Non-localization of eigenfunctions on large regular graphs
aa r X i v : . [ m a t h . D S ] D ec NON-LOCALIZATION OF EIGENFUNCTIONS ONLARGE REGULAR GRAPHS
SHIMON BROOKS AND ELON LINDENSTRAUSS
Abstract:
We give a delocalization estimate for eigenfunctions ofthe discrete Laplacian on large d + 1-regular graphs, showing that anysubset of the graph supporting ǫ of the L mass of an eigenfunctionmust be large. For graphs satisfying a mild girth-like condition, thisbound will be exponential in the size of the graph.1. Introduction
The extent to which eigenfunctions of the Laplacian can localize insmall sets is a question that has attracted much recent attention. Inthe case of compact negatively curved surfaces, it is conjectured (see,eg., [IS95]) that the sup-norms of Laplace eigenfunctions satisfy strongbounds (subexponential in the eigenvalue). Another notion of delo-calization is the Quantum Unique Ergodicity property, which roughlystates that eigenfunctions become equidistributed in the ambient space.This property has been conjectured by Rudnick and Sarnak [RS94] tohold for all compact manifolds of negative sectional curvature, andwas proved for certain arithmetic manifolds (and a natural but spe-cific choice of eigenbasis for the eigenfunctions) in [Lin06], [SV07] and[Sou09]. Other work (eg., [AN07]) proves entropy estimates for eigen-functions, a weaker notion of the inability to localize in small sets.In this paper, we investigate eigenfunctions of the discrete Laplacianon large d + 1-regular graphs, satisfying a mild condition (1) below(essentially asking that there not be too many short cycles through thesame point). We will use methods akin to the quantum chaos tools of[AN07] and others, and our results hold for all graphs satisfying (1)and any eigenfunction. It is likely much stronger results can be provenfor random graphs. An earlier version of our result was also written in[Bro09].To state the condition on our graphs, we let T d +1 be the d + 1-regulartree (the universal cover of our d +1-regular graphs). For f ∈ L ( T d +1 ), E.L. was supported in part by NSF grants DMS-0554345 and DMS-0800345. set ˜ S n ( f )( x ) = d − n/ X d ( x,y )= n f ( y )to be a normalized average over the sphere of radius n , and write S n forthe projection of this operator to L ( G ). We assume that there exist C and α >
0, such that the matrix coefficients of S n satisfy(1) sup x ∈G || S n δ x || ∞ ≤ Cd − αn for all n ≤ N For example, if we set N to be the radius of injectivity of the graph,then we may take C = 1 and α = 1 /
2. Ideally, we would like to have N & log |G| (see below); at the very least, we should ensure that N islarge relative to the other parameters, tending to ∞ with |G| .For random graphs, it is shown in [MWW04] that a large d + 1-regular graph G almost surely does not have 2 cycles of length at most (cid:0) − ǫ (cid:1) log d |G| that share an edge; in particular, this means that thecondition (1) holds for almost all graphs.More generally, we will assume control over the norm of S n as anoperator from L p ( G ) to L q ( G ), for some conjugate pair 1 ≤ p < Let ǫ > , and G a d + 1 -regular graph satisfying (2) || S n || L p ( G ) → L q ( G ) ≤ Cd − αn for all n ≤ N as an operator from L p ( G ) to L q ( G ) , for some conjugate ≤ p < and < q ≤ ∞ (i.e., satisfying p + q = 1 ). Then for any L -normalizedeigenfunction φ on G , any subset E ⊂ G satisfying X x ∈ E | φ ( x ) | > ǫ must be of size | E | & d δN as N → ∞ , where δ = δ ( ǫ, α, p ) can be taken to be δ = 2 − αp (2 − p ) ǫ . Theimplied constant depends on all parameters except N ; namely d , C , α , p , and ǫ . Note that if N & log d |G| , then the conclusion of Theorem 1 statesthat | E | & |G| δ ′ . ON-LOCALIZATION FOR GRAPH EIGENFUNCTIONS 3 Some Harmonic Analysis on the d + 1 -Regular Tree Throughout, we set G to be a d + 1-regular graph. We have thesymmetric operator T d f ( x ) = 1 √ d X d ( x,y )=1 f ( y )and an orthonormal basis { φ j } |G| j =1 of L ( G ) consisting of T d -eigenfunctions .(The discrete Laplacian on G can be written as ∆ f = (cid:16) √ dd +1 T d − (cid:17) f ,and so the eigenfunctions of T d are exactly the eigenfunctions of ∆.)The universal cover of G is the d + 1-regular tree, denoted T d +1 .Harmonic analysis on T d +1 has been well studied, see eg. [FTP83]. Forevery λ ∈ [ − d +1 √ d , d +1 √ d ], there exists a unique spherical function φ λ satisfying: • T d φ λ = λφ λ . • φ λ is radial; i.e., φ λ ( x ) = φ λ ( | x | ) for all x ∈ T d +1 , where | x | denotes the distance from x to the origin in T d +1 . • φ λ (0) = 1.The last condition is simply a convenient normalization.We distinguish two parts of this spectrum: the tempered spec-trum is the interval [ − , untempered spectrum is thepart lying outside this interval, i.e. ± (2 , d +1 √ d ]. We will find it convenientto parametrize the spectrum by λ = 2 cos θ λ , where: • θ λ ∈ [0 , π ] for λ tempered. • iθ λ = r λ ∈ (0 , log √ d ) for λ untempered and positive. • iθ λ + iπ = r − λ for λ untempered and negative.In this parametrization, we can write the spherical functions explicitlyas [Bro91]: φ λ ( x ) = d −| x | / (cid:18) d + 1 cos | x | θ λ + d − d + 1 sin ( | x | + 1) θ λ sin θ λ (cid:19) It will be convenient to use the Chebyshev polynomials P n (cos θ ) = cos nθQ n (cos θ ) = sin ( n + 1) θ sin θ The operator T d matches the operator defined above as S , though here we wishto emphasize the degree d rather than the radius of the sphere. SHIMON BROOKS AND ELON LINDENSTRAUSS of the first and second kinds, respectively. With this notation thespherical functions become(3) φ λ ( x ) = d −| x | / (cid:18) d + 1 P | x | ( λ/ 2) + d − d + 1 Q | x | ( λ/ (cid:19) For any compactly supported radial function k = k ( | x | ) on T d +1 , the spherical transform of k , denoted h k , is given by h k ( λ ) = X x ∈T d +1 k ( x ) φ λ ( x ) = k (0) + ( d + 1) ∞ X n =1 d n − k ( n ) φ λ ( n )for all λ ∈ [ − d +1 √ d , d +1 √ d ] (the sum is actually finite since k is compactlysupported).We have the Plancherel measure dm on [ − , 2] inverting the spher-ical transform on the tempered spectrum, i.e. Z π φ λ ( x ) dm ( θ λ ) = δ ( x )where δ is the δ function at 0 on T d +1 given by δ ( x ) = (cid:26) x = 00 x = 0The Plancherel measure is absolutely continuous (with respect toLebesgue measure on the semi-circle) and symmetric about π/ 2, andso its Fourier series is of the form dmdθ = ∞ X j =0 c j cos 2 jθ The Plancherel measure is then given explicitly by [FTP83, Theorem4.1] Z π cos (2 nθ ) dm = 1 − d d n for n > R π dm = 1).The spectrum of T d on L ( G ) is contained in [ − d +1 √ d , d +1 √ d ], and againwe distinguish between the tempered eigenvalues in [ − , 2] and theuntempered eigenvalues outside this interval. Eigenfunctions of T d arealso eigenfunctions of convolution with radial kernels; in fact, for a“point-pair invariant” k ( x, y ) = k ( d ( x, y )) on G × G , the eigenvalue for φ j under convolution with k depends only on λ j , and is given by thespherical transform h k ( λ j ) [TW03]. ON-LOCALIZATION FOR GRAPH EIGENFUNCTIONS 5 The Main Estimate Our result centers on the following estimate for matrix coefficients of P n ( T d ); recall that P n (cos θ ) = cos nθ are the Chebyshev polynomials(of the first kind). Lemma 1. Let δ be the δ -function supported at ∈ T d +1 , and n apositive even integer. Then P n ( T d / δ ( x ) = | x | odd or | x | > n − d d n/ | x | < n and | x | even d n/ | x | = n In particular, we have P n ( T d / δ ( x ) . d − n/ Proof: Write δ = R π φ λ dm ( θ λ ). Then since T d φ λ = cos θ λ φ λ , wehave P n ( T d / δ ( x ) = Z π (cos nθ λ ) φ λ ( x ) dm ( θ λ )= d −| x | / Z π (cos nθ λ ) (cid:18) d + 1 P | x | ( λ/ 2) + d − d + 1 Q | x | ( λ/ (cid:19) dm ( θ λ )by substituting (3) for the spherical functions.Now since n is even, both cos nθ λ and the Plancherel measure aresymmetric about π/ 2. But if | x | is odd, then both P | x | ( λ/ 2) = cos | x | θ λ Q | x | ( λ/ 2) = sin ( | x | + 1) θ λ sin θ λ = cos | x | θ λ + cos θ λ sin | x | θ λ sin θ λ = cos | x | θ λ + cos θ λ ( | x |− / X j =1 cos 2 jθ λ are odd functions with respect to π/ 2. Therefore the integral from0 to π/ π/ π , and P n ( T d / δ ( x )vanishes for | x | odd.Now consider | x | even, in which case we can write Q | x | ( λ/ 2) = sin ( | x | + 1) θ λ sin θ λ = 1 + 2 | x | / X j =1 cos 2 jθ λ We will also make repeated use of the identity2 cos α cos β = cos ( α + β ) + cos ( α − β ) SHIMON BROOKS AND ELON LINDENSTRAUSS If | x | > n , then we have d | x | / P n ( T d / δ ( x )= Z π (cos nθ λ ) (cid:18) d + 1 P | x | ( λ/ 2) + d − d + 1 Q | x | ( λ/ (cid:19) dm ( θ λ )The left part of the integral yields2 d + 1 Z π cos nθ λ cos | x | θ λ dm ( θ λ )= 1 d + 1 (cid:18)Z π cos ( | x | − n ) θ λ dm ( θ λ ) + Z π cos ( | x | + n ) θ λ dm ( θ λ ) (cid:19) = 1 d + 1 (cid:18) − d d ( | x |− n ) / + 1 − d d ( | x | + n ) / (cid:19) = 1 − dd + 1 (cid:18) d ( | x |− n ) / + 12 d ( | x | + n ) / (cid:19) The right part, on the other hand, is d − d + 1 Z π (cos nθ λ ) | x | / X j =1 cos 2 jθ λ dm ( θ λ )= d − d + 1 Z π cos nθ λ + | x | / X j =1 [cos ( n + 2 j ) θ λ + cos ( n − j ) θ λ ] dm ( θ λ )= d − d + 1 | x | / X j = −| x | / Z π cos ( n + 2 j ) θ λ dm ( θ λ )= d − d + 1 n/ − X j = −| x | / − d d n/ − j + 1 + | x | / X j = n/ − d d j − n/ = d − d + 1 (cid:18) d ( n + | x | ) / + 12 d ( | x |− n ) / (cid:19) since the sum telescopes. Putting the two halves together gives P n ( T d / δ ( x ) =0 for all | x | > n . ON-LOCALIZATION FOR GRAPH EIGENFUNCTIONS 7 If, however, | x | ≤ n , then the sum does not quite telescope as before.For | x | < n we have instead d | x | / P n ( T d / δ ( x )= 1 d + 1 (cid:18) − d d ( n −| x | ) / + 1 − d d ( n + | x | ) / (cid:19) + d − d + 1 ( n + | x | ) / X j =( n −| x | ) / − d d j = 1 − dd + 1 (cid:18) d ( n −| x | ) / + 12 d ( n + | x | ) / (cid:19) + d − d + 1 (cid:18) − d d ( n −| x | ) / + 12 d ( n + | x | ) / (cid:19) = 1 − dd + 1 (cid:18) d ( n −| x | ) / + d d ( n −| x | ) / (cid:19) = 1 − d d ( n −| x | ) / which finally gives P n ( T d / δ ( x ) = d −| x | / − d d ( n −| x | ) / = 1 − d d n/ for | x | < n and even.If | x | = n , then we replace − d d ( n −| x | ) / with 1, and the above calculationbecomes d | x | / P n ( T d / δ ( x )= 1 d + 1 (cid:18) − d d ( n + | x | ) / (cid:19) + d − d + 1 ( n + | x | ) / X j =1 − d d j = 1 d + 1 (cid:18) − d d ( n + | x | ) / (cid:19) + d − d + 1 (cid:18) 12 + 12 d ( n + | x | ) / (cid:19) = 1 d + 1 + d − d + 1)= d + 12( d + 1) = 1 / P n ( T d / δ ( x ) = d | x | / = d n/ for | x | = n , as required. (cid:3) Corollary 1. If G satisfies the hypothesis (2) of Theorem 1, then (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) P n (cid:18) T d (cid:19)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) L p ( G ) → L q ( G ) . d − αn for all even positive integers n ≤ N . SHIMON BROOKS AND ELON LINDENSTRAUSS Proof: Thanks to Lemma 1, we have that P n ( T p / 2) = n/ − X j =0 − d d n/ d j S j + 12 S n as operators on functions of G . Therefore, (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) P n (cid:18) T d (cid:19)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) L p → L q ≤ n/ − X j =0 d − d n/ || d j S j || L p → L q + 12 || S n || L p → L q ≤ d − n/ n/ X j =0 d j || S j ||≤ d − n/ n/ X j =0 Cd j (1 − α ) . d,α d − n/ · Cd n/ − αn . d − αn as required. (cid:3) Estimating the Mass of Small Sets Now we turn to the proof of Theorem 1. We first consider temperedeigenfunctions; it will become clear how the argument is applied tountempered eigenfunctions as well. Lemma 2. Let ǫ > . For any θ ∈ [0 , π ] , there exists a kernel k on T d +1 such that: • k is supported on a ball of radius N . • The operator of convolution with k , call it K θ ( f ) = f ∗ k , isbounded by || K θ || L p ( G ) → L q ( G ) . d,α Cd − αNǫ as an operator from L p ( G ) to L q ( G ) , where C and α are theparameters of the hypothesis (2). • The spherical transform of k satisfies h k ≥ − everywhere,and h k ( θ ) > ǫ − .Proof: Set M = ⌊ ǫ − ⌋ and R = ⌈ N ǫ ⌉ (one should think of N asbeing much larger than ǫ − , so that R is large). By Dirichlet’s Theo-rem, we can find a positive integer r ≤ R such that | rθ mod 2 π | < πR − ≤ πNǫ . There exists an even multiple of r , say r ′ = 2 lr , suchthat Rǫ ≤ r ′ ≤ R (if r ≥ Rǫ , we can simply take l = 1; oth-erwise there is a multiple of r between Rǫ and Rǫ , so take twice ON-LOCALIZATION FOR GRAPH EIGENFUNCTIONS 9 that multiple). Moreover, since we can choose 2 l ≤ Rǫ , we have | r ′ θ mod 2 π | < πǫ ≤ π M .We now set the spherical transform of k to be h k ( θ ) = F M ( r ′ θ ) − F M is the Fej`er kernel of order 2 M . Since r ′ θ mod 2 π ∈ (cid:2) − π M , π M (cid:3) is close enough to 0, we have F M ( r ′ θ ) = 12 M sin (2 M r ′ θ )sin ( r ′ θ ) > M + 2as long as M ≥ 4, and therefore the eigenvalue of φ θ under convo-lution with k will be > M + 1 ≥ ǫ − . Moreover, since F M is positive,the spherical transform of k is bounded below by − 1. It remains tocheck the first two properties.Now, by Corollary 1 of the main estimate, we see that the kernelwhose spherical transform is cos 2 jθ — i.e., the kernel of P j ( T d )—has norm . d − αj as a convolution operator from L p ( G ) to L q ( G ). Thespherical transform of k is a sum of terms of the form M − jM cos jr ′ θ ,where j = 1 , , . . . , M (note that we eliminated the j = 0 term bysubtracting off the constant contribution to F M ) and r ′ ∈ Z . Thus || K θ || L p ( G ) → L q ( G ) . M X j =1 d − αjr ′ . d,α d − αr ′ Then, since r ′ ≥ Rǫ ≥ N ǫ this concludes the proof of Lemma 2. (cid:3) We now wish to apply this convolution operator to examine the lo-calization of eigenfunctions in small sets. Proof of Theorem 1: Pick an eigenfunction φ j of eigenvalue λ j , anda set E satisfying(4) || φ j || L ( E ) = || φ j E || L ( G ) ≥ ǫ Define the operator K j = K θ λj of Lemma 2 (corresponding to θ = θ λ j )if λ j is tempered, or K j = K θ =0 if λ j is untempered. Observe that ineither case K j satisfies (cid:12)(cid:12) h K j ( φ j E ) , φ j E i (cid:12)(cid:12) ≤ || K j ( φ j E ) || q || φ j E || p ≤ || K j || L p → L q || φ j E || p By H¨older’s Inequality, we have || φ j E || p = || φ pj E || /p ≤ || φ pj || /p /p || E || /p / (2 − p ) = || φ j || · | E | − pp so that (cid:12)(cid:12) h K j ( φ j E ) , φ j E i (cid:12)(cid:12) ≤ || K j || L p → L q || φ j E || p ≤ || K j || L p → L q || φ j || · | E | − pp ≤ || K j || L p → L q · | E | − pp . d,α Cd − − αNǫ · | E | − pp (5)by Lemma 2.On the other hand, decompose φ j E spectrally as φ j E = h φ j E , φ j i φ j + g temp + g untemp where g temp and g untemp are the tempered and untempered componentsof φ j E , respectively, excluding the φ j component. Notice that since |h φ j E , φ j i| = || φ j E || ≥ ǫ we have || g temp || ≤ || φ j E || − |h φ j E , φ j i| = || φ j E || (1 − || φ j E || ) ≤ || φ j E || (1 − ǫ )(6)Now the K j -eigenvalue of any tempered eigenfunction is at least − K j must be positive on the untempered eigenfunctions,since each term in the Fourier expansion of the Fejer kernel is of theform cos 2 jθ = cos − i jr = cosh( − jr ) > θ = − ir , and similarlycos(2 jθ ) = cos( − i jr − jπ ) = cosh( − jr ) for θ = − ir − π . Therefore h K j ( φ j E ) , φ j E i ≥ |h φ j E , φ j i| h K j φ j , φ j i − || g temp || ≥ || φ j E || (cid:18) || φ j E || h K j φ j , φ j i − (1 − ǫ ) (cid:19) (7)If λ j is tempered, then Lemma 2 implies that the φ j -eigenvalue of K j is at least ǫ − , whereby h K j φ j , φ j i ≥ ǫ − || φ j || = ǫ − If λ j is untempered, then because cosh(2 jθ j ) > j (0)) we choseto use the same kernel as θ = 0 from the tempered case, and get that h K θ =0 φ j , φ j i ≥ ( F M (0) − || φ j || = 2 M − > ǫ − in the untempered case as well. Applying (4), we get from (7) that h K j ( φ j E ) , φ j E i ≥ || φ j E || ( || φ j E || · ǫ − − ǫ ) ≥ || φ j E || (1 − ǫ ) ≥ ǫ ( ǫ ) = ǫ (8) ON-LOCALIZATION FOR GRAPH EIGENFUNCTIONS 11 and combining (5) with (8) yields | E | − pp & d,α C − ǫ d − αǫ N & d − αǫ N which gives the bound of Theorem 1. (cid:3) References [AN07] N. Anantharaman and S. 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