aa r X i v : . [ m a t h . P R ] F e b SOME RESULTS ON RANDOM CIRCULANT MATRICES
MARK W. MECKES
Abstract.
This paper considers random (non-Hermitian) circulant matrices, and proves severalresults analogous to recent theorems on non-Hermitian random matrices with independent entries.In particular, the limiting spectral distribution of a random circulant matrix is shown to be complexnormal, and bounds are given for the probability that a circulant sign matrix is singular. Introduction
Given a sequence X , X , . . . of independent complex random variables, denote by C n the n × n random circulant matrix with first row X , . . . , X n − : C n = X X · · · · · · X n − X n − X n − X X X n − ... X n − X ...... . . . ... X X X X X · · · · · · X n − X . It is well-known (and easy to verify) that the eigenvalues of C n are n − X j =0 ω jkn X j , k = 0 , . . . , n − , where ω n = e πi/n . The eigenvalues of random circulant matrices have only recently been studiedexplicitly in the literature, e.g. in [4, 5, 19], which consider real symmetric circulant matrices andvariants of them. Other models of random matrices lying in some linear subspace of matrix spacehave also been studied recently in [5, 7, 9, 13, 19], among others. These papers mainly prove resultsanalogous to classical theorems for Wigner-type random matrices, i.e. symmetric (or Hermitian)random matrices whose entries are all independent except for the symmetry constraint.This paper mainly considers the eigenvalues of the random circulant matrices C n with no symme-try constraint, and prove results analogous to recent theorems about random matrices with all inde-pendent entries. In Section 2 we investigate the limiting spectral distribution of large-dimensionalrandom circulant matrices. In Section 3 we consider the joint distribution of eigenvalues of circulantmatrices with Gaussian entries and observe some consequences, especially for the distributions ofextreme eigenvalues. Finally, in Section 4 we investigate the probability that a circulant matrixwith ± Limiting spectral distribution
We denote by(2.1) λ k = 1 √ n n − X j =0 ω jkn X j for k = 0 , . . . , n − n − / C n , and by µ n = 1 n n − X k =0 δ λ k the empirical spectral measure of n − / C n . Theorems 1 and 2 show that as n → ∞ , µ n convergesto a universal limiting measure, namely the standard complex Gaussian measure γ C with density π e −| z | with respect to Lebesgue measure on C . Theorem 1 shows that this convergence holds inprobability under very weak assumptions on the random variables X j (including in particular thecase of i.i.d. random variables with finite variance); Theorem 2 strengthens this to almost sureconvergence under much stronger assumptions.Theorems 1 and 2 are analogues for circulant matrices of the circular law for eigenvalues ofrandom matrices with independent entries, which was recently proved by Tao and Vu in [25] in thealmost sure sense for i.i.d. entries with finite variance. It is likely that the conclusion of Theorem2 also holds in this level of generality. However, the result of [25] and other results leading up to itwere made possible by recent advances in controlling how close a random matrix with i.i.d. entriesis to being singular. As discussed in Section 4 below, random circulant matrices are frequentlymuch more likely to be singular; thus it may be difficult to prove the optimal result for circulantmatrices. (On the other hand, the fact that circulant matrices are normal may make such a resultapproachable via the classical moment method used extensively for Hermitian random matrices.) Theorem 1.
Suppose that the X j satisfy (2.2) E X j = 0 , E X j = α, E | X j | = 1 , for some α ∈ C and (2.3) lim n →∞ n n − X j =0 E (cid:0) | X j | | X j | >ε √ n (cid:1) = 0 for every ε > . Then µ n converges in expectation and weak- ∗ in probability to γ C , the standardcomplex Gaussian measure on C . Observe that the hypotheses cover the case of i.i.d. complex random variables with finite variance,as well as real random variables and rotationally invariant distributions satisfying the Lindebergcondition (2.3).
Proof.
Without loss of generality we may assume α ∈ R . Observe that for a measurable set A ⊆ C , E µ n ( A ) = 1 n n − X k =0 P [ λ k ∈ A ] = 1 n n − X k =0 P √ n n − X j =0 ω jkn X j ∈ A . We consider λ k as a sum of independent random vectors in R ∼ = C , so we will needCov( ω jkn X j ) = E (Re ω jkn X j ) E (Re ω jkn X j )(Im ω jkn X j ) E (Re ω jkn X j )(Im ω jkn X j ) E (Im ω jkn X j ) ! . The identities ( w + w ) z = (Re w )(Re z ) + i (Re w )(Im z ) , ( w − w ) z = − (Im w )(Im z ) + i (Re w )(Im z ) . (2.4) ANDOM CIRCULANT MATRICES 3 for w, z ∈ C are useful. Letting w = z = ω jkn X j , n − X j =0 E (Re ω jkn X j ) = 12 Re n − X j =0 E ( ω jkn X j + | X j | ) = 12 Re n − X j =0 ( αω jkn + 1) . Since ω kn is an n th root of unity, P n − j =0 ω jkn = 0 unless ω kn = 1, which is the case only if k = 0 or k = n/
2. Thus unless k = 0 or n/
2, 1 n n − X j =0 E (Re ω jkn ) = 12 . The other covariances are computed similarly and it follows that for k = 0 , n/ n n − X j =0 Cov( ω jkn X j ) = 12 I . Since | ω jkn X j | = | X j | , by (2.3) we can now apply a quantitative two-dimensional version ofLindeberg’s central limit theorem to the complex random variables { ω jkn X j | ≤ j ≤ n − } . Let A ⊆ C be measurable and convex and assume k = 0 , n/
2. By the proof of [3, Corollary 18.2]), thereis a function h ( n ) with lim n →∞ h ( n ) = 0, which depends on A and the rate of convergence in (2.3)but is independent of k , such that (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) P √ n n − X j =0 ω jkn X j ∈ A − γ C ( A ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ h ( n ) . Therefore(2.5) | E µ n ( A ) − γ C ( A ) | ≤ n n − X k =0 (cid:12)(cid:12) P (cid:2) λ k ∈ A (cid:3) − γ C ( A ) (cid:12)(cid:12) ≤ n − h ( n ) n n →∞ −−−→ . Thus the claimed convergence holds in expectation.Next observe that E µ n ( A ) = E n n − X k,ℓ =0 λ k ∈ A λ ℓ ∈ A = 1 n n − X k,ℓ =0 P (cid:2) ( λ k , λ ℓ ) ∈ A × A (cid:3) . As before, we can compute the sums of the relevant covariance matrices using (2.4), in this casewith w = ω jkn X j and z = ω jℓn X j . We obtain1 n n − X j =0 Cov( ω jkn X j , ω jℓn X j ) = 12 I except when k = 0 or n/ ℓ = 0 or n/ k + ℓ = 0 or n , or k = ℓ . These exceptional casesaccount for fewer than 6 n of the n possible values of ( k, ℓ ). Applying the four-dimensional case ofLindeberg’s theorem similarly to above, we have(2.6) (cid:12)(cid:12) E µ n ( A ) − γ C ( A ) (cid:12)(cid:12) ≤ n n − X k,ℓ =0 (cid:12)(cid:12) P (cid:2) ( λ k , λ ℓ ) ∈ A × A (cid:3) − γ C ( A ) (cid:12)(cid:12) n →∞ −−−→ . We are of course using here that standard Gaussian measure on C is the two-fold product of γ C .From (2.5) and (2.6) it follows that E (cid:2) µ n ( A ) − γ C ( A ) (cid:3) = (cid:2) E µ n ( A ) − γ C ( A ) (cid:3) − γ C ( A ) (cid:2) E µ n ( A ) − γ C ( A ) (cid:3) n →∞ −−−→ . MARK W. MECKES
Thus for every convex measurable A ⊂ C , the random variable µ n ( A ) converges to γ C ( A ) in L and hence in probability. (cid:3) Bose and Mitra [4] proved earlier a result analogous to Theorem 1 for real symmetric circulantmatrices, which is a circulant matrix analogue of Wigner’s semicircle law; the limiting distribution inthis case is a real Gaussian measure. This result was strengthened from convergence in probabilityto almost sure convergence by Massey, Miller, and Sinsheimer [19]. The main result of [4] is asimilar result for circulant Hankel matrices, which amounts to studying the singular values insteadof eigenvalues of C n (see [7, Remark 1.2] about the relationship between Hankel and Toeplitzmatrices). Since circulant matrices are normal, their singular values are simply the moduli of theireigenvalues; thus this latter result is essentially a corollary of Theorem 1. The proof of Theorem 1follows the basic outline of the proofs of [4], which assume the X j are i.i.d. with finite third absolutemoments; the greater generality of Theorem 1 is achieved by applying Lindeberg’s theorem, insteadof the Berry-Esseen theorem as in [4].The statement of Theorem 1 assumes that the same sequence of random variables X j is used toconstruct C n for every n . The proof shows however that the result generalizes directly to circulantmatrices constructed from a triangular array of random variables. The same comment applies toTheorem 2 below. Theorem 2.
Suppose that, in addition to (2.2) one of the following holds. (1)
There exists a
K > such that for each j , | X j | ≤ K almost surely. (2) There exists a
K > such that for each j , the distribution of X j satisfies a quadratictransportation cost inequality with constant K (see below for the meaning of this).Then µ n converges weak- ∗ to γ C almost surely. Recall that a probability measure µ on R d is said to satisfy a quadratic transportation costinequality with constant K > π ∈ Π( µ,ν ) Z Z | x − y | dπ ( x, y ) ≤ p KH ( µ | ν )for every probability measure ν on R d . Here Π( µ, ν ) is the class of probability measures on R d × R d with marginals µ and ν respectively, and H ( µ | ν ) is relative entropy. Such an inequality is satisfiedin particular if µ satisfies a logarithmic Sobolev inequality. See [18, Chapter 6] for background andreferences. Proof.
We begin with the assumption that a quadratic transportation cost inequality is satisfied.This assumption implies (and is essentially equivalent to, see [12]) the following concentrationinequality for Lipschitz functions of ( X , . . . , X n ). Let F : C n → R have Lipschitz constant | F | Lip = L . Then(2.7) P (cid:2) | F ( X , . . . , X n ) − E F ( X , . . . , X n ) | ≥ t (cid:3) ≤ Ce − ct /K L for every t >
0, where c, C > f : C → R , define F ( x , . . . , x n ) = 1 n n X j =1 f ( x j ) , so that F ( X , . . . , X n ) = R C f dµ n . Then | F ( x , . . . , x n ) − F ( y , . . . , y n ) | ≤ | f | Lip n n X j =1 | x j − y j | ≤ | f | Lip √ n vuut n X j =1 | x j − y j | , ANDOM CIRCULANT MATRICES 5 so | F | Lip ≤ n − / | f | Lip . Combining Theorem 1, (2.7), and the Borel-Cantelli lemma, we obtainthat for each compactly supported smooth function f : C → R , Z C f dµ n n →∞ −−−→ Z C f dγ C almost surely. Applying this for a countable dense family of such f proves the theorem.The case of bounded entries is treated similarly using Talagrand’s famous concentration inequal-ity for convex Lipschitz functions of bounded random variables [24] (also see [18, Section 4.2]); inthis case an extra step is required to handle non-convex test functions f : C → R .As above, let f : C → R be compactly supported and smooth. Let R > f ( x ) = 0for | x | ≥ R and let λ ≥ D f are bounded below by − λ . For x ∈ C define g ( x ) = ( λ | x | if | x | ≤ R,λR (cid:0) | x | − R (cid:1) if | x | ≥ R. Then | g | Lip = λR and so | f + g | Lip ≤ | f | Lip + λR . Furthermore, since D ( f + g ) ≥
0, both g and f + g are convex. Applying the above argument to f + g and g using Talagrand’s concentrationinequality in place of (2.7) implies that with probability 1, Z C ( f + g ) dµ n n →∞ −−−→ Z C ( f + g ) dγ C and Z C g dµ n n →∞ −−−→ Z C g dγ C and hence Z C f dµ n n →∞ −−−→ Z C f dγ C . Again, applying this for a countable dense family of such f proves the theorem. (cid:3) We remark that a slight generalization of the argument in the last paragraph shows that convexLipschitz functions on R d form a convergence-determining class for the family of probability mea-sures on R d with respect to which such functions are integrable. This fact is presumably well-knownto experts but we could not find a statement of it in the literature.3. Gaussian circulant matrices and extreme eigenvalues
The following result is a circulant analogue of classical formulas (found, e.g., in [21]) for the jointdistribution of eigenvalues of random matrices with independent complex Gaussian entries.
Proposition 3.
Let each X j have the standard complex normal distribution. Then the sequence λ , . . . , λ n − of eigenvalues of n − / C n is distributed as n independent standard complex normalrandom variables.Proof. The map ( X , . . . , X n − ) ( λ , . . . , λ n − ) defined by (2.1) is easily checked to be a unitarytransformation of C n , so it preserves the standard Gaussian measure on C n . (cid:3) An easy consequence of Proposition 3 is that in this setting, E µ n = γ C for every n , and not onlyin the limit n → ∞ as guaranteed by Theorem 1.If each X j is a standard real normal random variable, then the same observation implies thatthe sequence of eigenvalues λ , . . . , λ n − are jointly Gaussian random variables, but with singularcovariance since this sequence will lie in an n -dimensional real subspace of the 2 n -dimensionalspace C n . On the other hand, Proposition 3 does have the following simple analogue for complexHermitian circulant matrices with Gaussian entries. MARK W. MECKES
Corollary 4.
Let C Hn be a Hermitian random circulant matrix C Hn = Y Y · · · · · · Y n − Y n − Y n − Y Y Y n − ... Y n − Y ...... . . . ... Y Y Y Y Y · · · · · · Y n − Y , where Y , . . . , Y ⌊ n/ ⌋ are independent, Y j = Y n − j for j > n/ , Y and Y n/ (if n is even) havethe standard real normal distribution, and Y j has the standard complex normal distribution for ≤ j < n/ . Then the eigenvalues of n − / C Hn are distributed as n independent standard realnormal random variables.Proof. If C n is as in Proposition 3, then C Hn has the same distribution as √ ( C n + C ∗ n ). Because C n is normal, the eigenvalues of n − / C Hn are1 √ λ k + λ k ) = √ λ k for k = 0 , . . . , n −
1, which by Proposition 3 are independent standard real normal random variables. (cid:3)
The Gaussian Hermitian circulant matrix C Hn of Corollary 4 should be thought of as a circulantanalogue of the Gaussian Unitary Ensemble, which (up to a choice of normalization) is defined as √ ( G + G ∗ ), where G is an n × n random matrix whose entries are independent standard complexnormal random variables. Corollary 5.
Let C n and C Hn be as in Proposition 3 and Corollary 4. Let α ≥ · · · ≥ α n ≥ bethe eigenvalues of n − C n C ∗ n and let β ≥ · · · ≥ β n be the eigenvalues of n − / C Hn . Then α n has anexponential distribution with mean /n , and α − log n and p n (cid:18) β − p n − log log n + log 4 π √ n (cid:19) both converge in distribution as n → ∞ to the Gumbel distribution with cumulative distributionfunction e − e − x .Proof. Observe that since C n is normal, the eigenvalues of n − C n C ∗ n are | λ k | (though not usuallyin the same order), which by Proposition 3 are distributed as independent exponential randomvariables. The corollary follows by combinging this fact and Corollary 4 with classical theorems ofextreme value theory (see [17, Chapter 1]). (cid:3) The asymptotic distributions of β and α are circulant matrix analogues of famous results ofTracy and Widom [26] and Johnstone [15], respectively. Davis and Mikosch [10], who did notconsider random circulant matrices explicitly, proved (with slight modifications) what amounts toa universality result for α , which shows the conclusion follows for quite general distributions ofthe X j . Following their method Bryc and Sethuraman [8] proved an explicitly stated universalityresult for β ; these are then the circulant matrix analogues of the results of Soshnikov in [23, 22]respectively. The rough orders of magnitude of α and β follow in even greater generality fromwork of the author [20] and Adamczak [1] (see the remarks in [20, Section 3.1]). ANDOM CIRCULANT MATRICES 7 Singularity of circulant sign matrices
In this section we specialize to the case in which P [ X j = −
1] = P [ X j = 1] = 1 / j andconsider the probability that C n is singular. The corresponding problem for random n × n matrices M n with independent ± P [ M n is singular] = (cid:18)
12 + o (1) (cid:19) n , which is asymptotically the probability that two rows of M n are equal up to sign. The best resultcurrently known, proved by Bourgain, Vu, and Wood in [6], is P [ M n is singular] ≤ (cid:18) √ o (1) (cid:19) n . By contrast, the following result shows that the singularity probability of an n × n random circulantmatrix with ± n . Theorem 6.
Let P [ X j = −
1] = P [ X j = 1] = 1 / for each j . If n is even, then (4.1) c √ n ≤ P [ C n is singular ] ≤ c √ n , where c , c > are absolute constants. If n ≥ is odd, then (4.2) P [ C n is singular ] ≤ min ( c d ( n ) n , X
Begin by defining the random polynomial f ( t ) = n − X j =0 X j t j , so that the eigenvalues of C n are f ( ω kn ) for k = 0 , . . . , n −
1. Observe first that f (1) and f ( −
1) areidentically distributed and(4.4) P [ f (1) = 0] = P [ f ( −
1) = 0] = ( − n (cid:0) nn/ (cid:1) if n is even , n is odd . which implies the lower bound in (4.1) by Stirling’s formula. MARK W. MECKES
For each k > ω kn is a primitive root of unity of order n/m , where m = gcd( k, n ). The minimalpolynomial of ω kn over the rational numbers is thus the cyclotomic polynomial Φ n/m , so f ( ω kn ) = 0if and only if Φ n/m is a factor of f . (See e.g. [14, Section V.8] for background on cyclotomicpolynomials.) Therefore we need only consider the cases when k is a divisor of n , and(4.5) P [ C n is singular] ≤ P [ f (1) = 0] + X ≤ k
2; the calculations involved in applying the result of [11] are similarto those in the proof of Theorem 1. The upper bound of (4.1) and the first upper bound of (4.2)follow by combining (4.5), (4.4), and (4.6). To bound the number of terms in (4.5) in the evencase it is enough to use the trivial estimate d ( n ) < √ n (divisors of n occur in pairs k, n/k with k ≤ √ n ) rather than the much more delicate result (4.3).For n ≥ d ≥ P n,m be the set of polynomials g with rational coefficients of degree atmost n − m is a factor of g . With the substitution m = n/k , (4.5) becomes P [ C n is singular] ≤ X 1. Therefore some set of n − ϕ ( m ) coefficients of apolynomial g suffice to determine whether g ∈ P n,m , and so P (cid:2) f ∈ P n,m (cid:3) ≤ − ϕ ( m ) , which proves the second upper bound in (4.2). (cid:3) Acknowledgements The author thanks W lodek Bryc and Elizabeth Meckes for helpful discussions. References [1] R. Adamczak. A few remarks on the operator norm of random Toeplitz matrices. Preprint, available at http://arxiv.org/abs//0803.3111 .[2] T. M. Apostol. Introduction to Analytic Number Theory . Springer-Verlag, New York, 1976. Undergraduate Textsin Mathematics.[3] R. N. Bhattacharya and R. Ranga Rao. Normal Approximation and Asymptotic Expansions . Robert E. KriegerPublishing Co. Inc., Melbourne, FL, 1986. Reprint of the 1976 original.[4] A. Bose and J. Mitra. Limiting spectral distribution of a special circulant. Statist. Probab. Lett. , 60(1):111–120,2002.[5] A. Bose and A. Sen. Another look at the moment method for large dimensional random matrices. Electron. J.Probab. , 13:no. 21, 588–628, 2008.[6] J. Bourgain, V. Vu, and P. Wood. On the singularity probability of discretely random complex matrices. Preprint.[7] W. Bryc, A. Dembo, and T. Jiang. Spectral measure of large random Hankel, Markov and Toeplitz matrices. Ann. Probab. , 34(1):1–38, 2006.[8] W. Bryc and S. Sethuraman. On the maximum eigenvalue for circulant matrices. Manuscript in preparation.[9] S. Chatterjee. Fluctuations of eigenvalues and second order Poincar´e inequalities. Probab. Theory Related Fields ,143(1-2):1–40, 2009.[10] R. A. Davis and T. Mikosch. The maximum of the periodogram of a non-Gaussian sequence. Ann. Probab. ,27(1):522–536, 1999. ANDOM CIRCULANT MATRICES 9 [11] O. Friedland and S. Sodin. Bounds on the concentration function in terms of the Diophantine approximation. C. R. Math. Acad. Sci. Paris , 345(9):513–518, 2007.[12] N. Gozlan. A characterization of dimension free concentration in terms of transportation inequalities. Preprint,available at http://arxiv.org/abs/0804.3089 .[13] C. Hammond and S. J. Miller. Distribution of eigenvalues for the ensemble of real symmetric Toeplitz matrices. J. Theoret. Probab. , 18(3):537–566, 2005.[14] T. W. Hungerford. Algebra , volume 73 of Graduate Texts in Mathematics . Springer-Verlag, New York, 1980.Reprint of the 1974 original.[15] I. M. Johnstone. On the distribution of the largest eigenvalue in principal components analysis. Ann. Statist. ,29(2):295–327, 2001.[16] J. Kahn, J. Koml´os, and E. Szemer´edi. On the probability that a random ± J. Amer. Math.Soc. , 8(1):223–240, 1995.[17] M. R. Leadbetter, G. Lindgren, and H. Rootz´en. Extremes and Related Properties of Random Sequences andProcesses . Springer Series in Statistics. Springer-Verlag, New York, 1983.[18] M. Ledoux. The Concentration of Measure Phenomenon , volume 89 of Mathematical Surveys and Monographs .American Mathematical Society, Providence, RI, 2001.[19] A. Massey, S. J. Miller, and J. Sinsheimer. Distribution of eigenvalues of real symmetric palindromic Toeplitzmatrices and circulant matrices. J. Theoret. Probab. , 20(3):637–662, 2007.[20] M. W. Meckes. On the spectral norm of a random Toeplitz matrix. Electron. Comm. Probab. , 12:315–325 (elec-tronic), 2007.[21] M. L. Mehta. Random Matrices , volume 142 of Pure and Applied Mathematics (Amsterdam) . Elsevier/AcademicPress, Amsterdam, third edition, 2004.[22] A. Soshnikov. Universality at the edge of the spectrum in Wigner random matrices. Comm. Math. Phys. ,207(3):697–733, 1999.[23] A. Soshnikov. A note on universality of the distribution of the largest eigenvalues in certain sample covariancematrices. J. Statist. Phys. , 108(5-6):1033–1056, 2002. Dedicated to David Ruelle and Yasha Sinai on the occasionof their 65th birthdays.[24] M. Talagrand. Concentration of measure and isoperimetric inequalities in product spaces. Inst. Hautes ´EtudesSci. Publ. Math. , (81):73–205, 1995.[25] T. Tao and V. Vu. Random matrices: Universality of esds and the circular law. Preprint, available at http://arxiv.org/abs/0807.4898 . With an appendix by M. Krishnapur.[26] C. A. Tracy and H. Widom. Level-spacing distributions and the Airy kernel. Comm. Math. Phys. , 159(1):151–174,1994. Department of Mathematics, Case Western Reserve University, Cleveland, Ohio 44106, U.S.A. E-mail address ::