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?W H A T I S . . .
a Gaussian EntireFunction?
Fedor Nazarov and Mikhail Sodin
Random analytic functions have been attractingthe attention of mathematicians since the 1930s,though the focus of interest has been changingwith time. Just as the distribution of eigenvalues isthe essence of the random matrix theory, centralto the study of random analytic functions are theirzero sets. Our random functions are Gaussian andlive on the complex plane. The instance when therandom zero set is invariant in distribution withregard to (w.r.t., for short) isometries of the plane isthe most interesting one. Here we will introduce thereader to a remarkable model of Gaussian entirefunctions with invariant distribution of zeros.
A Gaussian entire function f (z) is the sum∑k ζkfk(z) of entire functions fk with independent
standard complex Gaussian random coefficientsζk (whose density w.r.t. the area measure in C is1π e
−|ζ|2 ). We assume that∑k
|fk(z)|2<∞ locally uniformly in C,
and also that the functions fk are linearly inde-pendent over `2, i.e.,
∑k akfk with {ak} ∈ `2 does
not vanish identically unless all ak = 0. The firstcondition implies that almost surely (a.s., for short)the random function f is entire.
Each Gaussian entire function can be uniquelyidentified with some Hilbert space H of entirefunctions (the image of the mapping
`2 3 {ak},∑k
akfk
with the scalar product borrowed from `2) so thatthe covariance function
Cf (z,w) = E{f (z)f (w)
}=∑k
fk(z)fk(w)
Fedor Nazarov is professor of mathematics at the Univer-sity of Wisconsin, Madison. His email address is [email protected].
Mikhail Sodin is professor of mathematics at Tel AvivUniversity. His email address is [email protected].
is the reproducing kernel inH ; i.e.,
g(w) = 〈g,C(·, w)〉H for every g ∈H , w ∈ C.
The functions fk form an orthonormal basis inH .Reversing the order, one can start with a HilbertspaceH of entire functions with the reproducingkernel CH , take an orthonormal basis {fk} in H ,and build a Gaussian entire function fH =
∑k ζkfk
with covariance CH . Since the Gaussian processis determined by its covariance function, thisconstruction does not depend on the choice of thebasis inH .
The properties of the (random) zero set Zf =f−1{0} are encoded in its (random) countingmeasure nf defined by nf (A) = #
(Zf ∩A
)for any
Borel set A. Recall that for every analytic functionf , we have
nf = 12π∆ log |f |
with the Laplacian taken in the sense of distri-butions. This makes it possible to use complexanalysis tools for the study of the distribution ofzeroes of Gaussian analytic functions. Using thisformula, and taking the expectation of both sides,we get Enf = 1
2π∆E log |f | . Note that f (z)√Cf (z,z)
is the
standard complex Gaussian random variable, so
E log |f | = 12 logCf (z, z)+ const .
This way, we arrive at the elegant Edelman-Kostlanformula
Enf (z) = 14π∆ logCf (z, z) .
The surprising Calabi rigidity tells us that themean Enf determines the distribution of Zf . Alas,this uniqueness gives us no hint as to how to findthe distribution of nf from its mean Enf .
All the aforementioned results are valid forGaussian analytic functions in other plane domains.It is the gaussianity that is crucial, not the domainof f .
March 2010 Notices of the AMS 375
Figure 1. Samples of the Poisson process (figure by B. Virág), limiting Ginibre process, and zeroesof a GEF (figure by M. Krishnapur). The last two processes are quite different, though the eye does
not easily distinguish them.
It is not at all obvious that there exist Gaussianentire functions with zeros having a translation-invariant distribution. It is not difficult to see thatGaussian entire functions cannot be translationinvariant themselves.1 Fortunately, a weaker prop-erty called projective invariance is sufficient for thetranslation invariance of zeros. Namely, if thereis a family of nonrandom functions φλ (λ ∈ C)without zeros such that the random functionsφλ(z)f (z+λ) and f (z) have the same distribution,then the distribution of Zf is translation invariant.
Letting fk(z) = zk/√k!, we get Cf (z,w) = ezw ,
which is the kernel for the classical Fock-Bargmannspace of entire functions, that is, the closure ofpolynomials in L2
(C, 1
π e−|z|2). The Gaussian entire
function associated with this Hilbert space isprojective invariant w.r.t. isometries of C. Therotation and reflection invariance are obvious. Toshow the translation invariance, note that theGaussian entire function
f (z + λ)e−zλ− 12 |λ|2 , λ ∈ C,
has the same covariance function as f .By the Edelman-Kostlan formula,
Enf = 14π∆|z|2 = 1
πm,
where m is the area measure (we treat the average
Enf as a measure). Replacing f by fL(z) = f (√Lπ z),
this average can be changed to Lm with anyL > 0. On the other hand, if zeros of a Gaussianentire function F have a translation-invariantdistribution, then the mean EnF is a translation-invariant measure on C. Hence, it is proportional tothe area measurem; i.e., EnF = Lm with a constantL > 0. Then by the Calabi rigidity, the zero setsZF and ZfL have the same distribution. In otherwords, the only freedom in this construction isthe scaling z, tz with t > 0, and the GaussianEntire Function (GEF, for short) with translation-invariant zeros is essentially unique. Geometers
1B. Weiss showed that, unexpectedly, there are translation-invariant random entire functions, not Gaussian, ofcourse.
know this in a different wording: z , {zk/√k!}k≥0
is an isometric embedding of the Euclidean planeinto the projective Hilbert space P
(`2
)equipped
with the Fubini-Study metric, and this embeddingis essentially unique.
The construction leading to projective invariancehas been known since the 1930s, though thecorresponding Gaussian functions were introducedonly in the 1990s by Kostlan, Bogomolny-Bohigas-Lebouef, Shub-Smale, and Hannay. It is worthmentioning that there are similar constructions forother domains with transitive groups of isometries(hyperbolic plane, Riemann sphere, cylinder, andtorus).
Few natural translation-invariant random pointprocesses on the plane are known. The mostwidely studied one is the Poisson process, wherefor any collection of disjoint subsets of theplane, the numbers of points in these subsets areindependent, and the mean number of points in a setis proportional to its area. This process is invariantw.r.t. all measure-preserving transformations ofthe plane, which is far more than we asked for.Another example is a one-component plasma ofcharged particles of one sign confined by a uniformbackground of the opposite sign. It contains as aspecial case the large N limit of Ginibre ensembleof eigenvalues of N×N matrices with independentstandard complex Gaussian entries.2 One moreexample is the random zero set Zf of GEF f .
The Poisson process can be easily recognizedsince its points can clump together while, incontrast, the Ginibre eigenvalue process and theGEF zero process have local repulsion betweenpoints: it is unlikely that one would see two pointsvery close to each other. The latter two look ratheralike, although some of their characteristics arequite different. For instance, as Forrester andHonner observed, if h is a smooth function with
2Though one-component plasma has been studied byphysicists for a long time, it seems that almost all rigor-ous mathematical results still pertain only to the specialcase of Ginibre ensemble.
376 Notices of the AMS Volume 57, Number 3
compact support, then the variance of the linearstatistics of zeros nf (r ;h) =
∑Zf h
( ar
)decays as
‖∆h‖2L2r−2 for r → ∞, while in the Ginibre case
the corresponding variance tends to the limitproportional to ‖∇h‖2
L2 (for the Poisson processthe variance grows with r as ‖h‖2
L2 r2).The decay of the variance of smooth linear
statistics for zeros of GEF yields another surprisingrigidity. We fix a bounded plane domain G andsuppose that we know the configuration of zeros off outside of G. Then taking any smooth compactlysupported test-function h that equals 1 in someneighborhood of the origin, we recover the numberof zeros of f inside G:
nf (G)= limr→∞
{r2
π
∫∫Chdm−
∑a∈Zf \G
h(ar
)}a.s.
At the end of this introductory tour, we will takea brief look at the random potential
Uf (z)= log |f (z)|− 12 |z|
2
and at its gradient field ∇Uf . Their distributionsare invariant w.r.t. isometries of C, and
12π∆Uf = div(∇Uf ) = nf − 1
πm.The potential Uf equals −∞ on Zf and has no otherlocal minima since its Laplacian is negative onC\Zf . The gradient curves oriented in the directionof decay of Uf and terminating at a ∈ Zf forma basin Ba. Different basins are separated by thegradient curves joining local maxima with saddlepoints. Remarkably, all bounded basins have the
same area π :
1− 1πm(Ba)=
12π
∫∫Ba∆Uf = 1
2π
∫∂Ba
∂Uf∂n=0 .
One can prove that the probability of a long gra-dient curve decays exponentially with its diameter,so, a.s., all basins are bounded. Thus, one obtains arandom partition of C into nice bounded domainsof equal area with many intriguing properties.
We hope that we have aroused the reader’scuriosity by now. Note that we have presented onlya tiny portion of results and questions concerningGaussian analytic functions and their zeros.
Further ReadingFor those new to this subject, we recommendthe book Zeros of Gaussian Analytic Functions andDeterminantal Point Processes, J. B. Hough, M. Krish-napur, Y. Peres, B. Virág, Amer. Math. Soc., 2009.The electronic version is available at stat-www.berkeley.edu/˜peres/GAF_book.pdf.
The lecture by M. Sodin at the 4th ECM,Stockholm, 2004 (arXiv:math/0410343), surveysresults obtained by that time. Further develop-ments can be found in recent papers written bythe authors with A. Volberg, by B. Tsirelson, andby A. Nishry, and posted in the arXiv.
Complex-geometry-oriented readers might beinterested in reading the papers by P. Bleher,M. Douglas, B. Shiffman, and S. Zelditch, which arealso posted in the arXiv.
Figure 2. Random partition of the plane into domains of equal area generated by the gradientflow of the random potential Uf (figure by M. Krishnapur). The lines are gradient curves of Uf , theblack dots are random zeros. Many basins meet at the same local maximum, so that two of themmeet tangentially, while the others approach it cuspidally, forming long, thin tentacles.
March 2010 Notices of the AMS 377
