LECTURE 4. PROOF OF IHARA S THEOREM, EDGE CHAOS. Ihara Zeta Function. ν(c) ζ(u,x) =(1-u ) det(i-au+qu t(ia )

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1 LCTUR 4. PROOF OF IHARA S THORM, DG ZTAS, QUANTUM CHAOS Ihara Zeta Function ν(c) ( ) - ζ(u,x)= -u [C] prime ν(c) = # edges in C converges for u complex, u small Ihara s Theorem. - 2 r- 2 ζ(u,x) =(-u ) det(i-au+qu t(ia ) A=adjacency matrix, Q +I = diagonal matrix of degrees, r=rank fundamental group.

2 dge Zetas Orient the edges of the graph. Multiedge matrix W has ab entry w ab in C, w(a,b)=w ab if the edges a and b look like a Otherwise set w ab =0. b and a is not the inverse of b For a prime C = a a 2 a s, define the edge norm N C wa a wa a wa a wa a ( ) = ( s, ) (, 2) ( 2, 3) ( s, s) Define the edge zeta for small w ab as [ C ] ( ) ζ ( W, X) = N ( C) Properties of dge Zeta Ihara ζ(u,x) = ζ (W,X) non-0 w(i,j)=u edge e deletion ζ (W,X-e)=ζ (W,X) 0=w(i,j), if i or j=e Determinant Formula For dge Zeta (, ) = det ( ) ζ ( W, X) I W From this Bass gives an ingenious proof of Ihara s theorem. Reference: Stark and T., Adv. in Math., Vol. 2 and 54 and 208 (996 and 2000 and 2007) 2

3 xample e 3 e 6 D=Dumbbell Graph e 2 e 5 e e 4 w w w w w 33 0 w35 0 ζ ( W, D) = det 0 w42 0 w w5 0 0 w w 65 w66 e 2 and e 5 are the vertical edges. Specialize all variables with 2 and 5 to be 0 and get zeta function of subgraph with vertical edge removed. Fission Diagonalizes the matrix. Proof of the Determinant Formula log ζ N ( P) j j = ν(p)=m #[P]=m. [ P] j log N ( P) k ζ m m P k ν ( P) = m = P prime path, not class [P] m log ζ = N ( C ) = Tr ( W ) C ν ( C) m m Here C need not be prime path, still closed, no backtrack, no tails k non-prime paths C are powers of primes C=P k for last = use the same sort of argument as in Lecture 2. 3

4 Using the matrix calculus exercise from Lecture 2 det(exp(b)=exp(tr(b)) gives W = Tr W = I W m log ζ ( ) ( ) logdet m m ( ) This proves (log( determinant formula)). ζ (, ) det( ) W X = I W ζ ( W, X ) det( I W ) = ζ 2 2 (, ) ( ) r V ux = u det( I Au+ Qu) 4

5 s ve Part of Bass Proof Define Define starting matrix S and terminal matrix T - both V x2 matrices of 0s and s 0 I J = I 0, if v is starting vertex of oriented edge e = 0, otherwise, if v is the terminal vertex of oriented edge e t ve = 0, otherwise Then, recalling our edge numbering system, we see that SJ = T, TJ=S since start t (end) of e is end (start) t) of e j j+ t t t A = S T, Q+I V = SS = TT Note: matrix A counts number of undirected edges connecting 2 distinct vertices and twice # of loops at each vertex. Q+I = diagonal matrix of degrees of vertices Part 2 of Bass Proof W matrix obtained from W by setting all non -zero w ij equal to t W+J=T S, where J compensates for not allowing edge e to feed into e Below all matrices are ( V +2 ) x ( V +2 ), with V x V st block. The preceding formulas imply that: 2 I V 0 I V ( u ) Su t T I2 0 I2 Wu 2 I I 0 Au + Qu Su V V = t t 0 I T S u I 2 Ju + 2 Then take determinants of both sides to see V 2 ( u ) det( I Wu) = det( I Au+ Qu ) det( I + Ju) j j± 5

6 nd of Bass Proof ( u ) det( I Wu) = det( I Au+ Qu ) det( I + Ju) 2 V 2 2 V 2 I Iu I 0 I Iu I + Ju = implies ( I + Ju) = 2 Iu I -Iu I 0 I( u ) So det(i+ju)=(-u 2 ) Since r-= - V V, for a connected graph, the Ihara formula for the vertex zeta function follows from the edge zeta determinant formula. A Taste of Random Matrix Theory / Quantum Chaos a reference with some background on the interest in random matrices in number theory and quantum physics: A.Terras, Arithmetical quantum chaos, IAS/Park City Math. Series, Vol. 2 (2007). In lecture we mentioned the experimental connections between statistics of spectra of random symmetric real matrices and the statistics of imaginary parts of s at poles of Ihara ζ(q -s ) (analogous to statistics of imaginary parts of zeros of Riemann ζ and spectra of Hermitian matrices). 6

7 from O. Bohigas and M.-J. Giannoni, Chaotic motion and random matrix theories, Lecture Notes in Physics, 209, Springer-Verlag, Berlin, 984: arrows mean lines are too close to distiguish The dichotomy RMT spacings (GO etc) ½πxexp( πx 2 /4) quantum spectra of system with chaos in classical counterpart Bohigas Giannonii Schmit Conjecture, 984 eigenvalues of Laplacian for nonarithmetic manifold zeros Riemann zeta Poisson spacings exp(-x) energy levels of quantum system with integrable system for classical counterpart t Berry Tabor Gutzwiller Conjecture, 977 eigenvalues of Laplacian for arithmetic manifold; e.g. H/SL(2,Z) Sarnak invented the term arithmetical quantum chaos to describe the 2nd row of our table. See the book of Katz and Sarnak for some proved results about zetas of curves over finite fields. See Rudnick What is quantum chaos? Notices AMS, Jan for definitions of some of these things in the context of billiards. None of these conjectures is proved as far as I know. 7

8 Now we wish to add a new column to earlier figure - spacings of the eigenvalues of the W matrix of a graph Here although W is not symmetric, we mean the nearest neighbor spacing (i.e., histogram of minimum distances between eigenvalues which lie in the complex plane not just the real axis). Reference on spacings of spectra of non-hermitian or non-symmetric matrices. P. LeBoef, Random matrices, random polynomials, and Coulomb systems,arxiv. J. Ginibre, J. Math. Phys. 6, 440 (965). Mehta, Random Matrices, Chapter 5. An approximation to the density for spacings of eigenvalues of a complex matrix (analogous to the Wigner surmise for Hermitian matrices) is: Γ s 3 4 4Γ 4 se Statistics of the poles of Ihara zeta or reciprocals of eigenvalues of the dge Matrix W Define W to be the 0, matrix you get from W by setting all non-0 entries of W to be. Theorem. ζ(u,x) - =det(i-w u). Corollary. The poles of Ihara zeta are the reciprocals of the eigenvalues of W. The pole R of zeta is: R=/Perron-Frobenius eigenvalue of W. 8

9 Properties of W ) A B, B and C symmetric real, A real W = T C A 2) jth row sums of entries are q j +=degree vertex which is start of edge j. Poles Ihara Zeta are in region q - R u, q+=maximum degree of vertices of X. Theorem of Kotani and Sunada If p+=min vertex degree, and q+=maximum vertex degree, non-real n lpl poles u of zeta satisfy tif u q p Kotani & Sunada, J. Math. Soc. U. Tokyo, 7 (2000) or see my manuscript on my website: Spectrum of Random Matrix with Properties of W -matrix A B W = C A T entries of W are nonnegative from normal distribution B and C symmetric diagonal entries are 0 Girko circle law for real matrices with circle of radius ½(+ 2) n symmetry about real axis Can view W as edge matrix for a weighted graph We used Matlab command randn(500) to get A,B,C matrices with random normally distributed entries mean 0 std dev 9

10 Nearest Neighbor Spacings vs Wigner surmise of Ginibre p w (s) = 4 Γ(5/4) 4 s 3 exp ( Γ(5/4) 4 s 4 ) See P. Leboef, Random matrices, random polynomials ls and Coulomb systems, ArXiv, Nov. 5, 999. Mehta, Random Matrices, Chapter 5 Note that we have normalized the histogram to have area. The spacings are also normalized to have mean. Matlab xperiments with igenvalues of Random W matrix of an Irregular Graph - Reciprocals of Poles of Zeta Circles have radii p blue / R green q turquoise RH approximately true region 2 dimensional but not even an annulus Looks very similar to the regions obtained for random covers of a small base graph. probability of edge

11 The corresponding nearest neighbor spacings for the preceding graph vs the Wigner surmise Matlab xperiment Random Graph with Higher Probability of an dge Between Vertices (dge Probability ) RH true

12 Spectrum W for a Z 6 xz 65 -Cover of 2 Loops + xtra Vertex are pink dots Circles Centers (0,0); Radii: 3 -/2, R /2,; R.47 RH very False: Lots of Pink outside green circle Level Spacing for the Previous W Matrix Comparing W spacings for this abelian cover with the random cover following looks like the dichotomy between spacings of eigenvalues of the Laplacian for arithmetic vs non-arithmetic groups. 2

13 Spec W for random 70 cover of 2 loops plus vertex graph in picture. The pink dots are at Spectrum W. Circles have radii q, / R, p, with q=3, p=, R RH approximately True. Level Spacing for the Previous W Matrix vs modified Wigner surmise: 3

14 References: 3 papers with Harold Stark in Advances in Math. Papers with Matthew Horton & Harold Stark on my website For work on directed graphs, see Matthew Horton, Ihara zeta functions of digraphs, Linear Algebra and its Applications, 425 (2007) work of Angel, Friedman and Hoory giving analog of Alon conjecture for irregular graphs, implying our Riemann Hypothesis (see Joel Friedman s website: ca/~jf) The nd 4