As we have discussed previously, the moments of the Gaussian distribution are given by

where we are using the Iverson bracket for indicator functions, and the double factorial is like the usual factorial, but going down in steps of two (e.g. ). Thus one can evaluate the integral of any polynomial against the standard Gaussian measure using linearity of the integral and this simple formula.

In dimensions, if one wants to be able to compute the integral

of a polynomial function on then it is necessary and sufficient to compute all correlation functions of the Gaussian measure, which simply means integrals of monomials of a given degree. Physicists have a nice notation for such integrals, and write them as

where Observe that in the case $latex $ is a constant function, our old formula still applies, since this reduces to the one-dimensional case. Another easy case is the integration of monomials of degree two, where we have

If this is

while if we instead get

The case of general -point correlation functions reduces to that of -point functions via the following combinatorial formula, which statisticians call Isserlis’ formula and physicists call Wick’s lemma. Wick’s lemma says that, for any and any index function the -point correlation function

is equal to the sum of -point correlators obtained by pairing the factors in the product in all possible ways. For example we have

because there aren’t any pairings of three objects, while

because is the set of fixed point free involutions in Hence equals a sum of delta functions,

Equivalently, one may say that is equal to the number of fixed point free involutions such that the given index function is constant on the cycles of

At the end of Lecture 15, we were considering the Gaussian probability measure on the real vector space of Hermitian matrices (i.e. complex matrices such that ). This real vector space is isomorphic to an isomorphism is gotten by reading the upper triangular part of the Hermitian matrix across rows, so that each matrix element with contributes two real parameters and for a total of real parameters, and each of the diagonal matrix contributes itself, , giving more real parameters. A natural scalar product on is given by the trace form, and one can define the corresponding Gaussian measure on to be the absolutely continuous (w.r.t to Lebesgue) measure with density

,

where

is a normalization constant which can be explicitly computed, but is not simply equal to

as you might expect. The reason for this is the identity

which corresponds to a scalar product on but not the standard scalar product (i.e. dot product) because of the factor of on terms coming from off-diagonal matrix elements. This is a minor difference, and the upshot is that the Wick formula still works, and reduces the computation of correlation functions

to two-point correlators, which are given by

For example, we have

the terms of which correspond, just as in the vector case, to the fixed point free involutions in and this reduces to the corresponding sum of delta functions of indices,

Let us use the matrix Wick lemma to compute a particularly interesting correlation function: On one hand, this has a spectral interpretation: we have

where are the eigenvalues of On the other hand, this is a sum of -point correlation functions, namely

which may more compactly be written as

where is the full forward cycle in the symmetric group In fact, it is convenient here to change the variance in the Gaussian measure, altering it to match the the in (you’ll see why in a moment). This means that we are instead computing

which has the effect of dividing the -point functions by i.e.

Now if we apply the Wick formula, the computation evolves as follows:

Now, to say that simply means that is a cycle of the involution which in turn just means that

so that

and we see that the product

is really the single condition at the level of equality of functions (a concatenation of functions means composition). We thus have

Now, the condition is not difficult to interpret: in order to return true, it must be the case that is constant on the cycles of the permutation To build such a function, we are free to assign any of the numbers for the value of on each cycle of and we thus obtain

And now, we know how to make topology out of this: the combinatorial map yields a surface by taking the single cycle of and writing it out as a -gon, and then glueing its sides in pairs using the involution When we do this, the former vertices and edges of the polygon become the vertices and edges of a graph drawn on the resulting surface: the number of vertices of this embedded graph is the number of cycles in the number of edges is and of course the number of faces is since we started with a single polygon. We thus have that

so that by the Euler formula the exponent of is exactly where is the genus of the surface obtained from We thus find that the matrix Wick formula has provided us for a generating function for the number of one-face maps obtained from gluing a -gon, sorted by genus:

where is the number of gluings of a -gon that produce a surface of genus And that’s the Harer-Zagier theorem. Next lecture, we’ll talk about what it’s good for.

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