MATRIX INTEGRALS AND MAP ENUMERATION 1

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1 MATRIX INTEGRALS AND MAP ENUMERATION 1 IVAN CORWIN Abstract. We explore the connection between integration with respect to the GUE and enumeration of maps. This connection is faciliated by the Wick Formula. Random matrix integration methods provide an important tool is deriving exact formulas and recurrence relations for certain important enumerative quantities. Connections exist with QFT and QCD which are not explored. Much of this material is from the article Matrix Integrals and Map Enumeration by A. Zvonkin, as well as from the maps book by Lando and Zvonkin. 1. Wick formula and GUE 1.1. Wick formula. (1) Given a (centered) Gaussian measure µ on R n it can be written in terms of its covariance matrix C as dµ(x) = cexp{ 1 2 (Bx,x)}dx where c = (2π) n/2 (det B) 1/2 and where B = C 1. (2) The matrix C gives covariances so that x i x j = c ij. (3) If f(x) is a monomial of odd degree, then f = 0 (by symmetry of µ). Theorem 1.1 (Wick formula). Let f 1,f 2,...,f 2k be a set of linear functions (not necessarily different) of x 1,...,x n. Then, f 1 f 2 f 2k = f p1 f q1 f p2 f q2 f pk f qk, where the sum is taken over all permutations p 1 q 1 p 2 q 2 p k q k of the set of indices 1,2,...,2k such that p 1 < p 2 < < p k and p i < q i for all i. There are a total of (2k 1)!! = (2k 1) terms to sum over, and each such permutation is known as a Wick coupling. (4) Compute the 2k th moment of the standard one-dimensional Gaussian measure to be (2k 1)!!. (5) Proof idea is to express the moment generating function in terms of the coefficients for the Taylor series of the log of the moment generating function which happens to terminate after quadratic terms The GUE. (1) The Gaussian Unitary Ensemble (GUE) is a Gaussian measure on the space of Hermitian matrices. Let H = (h ij ) be a N N Hermitian matrix (h ij = h ji ) and denote H N the space of all such matrices. As a vector space this is isomorphic to R N2 where we set x ij = R(h ij ) and y ij = I(h ij ) for 1 i < j N and x ii = h ii, where the x and y are real parameters. Let dv(h) be the Lebesgue measure on this real vector space N dv(h) := dx ii dx ij dy ij. Date: October 17, i<j 1

2 MATRIX INTEGRALS AND MAP ENUMERATION 1 2 We fix a quadratic form tr(h 2 ) and compute the matrix B for this form: tr(h 2 ) = N h ij h ji = i,j=1 N h ij h ij = i,j=1 N x 2 ii + i j (x 2 ij + y2 ij ) = N x i i i<j (x 2 ij + y2 ij ), so B has N ones on the diagonal, followed by N 2 N twos on the diagonal, and no off diagonal terms. Therefore detb = 2 N2 N. Using this quadratic form we have { dµ(h) = (2π) N2 /2 2 (N2 N)/2 exp 1 } 2 tr(h2 ) dv(h). Lemma 1.2. With respect to the measure µ above we have h ij h ji = 1 while all other second moments are zero. (2) Calculating tr(h 2n ) using Wicks formula: We can write tr(h 2n ) = h i1 i 2 h i2 i 3 h i2n 1 i 2n h i2n i 1 where we sum over all N 2n values for the indices. To calculate expectations we may use Wicks formula and the lemma above. (3) Example for n = 4: We first choose some Wicks coupling for the eight terms in the trace product. Lets couple as follows h i1 i 2 h i4 i 5 h i2 i 3 h i5 i 6 h i3 i 4 h i8 i 1 h i6 i 7 h i7 i 8. Now determine what conditions are necessary for this product not to be zero. From the Lemma, From this we can read off that h i1 i 2 h i4 i 5 = 1 i 1 = i 5, i 2 = i 4, h i2 i 3 h i5 i 6 = 1 i 2 = i 6, i 3 = i 5, h i3 i 4 h i8 i 1 = 1 i 3 = i 1, i 4 = i 8, h i6 i 7 h i7 i 8 = 1 i 6 = i 8, i 7 = i 7. i 1 = i 5 = i 3 = i 1, i 2 = i 4 = i 8 = i 6 = i 2, i 7 = i 7. Thus there are only N V combinations of indices, where V = 3 is the number of free indices. If we sum over all Wick couplings and determine the contribution of each, we get our desired answer. (4) We can represent this Wick coupling graphically as in the figure above. We will see that every Wicks coupling corresponds with a way of gluing a 2k polygon, and that the power of N will correspond with the number of vertices in the resulting map. This connection will allow us to solve certain combinatorial enumeration problems using the analytic tools of random matrix integrals we will soon develop. 2. Maps 2.1. Definition and properties. (1) A map is a graph imbedded into a compact oriented surface so that edges do not intersect, vertices are distinct, and the complement of the graph is a disjoint union of disks (the faces of the map).

3 MATRIX INTEGRALS AND MAP ENUMERATION 1 3 ded_figure Figure 1. A polygon representation for the Wicks coupling ross_roads (2) The degree of a vertex is the number of edges incident to it (a loop is incident twice) and the degree of a face is the number of boundary edges. (3) The quantity χ = V E + F = 2 2g depends on the genus g of the surface on which the map is drawn, and not on the actual map itself. (4) Can draw maps using darts, half edges, fat ribbons, or directed arrows. (5) From a map we can create a dual map (6) There are many ways of create a map: (a) You can start with a graph and then specify a cyclic ordering of the edges arond each vertex. This uniquely determines a map. There are v (d v 1)! such cyclic orderings (d v is the degree of vertex v). (b) You can start with a collection of oriented faces and then glue edges together respecting the orientation. (c) Dual to this you can start with a collection of round-abouts and glue these together respecting traffic flow direction. Figure 2. A round-about 2.2. Enumerating one face maps. (1) Consider a 2n-gon and fix an orientation (clockwise) and label vertices i 1,i 2,...,i 2n (or just fix a marked vertex). Could, dually, consider a 2n star. (2) There are (2n 1)!! different possible gluing and hence that many possible maps. We would like to enumerate them in terms of their genus. Clearly the maximal genus is n/2. Let ǫ g (n) := #(gluing of 2n-gon of genus g). We know n/2 g=0 ǫ g(n) = (2n 1)!!.

4 MATRIX INTEGRALS AND MAP ENUMERATION 1 4 (3) From the Euler characteristic, we know that F = 1, E = k so V = n + 1 2g. (4) What is ǫ 0 (n)? It is none-other-than the Catalan numbers C n = 1 ( ) 2n. 1 + n n A genus zero map correspond with a resulting graph which is a tree and come from non-crossing partitions of the edges Connection with GUE. (1) We graphically represent tr(h 2n ) by a polygon with 2n sides and vertices i 1 through i 2n labeled clockwise. For each of the (2n 1)!! Wick couplings we associate a gluing of edges. The number V of free indices in the Wick coupling is equal to the number of vertices of the resulting map. The contribution of the coupling is N V or equivalently N n+1 2g. This is also known as a Feynman diagram. Theorem 2.1. Let T n (N) be T n (N) := tr(h 2n ) = tr(h 2n )dµ(h). H N Then (2) Let T n (N) = n/2 g=0 n/2 ǫ g (n)n k+1 2g = N k+1 g=0 ( ) 1 g ǫ g (n) N 2, where ǫ g (n) is the number of label one-face maps of genus g with n edges. T(N,s) = 1 + 2Ns + 2s n=1 T n (N) (2n 1)!! sn. Then we can determine what the ǫ g (n) are from the following result Theorem 2.2. The generating function T(N,s) is equal to ( ) 1 + s N T(N,s) =. 1 s From this we have that the numbers ǫ g (n) satisfy the recurrence relation (n + 2)ǫ g (n + 1) = (4n + 2)ǫ g (n) + (4n 3 n)ǫ g 1 (n 1) where ǫ g (0) = 1 for g = 0 and 0 otherwise. We will show how to prove this next week. 3. Other enumeration problems and their relationships with matrix integrals 3.1. Arbitrary (finite) gluing. (1) We can either glue edges or glue pairs of arrows, since these are dual to each other. (2) Consider a collection of m polygons with α 1,...,α m edges each. Orient each one and fix a marked vertex (or a fixed set of indices). Consider the collection of gluing and the resulting maps. Not all maps will be connected. and therefore define the number of vertices for a (possibly non-connected) map resulting from a gluing σ to be the sum over all connected components of the number of vertices.

5 MATRIX INTEGRALS AND MAP ENUMERATION 1 5 Theorem 3.1. We have s N V (σ) = tr Hi α σ where the sum is over all possible gluing of edges. (3) Work out examples: A single square; two 2-gons; a square and a 2-gon, two loops, a square and a loop, etc. (have a sheet of these examples worked out) 3.2. Enumerating quadrangulations. (1) Consider gluing of n squares. The enumeration of the numbers of vertices in the resulting maps is given by F n (N) = tr(h 4 ) n = N F(σ). σ (2) If we want to enumerate just connected maps, this just won t do. However, there is a way to go from maps to connected maps. (3) Owing to a general enumerative combinatoric theorem, we can go between general and connected maps, however, if we look to the exponential generating function for F n (N), considered as a series in n. Let 1 F(s,N) = n! F n(n)s n be the exponential generating function for F n (N). Now define C n (N) to be n=0 C n (N) = σ c N F(σc) where σ c is restricted to all gluing which result in a connected map. Let 1 C(s,N) = n! C n(n)s n n=0 be the exponential generating function for C n (N). Then Proposition 3.2. The exponential generating functions F(s, N) and C(s, N) are related by log(f(s,n)) = C(s,N). (4) Observe that F( t,n) = e t tr H4. Therefore by studying this expectation and in particular its log, we can learn about enumeration connected maps which are quadrangulations. More on this next week. I. Corwin, Courant Institute of the Mathematical Sciences, New York University, 251 Mercer Street, New York, NY 10012, USA address: corwin@cims.nyu.edu