Bulk asymptotics of skew-orthogonal polynomials for quartic double well potential anduniversality in the matrix modelSaugata Ghosh Citation: Journal of Mathematical Physics 50, 063515 (2009); doi: 10.1063/1.3093266 View online: http://dx.doi.org/10.1063/1.3093266 View Table of Contents: http://scitation.aip.org/content/aip/journal/jmp/50/6?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Random matrix models, double-time Painlevé equations, and wireless relaying J. Math. Phys. 54, 063506 (2013); 10.1063/1.4808081 Moments of ratios of characteristic polynomials of a certain class of random matrices J. Math. Phys. 50, 043518 (2009); 10.1063/1.3119483 Double scaling limit for matrix models with nonanalytic potentials J. Math. Phys. 49, 033501 (2008); 10.1063/1.2884578 Random matrices, nonbacktracking walks, and orthogonal polynomials J. Math. Phys. 48, 123503 (2007); 10.1063/1.2819599 Chern-Simons matrix models and Stieltjes-Wigert polynomials J. Math. Phys. 48, 023507 (2007); 10.1063/1.2436734

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Bulk asymptotics of skew-orthogonal polynomials forquartic double well potential and universality in the matrixmodel

Saugata Ghosha�

F-252 Sushant Lok II, Gurgaon 122002, India

�Received 21 January 2009; accepted 6 February 2009; published online 26 June 2009�

We derive bulk asymptotics of skew-orthogonal polynomials �m���, �=1, 4, defined

with respect to the weight exp�−2NV�x��, V�x�=gx4 /4+ tx2 /2, g�0 and t�0. Weassume that as m ,N→�, there exists an ��0 such that �� �m /N���cr−�, where�cr is the critical value that separates skew-orthogonal polynomials with two cutsfrom those with one cut. Simultaneously we derive asymptotics for the recursivecoefficients of skew-orthogonal polynomials. The proof is based on obtaining afinite term recursion relation between skew-orthogonal polynomials and orthogonalpolynomials and using asymptotic results of orthogonal polynomials derived byBleher and Its �“Semiclassical asymptotics of orthogonal polynomials, Riemann-Hilbert problems, and universality in the matrix model,” Ann. Math. 150, 185�1999��. Finally, we apply these asymptotic results of skew-orthogonal polynomialsand their recursion coefficients in the generalized Christoffel–Darboux formula�Ghosh, S., “Generalized Christoffel-Darboux formula for skew-orthogonal poly-nomials and random matrix theory,” J. Phys. A 39, 8775 �2006�� to obtain leveldensities and sine kernels in the bulk of the spectrum for orthogonal and symplecticensembles of random matrices. © 2009 American Institute of Physics.�DOI: 10.1063/1.3093266�

I. INTRODUCTION

Skew-orthogonal polynomials are useful in the study of orthogonal ��=1� and symplectic��=4� ensembles of random matrices.1–12 In this paper, we derive asymptotics of skew orthogonalfunctions m

����x� and m����x� and their recursion coefficients,8,9 defined with respect to the weight

w�x� = exp�− 2NV�x��, V�x� =gx4

4+

tx2

2, g � 0, t � 0. �1.1�

Here, 2N is a large parameter, which, in the context of random matrix theory, is the size of thematrices.

We define skew-orthogonal functions,

n����x� =

1

�gn���

�n����x�exp�− NV�x��, �n

����x� = �k=0

n

ck�n,��xk, � = 1,4, �1.2�

n�4��x� ª n�

�4��x�, n�1��x� ª �

Rn

�1��y���x − y�dy, ��r� =�r�2r

, n � N , �1.3�

where gn��� are normalization constants. They satisfy skew-orthonormality relations,

a�Electronic mail: saugata135@yahoo.com.

JOURNAL OF MATHEMATICAL PHYSICS 50, 063515 �2009�

50, 063515-10022-2488/2009/50�6�/063515/7/$25.00 © 2009 American Institute of Physics

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�R

n�1��x�m

�1��x�dx = Zn,m, �R

n�4��x�m

�4��x�dx =Zn,m

2, Z = 0 1

− 1 0 � ¯ , n,m � N ,

�1.4�

where Zn,m are elements of the semi-infinite block-diagonal antisymmetric matrix Z withZ2n,2n+1=−Z2n+1,2n=1, and 0 otherwise.

Using these skew-orthogonal functions, we study the corresponding random matrix model,

P�,N�H�dH =1

Z�Nexp�− �2N Tr V�H���dH, � = 1,4, �1.5�

where the matrix function V�H� is a double well quartic polynomial of H and

Z�N ª �H�M2N

�exp�− �2N Tr V�H���dH = �2N� ! �

j=0

2N−1

gj�. �1.6�

Here, M2N��� is a set of all 2N�2N real symmetric ��=1� and quaternion real self-dual ��=4�

matrices. dH is the standard Haar measure.To study statistical properties of such matrix models, we need to study certain kernel

functions,5

S2N����x,y� ª �

j,k=0

2N−1

Zj,k j����x�k

����y� .

For quartic potential, �=1, this is given by8

�x − y�S2N�1��x,y� = R2N−4,2N

�1� �2N−3�1� �x�2N

�1��y� − �x ↔ y�� + R2N−2,2N+2�1� �2N−1

�1� �x�2N+2�1� �y� − �x ↔ y��

− R2N−3,2N+1�1� �2N−4

�1� �x�2N+1�1� �y� − �x ↔ y�� − R2N−1,2N+3

�1� �2N−2�1� �x�2N+3

�1� �y�

− �x ↔ y�� + R2N−2,2N�1� �2N−1

�1� �x�2N�1��y� − �x ↔ y�� − yP2N−3,2N

�1� �2N−4�1� �y�2N

�1��x�

− �x ↔ y�� − �R2N−1,2N+1�1� �2N−2

�1� �x�2N+1�1� �y� − �x ↔ y��

− yP2N−1,2N+2�1� �2N−2

�1� �y�2N+2�1� �x� − �x ↔ y��� − yP2N−2,2N+1

�1� �2N−1�1� �x�2N+1

�1� �y�

− �x ↔ y�� + yP2N−1,2N�1� �2N−2

�1� �x�2N�1��y� − �x ↔ y�� , �1.7�

where the recursion coefficients Pj,k�1� and Rj,k

�1� are defined as

j�1��x� ª �

k

Pj,k�1�k

�1��x�, x j�1��x� ª �

k

Rj,k�1�k

�1��x� . �1.8�

For �=4, j�1��x� and j

�1��x� are replaced by j�4��x� and j

�4��x�, respectively.8

To study the asymptotic behavior of these skew-orthogonal functions and their recursioncoefficients Pj,k

��� and Rj,k���, we expand the skew-orthogonal functions m

����x� in a suitable basis oforthogonal functions. Using skew orthonormality, we obtain finite term recursion relations be-tween skew-orthogonal and orthogonal functions. We solve them to obtain compact relations �Eqs.�1.17� and �2.6�� between skew-orthogonal functions and their orthogonal counterparts. Usingthese relations and the asymptotic properties of orthogonal functions,13 we derive asymptotics ofskew-orthogonal functions. Finally, we apply them in the generalized Christoffel–Darboux �GCD�formula �1.7� to study the corresponding matrix models.

II. Skew-orthogonal functions and orthogonal ensemble

Let us define normalized orthogonal function j�2��x� as

063515-2 Saugata Ghosh J. Math. Phys. 50, 063515 �2009�

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j�2��x� =

�xj + ¯��hj

exp�− NV�x��, � j�2��x�k

�2��x�dx = � j,k, �1.9�

where hj is normalization constant. V�x� is the quartic weight defined in �1.1�. Using these, weexpand skew-orthogonal functions j

�1��x� on the basis of j��2��x� such that m

�1��x�= m�m�m−3

��2��x�+�m−1 k

�m�k�1��x� for m�3. Multiplying by ��y−x� and integrating by parts, we have m

�1��x�= m

�m�m−3�2� �x�+�m−1 k

�m�k�1��x�. Using skew orthonormality �1.4�, we get the recursion relations

2m+1�1� �x� = 2m+1

�2m+1�2m−2�2� �x� + 2m−1

�2m+1�2m−1�1� �x�, m � 1, �1.10�

2m�1��x� = 2m

�2m�2m−3�2� �x� + 2m−2

�2m� 2m−2�1� �x� + 2m−4

�2m� 2m−4�1� �x�, m � 1, �1.11�

2m+1�1� �x� = 2m+1

�2m+1��P2m−2,2m+1�2� 2m+1

�2� �x� + ¯ P2m−2,2m−5�2� 2m−5

�2� �x�� + 2m−1�2m+1�2m−1

�1� �x� ,

�1.12�

2m�1��x� = 2m

�2m��P2m−3,2m�2� 2m

�2��x� + ¯ P2m−3,2m−6�2� 2m−6

�2� �x�� + 2m−2�2m� 2m−2

�1� �x� + 2m−4�2m� 2m−4

�1� �x� ,

�1.13�

We know from Ref. 13 that for even weight, with Rm= �hm /hm−1�, R0=0, orthogonal functionssatisfy

x j�2��x� = �Rj+1 j+1

�2� �x� + �Rj j−1�2� �x�,

d

dx j

�2��x� = �k=j−3

j+3

Pj,k�2�k

�2��x� ,

Pj,j+3�2� = − Ng�Rj+1Rj+2Rj+3, Pj,j+1

�2� = −�j + 1�2Rj+1

1/2 , Pj,j−1�2� =

j

2Rj1/2 , Pj,j−3

�2� = Ng�Rj−2Rj−1Rj .

�1.14�

Furthermore, using 2m−1�2� �0�=0 in �1.14�, we get

2m�2��0� = �− 1�m�R2m−1 ¯ R1

R2m ¯ R20

�2��0� . �1.15�

Using skew-orthogonality relations ��2m−2�1� �x� ,2m+1

�1� �x��=0�, ��2m−3�1� �x� ,2m

�1��x��=0�, and��2m−1

�1� �x� ,2m�1��x��=0� in �1.10�–�1.13�, we get

2m−1�2m+1� = − 2m+1

�2m+1� 2m−2�2m−2�P2m−5,2m−2

�2� , 2m−4�2m� = 2m

�2m� 2m−3�2m−3�P2m−6,2m−3

�2� ,

2m−2�2m� = 2m

�2m�� 2m−1�2m−1�P2m−4,2m−3

�2� + 2m−3�2m−1� 2m−3

�2m−3�P2m−6,2m−3�2� � , �1.16�

respectively. We choose 2m+1�2m+1�=1 and 2m−1

�2m+1�=−�R2m−2 /R2m−3. 2m−2�2m−2�, 2m−2

�2m� , and 2m−4�2m� can be

calculated from �1.14� and �1.16�. Here, we note that choice of 2m+1�2m+1� and 2m−1

�2m+1� is such that wecan use properties of orthogonal polynomials to obtain 2m+1

�1� �x� from �1.10�. However, this makesthe skew-orthogonal polynomials nonmonic.

Using �1.14� and �1.16�, we solve the recursion relation involving 2m+1�1� �x� in �1.10�, while

2m�1��x� is obtained using skew orthonormalization �1.4�,

063515-3 Skew orthogonal polynomials J. Math. Phys. 50, 063515 �2009�

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2m+1�1� �x� = �R2m−1�2m−1

�2� �x�x

, 2m�1��x� =

x2m−1�2� �x�

�R2m−1

, m � 1. �1.17�

Here, 2m+1�1� �x� and 2m

�1��x� can be obtained by taking derivative of 2m+1�1� �x� and using skew

orthonormality, respectively.Now, we derive asymptotics for skew-orthogonal functions using results for orthogonal

functions.13 In the limit m ,N→�, we assume ��0 such that ����=m /N���cr�=t2 /4g�. Thisensures that m

����x� and m����x� are concentrated on two intervals �−x2 ,−x1� and �x1 ,x2� and

exponentially small outside. Using results from Ref. 13 for large m,

m�2��x� =

2Cm�x

�sin ��cos�fm���� + O�N−1��, fm��� = m + 1/2

2 sin�2��

2− � − �− 1�m�

4+

�

4,

�1.18�

R2m+1,R2m � R,L + O�N−2� =− t � �t2 − 4�g

2gwith � �

m

N= gRL, t + g�R + L� = 0,

�1.19�

where �1.19� signifies that in the asymptotic limit, Rj oscillates between two constants R and L forj odd and even, respectively. Thus we have for x1,2=��−t�2��g� /g, in the range x1+��x�x2−�, ��0 and m�1,

2m+1�1� �x� � 2C2m−1� R

x sin ��cos�f2m−1���� + O 1

N ,

2m�1��x� �

C2m−1

N�� Lx

sin3 ��sin�f2m−1���� + O 1

N ,

2m+1�1� �x� � − 4N�C2m−1�x sin �

L�sin�f2m−1���� + O 1

N ,

2m�1��x� � 2C2m−1� x3

R sin ��cos�f2m−1���� + O 1

N ,

where ��=�+1 /2N, and

q = cos � =gx2 + t

2���g, cos � =

2���g − tq

2���gq − t, Cm =

1

2�� g

�1/4

�1 + O�N−1�� . �1.20�

Here, 2m�1��x� is calculated by expressing the asymptotic result of 2m

�1��x� as a derivative andintegrating by parts, while 2m+1

�1� �x� is calculated from 2m+1�1� �x� by simply taking its derivative. We

note that for a given x, � varies with m. We have chosen ���2m−1 such that

2m+1�k�1� �x� � 2C2m−1� R

x sin ��cos�f2m−1��� � �k��/2� + O�1/N��, m � k , �1.21�

and so on for 2m�k�1� �x�, 2m+1�k

�1� �x�, and 2m�k�1� �x�.

For the recursion coefficient �1.8�, we simply read off from �1.10�–�1.13�. We get for large m

P2m,2m+3�1� =�L

R, P2m+1,2m+4

�1� = Ra2, P2m,2m+1�1� = −

t

gR, P2m+1,2m+2

�1� = 0,

063515-4 Saugata Ghosh J. Math. Phys. 50, 063515 �2009�

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R2m,2m+4�1� = R2m+1,2m+5

�1� = − N�, R2m,2m+2�1� = R2m+1,2m+3

�1� = Nt�RL , �1.22�

where a=−N�g�.To calculate S2N

�1��x ,y�, we use �1.20� and �1.21� in �1.7�. The first four terms �modulo O�N−1��in �1.7� give −�2 cos 2� sin��2N−1�� / �� sin2 ��, the next four terms give�2 cos2 � sin��2N−1�� / �� sin2 ��, while the last two terms give �sin��2N−1�� /�, where �� j

=����f j���� /���. Combining all, we get for x=y+�y

�x − y�S2N�1��x,y� =

sin��2N−1��

+ O�N−1� � S2N�1��x,y� =

sin��y2N�gy�1 − q2���y

+ O�N−1� .

�1.23�

To obtain the level density, we take the limit:

lim�y→0

1

2NS2N

�1��x,y� =1

2NS2N

�1��y,y� =�g

��y��1 − q2 + O�N−1� =

�y���g − gy2 + t

22

+ O�N−1� ,

�1.24�

while S2N�1��x ,y� /S2N

�1��y ,y� gives the sine kernel,

S2N�1��x,y�

S2N�1��y,y�

=sin �r

�r, r = �yS2N

�1��y,y� . �1.25�

III. SKEW-ORTHOGONAL FUNCTIONS AND SYMPLECTIC ENSEMBLE

For skew-orthogonal functions with quartic weight, we expand m�4��x�= m

�m�m�2��x�

+�m−1 k�m�k

�4��x�. Using skew orthonormality �1.4�, we get recursion relations,

2m+1�4� �x� = 2m+1

�2m+1�2m+1�2� �x� + 2m−1

�2m+1�2m−1�4� �x� , �2.1�

2m�4��x� = 2m

�2m�2m�2��x� + 2m−2

�2m� 2m−2�4� �x� + 2m−4

�2m� 2m−4�4� �x� , �2.2�

2m+1�4� �x� = 2m+1

�2m+1��P2m+1,2m+4�2� 2m+4

�2� �x� + ¯ P2m+1,2m−2�2� 2m−2

�2� �x�� + 2m−1�2m+1�2m−1

�4� �x� , �2.3�

2m�4��x� = 2m

�2m��P2m,2m+3�2� 2m+3

�2� �x� + ¯ P2m,2m−3�2� 2m−3

�2� �x�� + 2m−2�2m� 2m−2

�4� �x� + 2m−4�2m� 2m−4

�4� �x� ,

�2.4�

where j�2��x� are orthogonal functions defined in �1.9� and �1.14�.

Using skew-orthogonality relations ��2m−2�4� �x� ,2m+1

�4� �x��=0�, ��2m−3�4� �x� ,2m

�4��x��=0�, and��2m−1

�4� �x� ,2m�4��x��=0� in �2.1�–�2.4�, we get

2m−1�2m+1� = 2 2m+1

�2m+1� 2m−2�2m−2�P2m−2,2m+1

�2� , 2m−4�2m� = − 2 2m

�2m� 2m−3�2m−3�P2m−3,2m

�2� ,

2m−2�2m� = − 2 2m

�2m�� 2m−1�2m−1�P2m−1,2m

�2� + 2m−3�2m−1� 2m−3

�2m−3�P2m−3,2m�2� � , �2.5�

respectively. We choose 2m+1�2m+1�=1 /�2 and 2m−1

�2m+1�=−�R2m+1 /R2m. 2m−2�2m−2�, 2m−2

�2m� , and 2m−4�2m� can be

calculated from �1.14� and �2.5�. Choice of 2m+1�2m+1� and 2m−1

�2m+1� is such that we can use propertiesof orthogonal polynomials to solve �2.1�. However, this makes the skew-orthogonal polynomialsnonmonic.

Using �1.14�, �1.15�, and �2.5� we solve �2.1�. 2m�4��x� is obtained using skew orthonormaliza-

tion �1.4�,

063515-5 Skew orthogonal polynomials J. Math. Phys. 50, 063515 �2009�

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2m+1�4� �x� =

�R2m+2

x�2�2m+2

�2� �x� − 2m+2�2� �0�exp�− NV�x���, 2m

�4��x� = −x2m+2

�2� �x��2R2m+2

. �2.6�

Here, 2m�4��x� and 2m+1

�4� �x� can be derived by integrating and differentiating 2m�4��x� and

2m+1�4� �x�, respectively.

With m ,N→� and in the range x1+��x�x2−�, ��0, and neglecting the m independentterm, we get

2m+1�4� �x� � C� 2L

x sin ��cos�f2m+2���� + O�1/N�� ,

2m�4��x� � −

C

N��� Rx

2 sin3 ��sin�f2m+2���� + O�1/N�� ,

2m+1�4� �x� � − �4NC��

�2R �x sin ��sin�f2m+2���� + O�1/N�� , �2.7�

2m�4��x� � − C� 2x3

L sin ��cos�f2m+2���� + O�1/N�� ,

where fm���, Cm�C, �, �, and � are defined in �1.18� and �1.20�. Here, we note that for a givenx, � varies with m. We have chosen ���2m+2 such that

2m+1�k�4� �x� � C� 2L

x sin ��cos� f2m+2��� �

k�

2 + O�1/N� , m � k , �2.8�

and so on for 2m�k�4� �x�, 2m+1�k

�4� �x�, and 2m�k�4� �x�.

For the recursion coefficient �1.8�, we simply read off from �2.1�–�2.4� and use �1.14�. Forlarge m, we have

P2m,2m+3�4� = −�R

L, P2m+1,2m+4

�4� = − a2L, P2m,2m+1�4� =

t

gL, P2m+1,2m+2

�4� = 0,

R2m,2m+4�4� = R2m+1,2m+5

�1� = − N�, R2m,2m+2�4� = R2m+1,2m+3

�1� = Nt�RL . �2.9�

To calculate S2N�4��x ,y�, we use �2.7�–�2.9� in �1.7� for �=4. The first four terms �modulo O�N−1��

in �1.7� for �=4 give �cos 2� sin��2N+2�� / �� sin2 ��, the next four terms give−�cos2 � sin��2N+2�� / �� sin2 ��, while the last two terms give �sin��2N+2�� /2�. Combining all, weget for x=y+�y,

�y − x�S2N�4��x,y� = −

sin��2N+2�2�

+ O�N−1� � S2N�4��x,y� =

sin��y2N�gy�1 − q2�2��y

+ O�N−1� .

�2.10�

For level density, we take the limit

lim�y→0

1

NS2N

�4��x,y� =1

NS2N

�4��y,y� =�g

��y��1 − q2 =

�y���g − gy2 + t

22

+ O�N−1� �2.11�

such that we get the “universal” sine kernel

063515-6 Saugata Ghosh J. Math. Phys. 50, 063515 �2009�

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S2N�4��x,y�

S2N�4��y,y�

=sin 2�r

2�r, r = �yS2N

�4��y,y� . �2.12�

IV. CONCLUSION

In this study, we prove “universality” that exists in the orthogonal and symplectic ensemblesof random matrices for quartic double well potential using methods of skew-orthogonal functions.To do so, we obtain compact relations between skew-orthogonal and orthogonal functions. Mehtaand Mahoux14 made a similar effort while studying the partition functions with quartic weight.However, to get compact relations �and not a series� which are easily amenable to asymptoticanalysis of the skew-orthogonal functions, one needs to make a very careful choice of the basis ofexpansion. This is also evident �as pointed out by Mehta in Ref. 4� in the case of other weightfunctions.

We also note that universality for these matrix models is obtained in Refs. 15–17, where thekernel functions for �=1 and 4 are expanded in terms of that of �=2. In the process skew-orthogonal functions are avoided. This is a cumbersome process as compared to the GCD formu-lation discovered by Ghosh8 and, consequently, used in this paper.

The key achievement of this paper is the derivation of Eqs. �1.17� and �2.6�. This enables usto use asymptotic results of orthogonal functions13 to derive bulk asymptotics of skew-orthogonalfunctions with quartic weight. Simultaneously �1.10�–�1.13� and �2.1�–�2.4� give us the recursioncoefficients Pj,k

��� and Rj,k���. These results for skew-orthogonal functions are applied in the GCD

formula8 to prove “bulk universality” in the corresponding orthogonal and symplectic ensemblesof random matrices. We note that asymptotics for skew-orthogonal functions away from the bulkcan be trivially obtained from �1.17� and �2.6� since the corresponding results for orthogonalfunctions are already known. However proving the universality at the edge requires a good un-derstanding of certain integrals involving Airy function. We wish to come back to it in a laterpublication.

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12 Pierce, V. U., “A Riemann-Hilbert problem for skew-orthogonal polynomials,” J. Comput. Appl. Math. 215, 230 �2008�.13 Bleher, P. M. and Its, A. R., “Semiclassical asymptotics of orthogonal polynomials, Riemann-Hilbert problems, and

universality in the matrix model,” Ann. Math. 150, 185 �1999�.14 Mehta, M. L. and Mahoux, G., “A method of integration over matrix variables. III,” Indian J. Pure Appl. Math. 22, 531

�1991�.15 Stojanovic, A., “Universality in orthogonal and symplectic invariant matrix models with quartic potential,” Math. Phys.,

Anal. Geom. 3, 339 �2000�.16 Stojanovic, A., “Errata: Universality in orthogonal and symplectic invariant matrix models with quartic potential,” Math.

Phys., Anal. Geom.. 7, 347�E� �2004�.17 Stojanovic, A., “Une approche par les polynÙmes orthogonaux pour des classes de matrices alÈatoires orthogonalement

et symplectiquement invariantes: Application à l’universalité de la statistique locale des valeurs propres,” BoielefeldReport No. 00-01-006, 2000.

063515-7 Skew orthogonal polynomials J. Math. Phys. 50, 063515 �2009�

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