Lecture 3. A Proof of Ihara s Theorem, Edge & Path Zetas, Connections with Quantum Chaos

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The 4-regular tree T 4 can be identified with the 3-adic quotient SL(2,Q 3 )/SL(,Z 3 ) Ihara Zeta Function ν(c) (

1 Lecture 3. A Proof of Ihara s Theorem, dge & Path Zetas, Connections with Quantum Chaos Audrey Terras Correction to lecture The 4-regular tree T 4 can be identified with the 3-adic quotient SL(2,Q 3 )/SL(,Z 3 ) 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 Basic Assumptions graphs are connected, with r=rank fundamental group >, no degree vertices (called leaf vertex, hair, danglers,...) Outline of Talk: ) Bass s proof of Ihara s theorem. It involves defining an edge zeta function with more variables coming from pairs of directed edges of the graph 2) Path zeta function which depends only on variables from the edges corresponding to generators of the fundamental group of the graph 3) a bit of quantum chaos for the W matrix 2

3 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 = 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 3

4 Determinant Formula For dge Zeta ζ ( ) d t W X = I W (, ) det ( ) 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) 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. 4

no tails though. k An exercise in matrix calculus gives ( ) m m Llog det I W Tr( W ) = This proves L(log( determinant formula)).

5 Proof of the Determinant Formula j N ( C) logζ = j L= wij w [ C] j L L N C k ( ) log ζ m k m C k ( ), ν ( C) = m = C primitive, no backtrack, no tails i, j L N C Tr W ij k LN ( C) = kmn ( C) m log ζ = ( ) = ( ) C m Here C need not be primitive, still no backtrack, no tails though. k An exercise in matrix calculus gives ( ) m m Llog det I W Tr( W ) = This proves L(log( determinant formula)). So we get the formula ζ (, ) det( ) W X = I W up to a multiplicative constant. The proof ends by noting that both sides are when all the w ij are 0. 5

6 ζ ( W, X ) det( I W ) = ζ 2 2 (, ) ( ) r V ux = u det( I Au+ Qu) Part of Bass Proof Define starting matrix S and terminal matrix T s ve Define, if v is starting vertex of edge e = 0, otherwise, if v is the terminal vertex of 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 0 I J = I 0 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 6

7 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± 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. 7

Next we define a zeta function invented by Stark which has several

It can be used to compute the edge zeta using smaller determinants.

spanning trees A tree is a connected graph without cycles.

8 Next we define a zeta function invented by Stark which has several advantages over the edge zeta. It can be used to compute the edge zeta using smaller determinants. It gives the edge zeta for a graph in which an edge has been fused. spanning trees A tree is a connected graph without cycles. A spanning tree for a graph X is a subgraph which is a tree and which contains all the vertices of X. the red graph is a spanning tree for K 4 8

9 Fundamental Group of X can be identified with group generated by edges left out of a spanning tree e,... e r, e,...,, e r 2r 2r multipath matrix Z has ij entry z ij in C if ej e i, z ij =0, otherwise. Imitate the definition of the edge zeta function. Define for a prime path C = a a where a e e, h { ± s j,, ± r } s the path norm NP( C) = z( as, a) z( ai, ai+ ) Define the path zeta function i= [ C ] ( ) ζ ( Z, X) = N ( C) P P edges left out of a spanning tree T of X are inverse edges are e,... er e = + e,..., e = e r 2r r t,...,, t X edges of the spanning tree T are with inverse edges t X,... t2 X 2 If ei e j, write the part of the path between e i and e j as the (unique) product tk t kn A prime cycle C is first written as a reduced product of generators of the fundamental group e j and then a product of actual edges e j and t k. Now specialize the multipath matrix Z to Z(W) with entries Then n- z = w(e,t )w(t,e ) w(t,t ) ij i k kn j kν kν+ ν= ζ (Z(W),X) = ζ (W,X) P 9

10 c f Recall that the edge zeta involved a 6x6 determinant. b e The path zeta is only 4x4. Maple computes ζ much faster than the 6x6. Fusion: shrink edge b to a point. The edge zeta of the new graph obtained by setting w xb w by =w xy in specialized path zeta & same for e instead of b. waa wabwbc 0 wabwbf wcewea wcc wcewed 0 0 wdbwbc wdd wdbw bf wfewea 0 wfewed wff a d xercises ) Fill in the details of the proof that /ζ (W,X)=det(I-W). 2) Fill in the details of the proof that the formula in exercise 4 implies Ihara s 3-term determinant formula for the vertex zeta. 3) Write a Mathematica (or whatever) program to do the process that specializes the path zeta to get the edge zeta. 0

11 4) Prove that if L= wij w i, j ( ) m L log det I W Tr ( W ) ij = m Hint: Use the fact that you can write the matrix W (which is not symmetric) as a product W=U - TU, where U is orthogonal and T is upper triangular by Gram-Schmidt. 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

Ihara ζ(q -s ) (analogous to statistics of

Giannoni, Chaotic motion and random matrix

12 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). 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 2

different origins, as illustrated in the Figure, each column with 50 levels... Note that the spectra have been rescaled to the same vertical axis from 0 to 49. (a) Poisson spectrum, i.e., of a random variable with spacings of probability density e x.

(d) from eigenvalues corresponding to transverse vibrations of a membrane whose boundary is the Sinai billiard which is a square with a circular hole cut out centered at the center of the square.

13 O. Bohigas and M.-J. Giannoni, Chaotic motion and random matrix theories, Lecture Notes in Physics, 209, Springer-Verlag, Berlin, 984: The question now is to discover the stochastic laws governing sequences having very different origins, as illustrated in the Figure, each column with 50 levels... Note that the spectra have been rescaled to the same vertical axis from 0 to 49. (a) Poisson spectrum, i.e., of a random variable with spacings of probability density e x. (b) primes between and (c) resonance energies of compound nucleus observed in the reaction n+66r. (d) from eigenvalues corresponding to transverse vibrations of a membrane whose boundary is the Sinai billiard which is a square with a circular hole cut out centered at the center of the square. (e) the positive imaginary parts of zeros of the Riemann zeta function (from the 55th to the 600th zero). (f) is equally spaced - the picket fence or uniform distribution. ib ti (g) from P. Sarnak, Arithmetic quantum chaos, Israel Math. Conf. Proc., 8 (995), (published by Amer. Math. Soc.) : eigenvalues of the Poincaré Laplacian on the fundamental domain of the modular group SL(2,Z), 2 2 integer matrices of determinant. (h) spectrum of a finite upper half plane graph for p=53 (a = δ = 2), without multiplicity (see my book Fourier Analysis on Finite Groups) The Figure is from from Bohigas, Haq, and Pandey, Fluctuation properties of nuclear energy levels and widths: comparison of theory with experiment, in K.H. Bockhoff (d.), Nuclear Data for Science and Technology, Reidel, Dordrecht, 983) Level spacing histogram for (a) 66r and (b) a nuclear data ensemble. 3

Wigner surmise for spacings of spectra of random symmetric real matrices This means that you arrange the eigenvalues) i in decreasing order: 2 n.

Wigner s Surmise from 957 says the level (eigenvalue) spacing histogram is the graph of the function ½πxexp( πx 2 /4),if the mean spacing is.

Note the level repulsion indicated by the vanishing of the function at the origin.

14 Wigner surmise for spacings of spectra of random symmetric real matrices This means that you arrange the eigenvalues) i in decreasing order: 2 n. Assume that the eigenvalues are normalized so that the mean of the level spacings i i+ is. Wigner s Surmise from 957 says the level (eigenvalue) spacing histogram is the graph of the function ½πxexp( πx 2 /4),if the mean spacing is. In 960, Gaudin and Mehta found the correct distribution function which is close to Wigner s. The correct graph is labeled GO in the Figure preceding. Note the level repulsion indicated by the vanishing of the function at the origin. Also in the preceding Figure, we see the Poisson density which is e x. A reference is Mehta, Random Matrices. Now we wish to add a new column to earlier figure - spacings of the eigenvalues of the W matrix of a graph Call this exercise 5 or, more accurately perhaps, research project. 4

Here although W is not symmetric, the nearest neighbor spacing (i.e., histogram of minimum distances between eigenvalues) is also of interest.

He studies the ensemble of matrices introduced by J. Ginibre, J. Math. Phys. 6, 440 (965).

this will probably not be the correct Wigner surmise. I haven t done this experiment yet.

15 Here although W is not symmetric, the nearest neighbor spacing (i.e., histogram of minimum distances between eigenvalues) is also of interest. many references on the study of spacings of spectra of non- Hermitian or non-symmetric matrices. I did find one: P. LeBoef, Random matrices, random polynomials, and Coulomb systems. He studies the ensemble of matrices introduced by J. Ginibre, J. Math. Phys. 6, 440 (965). An approximation to the distribution of spacings of eigenvalues of a complex matrix (analogous to the Wigner surmise for Hermitian matrices) is: Γ s 3 4 4Γ se 4 Since our matrix is real, this will probably not be the correct Wigner surmise. I haven t done this experiment yet. In what follows, I just plot the reciprocals of the eigenvalues of W - the poles of Ihara zeta for various graphs. The distribution looks rather different than that of a random real matrix with the properties of W. 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. 5

16 Properties of W ) A B, B and C symmetric real, A real W = T C A 2) 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 spectrum W=A B;C trn(a) with A rand norm, B,C rand symm normal r =sqrt(n)*(+2. 5)/2 and n=000 A B W = C A T 20 B and C symmetric Girko circle law with a symmetry with respect to real axis since our matrix is real. (Girko, Theory Prob. Appl. 22 (977)) We used Matlab command randn(000) to get A,B,C matrices with random normally distributed entries mean 0 std dev. 6

17 What is the meaning of the RH for irregular graphs? For irregular graph, natural change of variables is u=r s, where R = radius of convergence of Dirichlet series for Ihara zeta. Note: R is closest pole of zeta to 0. No functional equation. Then the critical strip is 0 Res and translating back to u- variable. In the q+-regular case, R=/q. Graph theory RH: ζ(u) is pole free in R < u < R xperiment on Locations of 0.6 Poles of Ihara Zeta of Irregular Graphs 0.4 All poles but - of ζ X (u) for a random graph with 80 vertices denoted by 0.2 using Mathematica: RandomGraph[80,/0] circles centered at 0 with radii R, q -/2, R /2, (pq) -/4, p -/2 q+=max degree, p+=min degree, R=radius of convergence of uler product for ζ X (u) RH is false but poles are not far inside circle of radius R /2 RandomGraph[80,/0] means probability of edge between 2 vertices =/0. 7

xperiment on Locations of Poles of Ihara Zeta of Irregular Graphs

are denoted, using Mathematica RandomGraph[00,/2] Circles centered

example with probability /0 of an edge rather than ½.

degree, p+=min degree R=radius of convergence of product for ζ X

it Poles of Ihara zeta are boxes on right.

18 xperiment on Locations of Poles of Ihara Zeta of Irregular Graphs All poles except - of ζ X (u) for a random graph with 00 vertices are denoted, using Mathematica RandomGraph[00,/2] Circles centered at 0 with radii RH is false maybe not as false as in previous example with probability /0 of an edge rather than ½. Poles clustering on RH circle (green) R, q -/2, R /2, p -/2 q+=max degree, p+=min degree R=radius of convergence of product for ζ X (u) Matthew Horton n s Graph has /R e to 7 digits. it Poles of Ihara zeta are boxes on right. Circles have radii R,q -½,R ½,p -½, if q+=max deg, p+=min deg. Here The RH is false. Poles more spread out over plane. 8

Poles of Ihara Zeta for a Z 6 xz 62 -Cover of 2 Loops +

8 0.6 0.4 0.2 0 0.2 0.4 0.6 0.8.2.2 0.8 0.6 0.4 0.2 0 0.2

2 Z is random 700 cover of 2 loops plus vertex graph in

19 Poles of Ihara Zeta for a Z 6 xz 62 -Cover of 2 Loops + xtra Vertex are pink dots Circles Centers (0,0); Radii: 3 -/2, R /2,; R.47 RH very False.2 random 700 cover of 2loops with extra vertex on loop Z is random 700 cover of 2 loops plus vertex graph in picture. The pink dots are at poles of ζ Z. Circles have radii q -/2, R /2, p -/2, with q=3, p=, R RH approximately True. 9

References: 3 papers with Harold Stark in Advances in Math.

Contemporary Mathematics, Volume 45 (2006) Quantum Graphs and Their

See my draft of a book: www.math.ucsd.edu/~aterras/newbook.

of digraphs, Linear Algebra and its Applications, 425 (2007) 30 42.

20 References: 3 papers with Harold Stark in Advances in Math. Paper with Matthew Horton & Harold Stark in Snowbird Proceedings, Contemporary Mathematics, Volume 45 (2006) Quantum Graphs and Their Applications, Contemporary Mathematics, v. 45, AMS, Providence, RI See my draft of a book: Draft of new paper joint with Horton & Stark: also on my website There was a graph zetas special session of this AMS meeting many interesting papers some 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: The nd 20

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 )

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 ) 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 )