A combinatorial trace formula

Transcription

1 A combinaorial race formula F. R. K. Chung Universiy of Pennsylvania Philadelphia, Pennsylvania 1914 S.-T. Yau Harvard Universiy Cambridge, Massachuses 2138 Absrac We consider he race formula in connecion wih he Laplacian of a graph. This can be viewed as he combinaorial analog of Selberg s race formula, which has influenced many areas of number heory, represenaion heory and geomery. We sar wih a general graph wih any specified se of edges. Therefore he race formula we consider is in full generaliy. In paricular, we consider laice graphs, heir subgraphs and higherdimensional generalizaions. By using he analog of he Poisson inversion formula, several hypergeomeric equaliies are derived. 1 Inroducion We consider a graph G =(V,E) wih verex se V = V (G) andedgesee = E(G). We assume G conains no loops or muliple edges (he definiions can be easily generalized o weighed graphs wih loops, cf. [5]). We will mainly consider finie graphs (while an infinie graph is viewed as aking he limi of a class of finie graphs). We define he marix L wih rows and columns indexed by verices of G as follows. d v if u = v L(u, v) = 1 if u and v are adjacen oherwise where d v denoes he degree of v. Le T denoe he diagonal marix wih he (v, v)-enry having value d v. The Laplacian L of G is defined o be 1 if u = v and d v L(u, v) = 1 if u and v are adjacen du d v oherwise This paper has appeared in Tsing Hua Lecures on Geomery and Analysis, (ed. S.-T. Yau), Inernaional Press, Cambridge, Massachuses, 1997,

2 When G is k-regular (i.e., d v = k for all v), i is easy o see ha L = I 1 k A where A is he adjacency marix of G. In general, he eigenvalues of L and L can be quie differen. The eigenvalues of L are denoed by = λ λ 1 λ 2... λ n 1.IfG is conneced, we have <λ 1. Various properies of he λ 1 s can be found in [3]. For a graph G, we consider he hea kernel H, which is defined for as follows: H = i e λ i P i = e L = I L L2... (1) where P i denoes he projecion ino he eigenspace associaed wih eigenvalue λ i. In paricular, H = I. and H saisfies he hea equaion H = LH Here we sae some useful facs abou he hea kernel. Fac 1: For any wo verices x and y, wehave H (x, y) Fac 2: For any wo verices x and y, wehave H (x, y) = j e λ j φ j (x)φ j (y) where φ k denoes he eigenfuncion corresponding he he eigenvalue λ k of he Laplacian L and φ k s are orhonormal. We omi he proofs for boh facs since i can be found in [6]. Fac 2 looks hard bu i can be easily proved using he definiions while Fac 1 looks easy bu is proof is quie involved. We remark ha we can consider induced subgraphs of a graph and hea kernels wih he Dirichle boundary condiion or Neumann boundary condiions [5]. However, for simpliciy of exposiion, we will deal wih graphs wih no boundaries in his paper. 2

3 2 The race formula For a graph G wih hea kernel H and eigenvalues λ i, he simples version of he race formula is: Tr (H )= x H (x, x) = j e λ j (2) In general, he hea kernel is no easy o deermine. However, he race can be compued in polynomial ime (of he same order of complexiy as marix muliplicaion). We remark ha a good par of specral geomery is in esimaing he hea kernel from which we can exrac various invarians of he Riemannian manifold. In a way, he race formula capures he essenial par of he hea kernel and i can ofen serve he same purpose. The combinaorial race formula provides an effecive ool o capure he major invarians and properies of a graph jus as he Selberg s race formula does for algebraic srucures and geomeric surfaces. If he graph is verex ransiive, (i. e. for any wo vericies u and v, here is an edgepreserving auomorphism mapping u o v.) hen for all v, H (x, x) has he same value and hus is easier o evaluae. Perhaps he simples verex-ransiive graph is an n-cycle. As i urns ou, he hea kernel for an n-cycle can be deduced from he hea kernel of an infinie pah. In fac, he hea kernel H(x, x) of an infinie pahs is fundamenal in he sense ha i can be used o deermine he hea kernels of numerous graphs such as n-cycles, n-pahs P n,higher dimensional laice graphs and some of heir induced subgraphs. The imporance of he hea kernel of an infinie pah is perhaps due o he fac ha i is he discree analog of a line which he higher dimensional Riemannian manifolds are based upon. We will firs derive he hea kernel for an infinie pah. Based on i, we will derive race formula for cycles, pahs, laice graphs and quoien graphs which we will describe in Secion 6. Before we proceed, we sae a simple bu useful combinaorial observaion. Lemma 1 In a k-regular graph G, supposew s =(v,,v s ) is a walk of lengh s from verex v, hrough verices v j o v s such ha v j = v j+1 or {v j,v j+1 } is an edge. Then he hea kernel H (x, y) of G saisfies H (x, y) = s s s! ( r ( 1 k )r ρ(s, r)) where ρ(s, r) denoes he number of walks of lengh s from x = v o y = v s conaining exacly r edges. 3

4 Proof: This follows from he fac ha H = I L L2 + and L s (x, y) is exacly he sum of weighs over all walks of lengh s from x o y where he weigh of a pah of lengh s conaining exacly r edgesisjus s 1 j= L(v j,v j+1 )=( 1 k )r. 3 The hea kernel for he one-dimensional laice graphs Firs we consider he one-dimensional case. The hea kernel for an infinie pah can be viewed as aking he limi of an n-cycle, C n. We use he noaion ha verices are labeled by inegers and x is adjacen o x +1andx 1. Theorem 1 In an infinie pah, he hea kernel saisfies: H (x, x) = ( 2k ) k k! ( 2 )k (3) k Proof: We use Lemma 1 and also we use he fac ha L = I Y/2 Y 1 /2whereY is he cyclic operaor Y (x) =x + 1. The coefficien of k in he expansion of H (x, x) in(1) is exacly 1/k! muliplied by he coefficien of Y in (I Y/2 Y 1 /2) k. By collecing he coefficiens erm by erm, Theorem 1 is proved. From Theorem 1, we can see ha H (x, x) is exacly a confluen hypergeomeric series 1F 1 (1/2; 1; 2) and is closely relaed o he modified Bessel funcions I ν (z). Tha is H (x, x) =F (1/2; 1; 2) =e I ( ) (4) I is known ha he above confluen hypergeomeric series and he modified Bessel funcion do no have a closed form formula. The usual asympoical expansions for I v or for he confluen hypergeomeric series (e.g. [5]) do no seem o have he following useful forms as saed in he following wo heorems: Theorem 2 In an infinie pah, he hea kernel saisfies: 2 H (x, x) = π 2 e y2 dy 1 y2 2 4

5 Proof: For an n-cycle, he eigenvalues of he Laplacian are λ k =1 cos 2πk n for k =, 1,,n 1. Therefore we have n 1 1 H (x, x) = lim n n 1 = lim n n = 1 π π π/2 = 2 π 2 = π πk =2sin2 n k= n 1 k= e λ k 2πk (1 cos e n ) e 2 sin2 x dx 2 e 2 sin2 x dx e y2 dy 1 y2 2 For large and a fixed, we can use he asympoic expansion of I ( ): e I ( ) (2m 1) 2 2π ( 2) m 1 m Therefore we have Theorem 3 When is sufficienly large, he hea kernel of an infinie pah saisfies H (x, x) = 1 (1 + O( 1 2π )) Nex, we consider he off-diagonal erms of he hea kernel of he infinie pah, ha is H (x, y). Theorem 4 In an infinie pah, he hear kernel saisfies, for any ineger a, H (x, x + a) =H (x, x a) =( 1) a ( 2k ) k+a ( k! 2 )k k a Proof: The proof is quie similar o ha of Theorem 1. We here focus on he coeffien of Y a in L k =(I Y/2 Y 1 /2) k for each k. We noe ha ( m n) = for inegers n>m. So he sum in he saemen of Theorem 4 can be aken over all non-negaive inegers k. Using Theorem 1, we here sae he relaion of H (x, x+a) o a confluen hypergeomeric series and he modified Bessel funcions I ν (z). 5

6 Theorem 5 H (x, x + a) =(/2) a /a! 1 F 1 (1/2+a;1+2a; 2) =( 1) a e I a ( ) Again he above hypergeomeric series in Thereom 4 has no closed formula. The inegral similar o ha in Theorem 2 is somewha complicaed. However, he asympoical esimaes for large >>a 2 have a very simple and elegan form: Theorem 6 The hea kernel of an infinie pah saisfies H (x, x + a) = 2 π π 2 e 2 sin2 x cos 2axdx Proof: We follow he proof of Theorem 2. In an n-cycle, he eigenfuncion φ k associaed wih he eigenvalue λ k =1 cos 2πk n for k =, 1,,n 1isφ k(j) =exp( 2πikj n )/ n.we have 1 H (x, x + a) = lim n n = 1 π = 2 π π π 2 Theorem 7 For a fixed a and >>a 2, we have n 1 k= e λ k e 2πika n e 2 sin2 x+2iax dx e 2 sin2 x cos 2axdx H (x, x + a) = e a 2 2 (1 + O( a )) 2π Proof: Under he assumpion ha a is fixed and >>a 2, we use he known esimaes on Bessel funcions I ν (z) andj ν (z) [8]: We choose I a ( ) = ( i) a J a ( i), J ν (ν sech α) = 1 (1 + O( )). 2πν anh α ν anh α eν(anh α α) a sech α = i, a anh α = 2 + 2, a α = 3πi a 2 +ln a 6

7 By subsiuing ino H (x, x + a), we ge H (x, x + a) = e a 2 2 (1 + O( a )) 2π Using Theorem 4 and 6, we deduce he hea kernel H (x, y, C n )for an n-cycle: Theorem 8 In he n-cycle, he hea kernel saisfies H (x, y, C n )= k= H (x, y + kn) Proof: Using he observaion in Lemma 1, we noe ha all walks from x o x in he cycle C n can be idenified wih walks in he infinie pahs from x o x + nk, forsomek. The sum of weighs from walks from x o x + nk conribue o H (x, x + kn). As an immediae consequence of Theorem 4 and 8, we have he following race formula for an n-cycle: Theorem 9 k= j= ( 2k ) ( 1) nj k+nj ( 2 )k k! = 1 n n 1 k= πk 2 sin2 e n 4 The hea kernel for k dimensional laice graphs We define he laice graph C n (k) o be he caresian produc of k copies of an n-cycle. The infinie laice graph P (k) is jus by aking he limi of C n (k) as n approaches infiniy. We firs consider he 2 dimensional cases. For P (2), each verex is labelled by (x, y),x,y Z. (x, y) isadjaceno(x +1,y), (x 1,y), (x, y +1)and(x, y 1). Theorem 1 In an infinie 2 dimensional laice graph, he hea kernel H (2) saisfies ( )( k 2j j j + a H (2) ((x, y), (x + a, y + b)) = ( 1) a+b k j = H /2 (x, x + a)h /2 (y,y + b) where H is he hea kernel for an infinie pah. )( ) 2(k j) ( /4) k k j + b k! Proof: The proof follows from he fac ha L = I 1 4 Y Y Y Y 2 1 7

8 where Y 1 (x, y) =(x +1,y), Y 2 (x, y) =(x, y +1) Using he expansion of e L, and Theorem 1, we see ha H (2) ((x, y), (x + a, y + b)) can be facored ino H /2 (x, x + a) andh /2 (y,y + b). Therefore we have Theorem 11 For wo verices u and v in a 2 dimensional laice graph, we have u v 2 H (2) (u, v) = e u v (1 + O( )) π where denoes he L 2 norm. Proof: By choosing u =(x, y) andv =(x + a, y + b), Theorem 11 is a consequence of Theorem 1 and Theorem 5. In a similar way, we have Theorem 12 The hea kernel for P k saisfies, for a =(a 1,...,a k ), a i Z, K (a) =K (a 1 )...K (a k ) (5) k k where K (a) =H (k) (x, x + a). Similar o he proof of Theorem 11, we have Theorem 13 H (k) u v 2 (u, v) = e k 2 v (1 + O(k u )) (2π/k) k/2 5 The race formula for laice graphs Le L denoe a subse of Z k and we wrie L = { x n x x : x L, n x Z} Le P k /L denoe he graph formed from P k by idenifying verices x and x + y in L for any y. Furhermore, we define We will prove he following: L = {y (R/Z) k : x, y Z for all x L} 8

9 Theorem 14 (Discree Poisson Formula) P k /L y <L> K (y) = exp( 2 k x L k sin 2 πx j )) j=1 where K (y) is equal o he hea kernel H (x, x+y) of an infinie pah and x =(x 1,...,x k ). Proof: To prove he race formula for he graph P k /L, he hea kernel H(x, x) are all equal and H(x, x) = K (y) where K is defined by he hea kernel of P k. y L k We need o show ha he eigenvalues of P k /L is exacly 2 k sin 2 πx j for x =(x 1,...,x k ) j=1 in L. This can be shown by considering he eigenfuncion f x for x =(x 1,...,x k )inl, defined as follows: I is sraighforward o check ha f x (y) =e 2πi<x,y> for y =(y 1,...,y k ) P k,y i Z Lf x (y) = 1 2k k (2 2cos2πx j )f x (y) j=1 Example 1: Fork =1andL = {n}, P/L is jus an n cycle and we ge k= k = j= j = k= j= ( 2k ) ( 1) nj k+nj ( 2 )k k! = 1 n Example 2: Fork =2andL = {(3, 5), (4, )}, wehave ( )( 2k 2k ) ( 4 )k+k k +3j +4j k +5j k! k! n 1 k= = 1 2 πk 2 sin2 e n 3 k= k = 4 e (sin2 kπ 4 (k 3k )π +sin2 4 ) 5 Example 3: Fork =3andL = {(1, 1, 1)}, P 3 /L is he riangular laice and we have ( )( )( ) ( 1) a+b+c 2k1 2k2 2k3 ( 6 )k 1+k 2 +k 3 k 1 + a k 2 + b k 3 + c k 1!k 2!k 3! a+b+c= k 1 = k 2 = k 3 = π π = 1 π e 2 3 (sin2 x+sin 2 y+sin 2 (x+y)) dxdy 9

10 References [1] B. Bollobás, Exremal Graph Theory, Academic Press, London (1978). [2] R. Brooks, The specral geomery of k-regular graphs, Journal D analyse Mahémaique, 57 (1991) [3] F. R. K. Chung and S. -T. Yau, Eigenvalues of graphs and Sobolev inequaliies, Combinaorics, Probabiliy and Compuing, 4 (1995) [4] F. R. K. Chung and S. -T. Yau, A Harnack inequaliy for homogeneous graphs and subgraphs, Communicaions in Analysis and Geomery, 2, (1994), [5] F. R. K. Chung and S. -T. Yau, Coverings of graphs and heir hea kernels, preprin. [6] F.R.K.Chung,Specral Graph Theory, CBMS Lecure Noes, 1996, AMS Publicaion. [7] S. T. Yau and Richard M. Schoen, Lecures on Differenial Geomery, (1994), Inernaional Press, Cambridge, Massachusees [8] G. N. Wason, A Treaise on he Theory of Bessel Funcions, Cambridge Universiy Press,