( k ) is a polynomial in n , where M n ( k ) = 0 when k is odd, while deg n M n ( k ) = k when k is even. In these cases the coefficient of n k/ in M n ( k ) is given by the Catalan number C k/ .In particular, M n ( k ) = ( n k/ C k/ + f ( n ) , k is even; , k is odd, (13) where C k/ = ( kk/ ) k/ and f is a polynomial of degree at most k/ . Before the proof of theorem 2 we state a well known result without proof:
Proposition 1 (See [13] p. 106. eqn. 5.5.4.)
Let us denote by H n ( x ) = P nj =0 a ( n ) j x j the Hermitepolynomial of degree n . Then a ( n ) n − k = ( ( − k/ ( nk ) k !( k/ k/ k is even; k is odd. (14) Proof: (of Theorem 2). First let us note that a ( n ) n − k = 0 when k is an odd number or k > n , thus byinduction we obtain that M n ( k ) = 0 for k = 1 , , . . . . Using this fact let us write Newton’s identities (4)in the following matrix form: . . . . . . a ( n ) n − . . .
0. . . .... . . 0 a ( n ) n − k − . . . . . . a ( n ) n − M n (2) M n (4)...... M n (2 k ) = − a ( n ) n − − a ( n ) n − ...... − ka ( n ) n − k . (15)Since the determinant of the matrix standing on the left hand side is 1 we obtain that M n (2 k ) = det . . . − a ( n ) n − a ( n ) n − . . . − a ( n ) n − . . . . . . a ( n ) n − k − a ( n ) n − k − . . . a ( n ) n − − ka ( n ) n − k . (16)In order to compute the determinant above let us introduce the following function of variable x : A ( k, l ) := det . . . ( x ) l +1 − x ( x − . . . − ( x ) l +3 . . . . . . 1 ... ( − k − ( x ) k − k − ( k − . . . . . . − x ( x − − k − ( x ) l +2( k − ( k − k − (17)for k ≥ , l ≥
1, where ( x ) l = x ( x − · · · ( x − l + 1) and ( x ) = 1.Observe that M n (2 k ) = A ( k,
1) with x = n . Multiply the first column by ( x ) l +1 and subtract it from the6ast one, then the j th element ( j ≥
2) of the last column is given as:( − j − j − ( j − x ) l +2( j − − ( − j − j − ( j − x ) l +1 ( x ) j − == ( − j − j − ( j − x ) j − [( x − j + 2) l +1 − ( x ) l +1 ] == ( − j − j − ( j − x ) j − [ − j − l X h =0 [( x − j + 2) h × ( x − h − l − h ]= ( − j − j − ( j − l X h =0 (cid:2) ( x ) j − h × ( x − h − l − h (cid:3) . due to the fact that l Y j =0 α j − l Y j =0 β j = l X h =0 Y j : j
1) = x and A (0 , l ) = 0 for l ≥ k = 2 and k = 1 as well. Lemma 2 deg A ( k, l ) = k + l , for k ≥ , l ≥ .Proof of the lemma: Trivially deg A (1 , l ) = l + 1 and the highest degree coefficients are positive. Supposethe claim above is true for k − l ≥
1, thendeg A ( k, l ) = deg l +1 X i =1 l Y j = i ( x − j ) A ( k − , i ) = k + l , because by induction the highest degree coefficients of A ( k − , l ) and that of the multipliers ( x − i ) l +1 − i for l = 1 , . . . , l + 1 are positive. This concludes the proof of Lemma 2. (cid:3) In particular deg n M n (2 k ) = deg A ( k,
1) = k + 1. For example, when k = 2 , , x = n it is easyto see that M n (2) = − a n − = n − n = C · n − n (19) M n (4) = 2 n − n + 3 n = C · n − n + 3 n (20) M n (6) = 5 n − n + 32 n − n = C · n − n + 32 n − n (21)7 emark 1 Let us point out that from this it follows that lim n →∞ M n (2 k ) /n k +1 equals to the leadingcoefficient of A ( k, , consequently Theorem 1 implies that this has to be C k . But to provide a self-contained proof we show that it is possible to determine this leading coefficient using a simple graphtheoretic argument. Thus, we are going to prove that the leading coefficient – that is the coefficient of x k +1 – in A ( k,
1) is C k = (cid:0) kk (cid:1) / ( k + 1).Since in the recursive formula for A ( k, l ) the factors for A ( k − , i ) are with leading coefficient one,the leading coefficient of A ( k, l ) can be obtained as the sum of that of A ( k − , , . . . , A ( k − , l + 1).Applying now the recursive formula for the elements A ( k − , i ) and so on, we obtain that the leadingcoefficient of A ( k,
1) is given by the number of A (1 , t ) terms in the representation of A ( k,
1) by theseelements. This question can be translated in the following graph theoretical question:Let us consider the following (directed) graph G := (( Z ≥ ) , E ): For a := ( i , j ) , b := ( i , j ) ∈ ( Z ≥ ) there is an edge from a to b , i.e. ( a, b ) ∈ E if and only if i = i + 1 and j = j − h for h ≥ a ( j ) denote a ’s j th coordinate for j = 1 ,
2. The number of simple (directed) paths from the origin to( k,
0) is exactly the coefficient of x k +1 in A ( k,
1) = M n (2 k ). It can be checked easily that for k = 1 it is1, for k = 2 it is 2, for k = 3 it is 5. Lemma 3
In the graph G the number of simple paths from the origin to ( k, is exactly C k .Proof of Lemma 3: By induction on k . Denote by d k +1 the number of simple paths from the origin to( k + 1 , d := 1, denote by o the origin and denote by P k +1 the collection of (directed) simplepaths from (0 ,
0) to ( k + 1 , P k +1 := { ( o, a , a , . . . , a k +1 ) | a i ∈ ( Z ≥ ) ,a i (1) = i, ( a i − , a i ) ∈ E, ≤ i ≤ k + 1 , a k +1 (2) = 0 } ., For any path P = ( o, . . . , a m ) set t ( P ) := inf { j ≥ | a j = ( j, } and let P k +1 ( i ) := { P ∈ P k +1 | t ( P ) = i + 1 } . Note that t ( P ) = i + 1 means that the first node ofthe path P whose second coordinate is 0 and differs from the origin is a i +1 . It is easy to see that P k +1 = ∗ S ≤ i ≤ k P k +1 ( i ) , hence |P k +1 | = P ki =0 |P k +1 ( i ) | . We want to show that |P k +1 ( i ) | = |P i ||P k − i | = d i d k − i . Note that P = { ( o ) } and P = { ( o, (1 , } . Given two paths P = ( o, a , . . . , a i ) ∈ P i and P =( o, b , . . . , b k − i ) ∈ P k − i one can make a path P ∈ P k +1 ( i ) as follows: let us construct P = ( o, c , . . . , c k +1 )in such a way that c j (1) := a j (1), c j (2) := a j (2) + 1 for 1 ≤ j ≤ i and c j (1) := b j − i − (1) + i + 1 , c j (2) := b j − i − (2) for i < j ≤ k + 1. Remember that b = o . It is trivial that c k +1 = ( k + 1 ,
0) and dueto the definition of the graph ( c j , c j +1 ) ∈ E for 0 ≤ j ≤ k + 1. Since a j (2) ≥ ≤ j ≤ i , so c j (2) ≥ c i +1 = ( i + 1 ,
0) thus we obtain that t ( P ) = i + 1, therefore P ∈ P k +1 ( i ). Nowtake a path P = ( o, c , . . . , c k +1 ) ∈ P k +1 ( i ). Let P = ( o, a , . . . , a i ) be defined by a j (1) := c j (1) and a j (2) := c j (2) − ≤ j ≤ i . Since t ( P ) = i + 1, one has (for i ≥
1) that c j ≥ ≤ j ≤ i , therefore a j (2) ≥ ≤ j ≤ i . Furthermore, for i ≥ c i = ( i,
1) – because of c i +1 = ( i + 1 ,
0) – implying that a i = ( i, P is a valid path (due to the structure of the graph) and its last node is ( i, d i . Define P = ( o, b , . . . , b k − i ) by b j (1) = c j + i +1 (1) − i − b j (2) = c j + i +1 (2). Obviously b = o = (0 , b k − i (1) = c k +1 (1) − i − k − i , b k − i (2) = c k +1 (2) = 0,hence P ∈ P k − i . The number of such paths is d k − i . (Note that in case of i = 0, P = { ( o ) } , P = ( o )and P = ( o, b , . . . , b k ), so one only has to concatenate the two paths in such a way that the coordinatesof the nodes of the path are valid, i.e. c = o , c = (1 , c j +1 = ( b j (1) + 1 , b j (2)) for 1 ≤ j ≤ k .If P ∈ P k +1 (0) is given, then P = ( o, (1 , , c , . . . , c k +1 ) thus P := ( o ) and P = ( o, b , . . . , b k ) with b j (1) = c j +1 (1) − b j (2) = c j +1 (2) for 1 ≤ j ≤ k .) Now it is easy to see that we found a bijectionbetween P k +1 ( i ) and P i × P k − i , hence d k +1 = k X i =0 d i d k − i . (22)8hus the sequence d , d , . . . satisfies the same recursion which is valid for the Catalan numbers. Sinceas we pointed out above the first two terms of these sequences coincide we obtain by induction that d k = C k = (cid:0) kk (cid:1) / ( k + 1) for k ≥
0, thus Lemma 3 is proved. (cid:3)
Summarizing what we know until this point we arrive at A ( k,
1) = C k x k +1 + f ( x )where f is a polynomial of degree k . This concludes the proof our Theorem 2. (cid:3) We would like to point out that this methodology enables us to also compute the coefficient of thesecond highest degree term as well.
Proposition 2
Let A ( k,
1) = C k x k +1 + s k x k + g ( x ) , where g is a polynomial of degree k − at most.Then s k = − (cid:18) k − − (cid:18) k − k (cid:19)(cid:19) . (23) Proof:
First observe that s = 0, due to the identity A (0 ,
1) = x . Next we are going to show that thefollowing recursion holds: s k = k X j =1 ( s k − j C j − + C k − j ( s j − − jC j − )) for k ≥ . (24)Using the notations of Lemma 3 we have P k = k − [ j =0 P k ( j )We showed in Lemma 3 that |P k | is equal to the highest degree coefficient of A ( k,
1) that is C k . Let uswrite on any edge ( a, b ) of the graph the following polynomials:( a, b ) ( b (2) = a (2) − Q b (2)+1 j = a (2)+1 ( x − j ) if b (2) ≥ a (2) (25)and assign the polynomial x to the origin. Using this we can assign polynomials to each path P in thegraph starting at the origin as the product of the polynomials assigned to the edges along the path –denote this by p ( x ; P ) – and the one written on the origin. We obtain xp ( x ; P ). Recursion (18) impliesthat A ( k, l ) equals the sum of the polynomials corresponding to paths leading from the origin to ( k, l − A ( k,
1) = x X P ∈P k p ( x ; P ) = x k X j =1 X P ∈P k ( j − p ( x ; P ) . Let us observe, that the second highest degree coefficients are always negative. Furthermore, for any P ∈ P k ( j −
1) one has that p ( x ; P ) = q ( x ) q ( x ), where the polynomial q ( x ) corresponds to a pathstarting from the origin, ending in ( j,
0) and never touching the x-axis before that, while the polynomial q ( x ) corresponds to the path from ( j,
0) to ( k, q ( x ) coincides with a polynomial corresponding to a path from the origin to ( k − j, A ( k,
1) = x k X j =1 X P ∈P j ( j − Q ∈P k − j p ( x ; P ) p ( x ; Q ) = x k X j =1 X P ∈P j ( j − p ( x ; P ) X Q ∈P k − j p ( x ; Q ) = k X j =1 X P ∈P j ( j − p ( x ; P ) A ( k − j, . (26)Let us recall that in the proof of Lemma 3 we have constructed a bijection between P j − and P j ( j − P j − keeping the origin as a starting point but increasing thesecond coordinates of the other points along the path by one and finally adding a last edge from ( j − ,
1) to( j,
0) we obtained the corresponding trajectory. The map (25) shows that as a result of this construction9ll the roots of the polynomial corresponding to the path in P j − will be increased by 1. Since its degreeis j the sum of the roots increases by j . Taking the summation with respect to all paths in P j − weobtain that the highest degree coefficients of P P ∈P j ( j − p ( x ; P ) and A ( j − ,
1) are equal, while thedifference in the second highest degree coefficient is jC j − , thus equation (24) holds.This recursion leads to the generating function S ( z ) = ∞ X k =1 s k z k = − zc ( z ) ddz ( zc ( z ))1 − zc ( z ) = − z − z c ( z ) == − ∞ X k =1 k X j =1 C k − j j − z k . In order to determine this value more explicitely let us consider the symmetric walk on Z with 2 k − k − . Write the set of possible trajectories as the disjointunion of paths that enter the negative region in the (2 j + 1) th step first with 0 ≤ j ≤ k − k − X j =0 C j k − j − (27)one has that this sum counts the trajectories of the former type, while the number of trajectories of thelatter type is given by (cid:0) k − k (cid:1) (see e.g. [3] p.71), hence s k = − (cid:18) k − − (cid:18) k − k (cid:19)(cid:19) and so the proof is complete. (cid:3) Remark 2
Note that Proposition 2 implies that the convergence rate in Theorem 1 cannot be faster than O (1 /n ) . In the introduction we have seen, that even under general conditions on the random symmetric matrixthe expectation of its characteristic polynomial is the monic Hermite polynomial of appropiate degree.The limiting distribution of the roots of the H n is the semicircle law as it is shown in Theorem 1. Itis also known that the limiting distribution of the eigenvalues of a properly scaled Wigner matrix isgiven by the semicircle law [1] in the same sense as above, hence there is a deep connection betweenthe Hermite polynomials, random symmetric matrices with independent elements and the semicircle law.This suggests that studying the Hermite polynomials and their roots could give us a deeper insight onthe behavior of the eigenvalues of a Wigner random matrix. References [1] L. Arnold. On the asymptotic distribution of the eigenvalues of random matrices.
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