Stochastic geometry and topology ofnon-Gaussian fieldsThomas H. Beumana, Ari M. Turnerb, and Vincenzo Vitellia,1

aInstituut-Lorentz for Theoretical Physics, Leiden University, NL 2333 CA Leiden, The Netherlands; and bInstitute for Theoretical Physics, University ofAmsterdam, NL 1090 GL Amsterdam, The Netherlands

Edited by David R. Nelson, Harvard University, Cambridge, MA, and approved October 12, 2012 (received for review July 16, 2012)

Gaussian random fields pervade all areas of science. However, it isoften thedepartures fromGaussianity that carry the crucial signatureof the nonlinear mechanisms at the heart of diverse phenomena,ranging from structure formation in condensed matter and cosmol-ogy to biomedical imaging. The standard test of non-Gaussianity is tomeasure higher-order correlation functions. In the present work, wetake a different route. We show how geometric and topologicalproperties of Gaussian fields, such as the statistics of extrema, aremodified by the presence of a non-Gaussian perturbation. Theresulting discrepancies give an independent way to detect andquantify non-Gaussianities. In our treatment, we consider both localand nonlocal mechanisms that generate non-Gaussian fields, bothstatically and dynamically through nonlinear diffusion.

extrema statistics | random surfaces | umbilics | cosmic microwavebackground

Random fields are ubiquitous.A disparate class of phenomena—ranging from the cosmic background radiation (1) and surface

roughness (2) to medical images of brain activity (3) and opticalspeckle patterns (4)—produce data that can be regarded as randomfields. The statistics of geometrical features of these fields, such asthe density of extrema of various types, can be used to characterizethem (5, 6). When the fields can be approximated as Gaussianfields, the physical meaning of these statistical properties is gen-erally well understood (7, 8); the statistics of extrema, for instance,reflects the amount of field fluctuations at short distances.Although analytical investigations are often restricted to

Gaussian fields, phenomena described by nonlinear laws (such asthe dynamics of inflation that produced the cosmic backgroundradiation) produce non-Gaussian signals. Quite often, the observ-able signal is averaged over a large scale, producing approximatelyGaussian statistics on account of the central limit theorem, thusmasking the nonlinearity. Nevertheless, the surviving tiny depar-tures fromGaussianity can carry a crucial signature of the nonlinearmicroscopic mechanisms at the heart of the phenomena. As anillustration, consider a low-resolution measurement of the spatialmagnetization of a material well above the critical temperature.The magnetization fluctuates like a Gaussian random variable—each region contains many domains oriented up or down in arbi-trary proportion. However, a small non-Gaussian contributionremains, because there is a maximum possible magnetization perunit area that can be traced all of the way down to the quantizationof the spin of the electrons, and hence the probability distributioncannot exhibit Gaussian tails.To unveil such elusive effects, one needs an indicator that is

sensitive to both short distances and small signals. The most com-mon tool used to probe the statistics of a random field is tomeasureits correlation functions. For example, the statistical properties ofa random scalar field, hðx; yÞ, with Gaussian statistics, are entirelydetermined by its two-point correlation function hhðx; yÞhðx′; y′Þi,and its higher-order correlation functions can be written simply asthe sum of products of two-point correlation functions. The non-factorizability of these higher-order correlation functions is one ofthe standard indicators of non-Gaussian statistics.

Here we focus on a more geometric approach: view the scalarfield as the height of a surface and study its random topographyto infer the statistical properties of the signal (Fig. 1, Inset). Thedensities of peaks and troughs, or of topological defects in thecurvature lines known as umbilics (Fig. 1), are sensitive indica-tors of how jagged the height field is at short distances; as weshall see, they provide an independent pipeline to detect non-Gaussianities, distinct from multiple-point correlation functions.This geometric approach has been applied successfully to trackthe power spectrum of a Gaussian field, and it has been thesubject of extensive theoretical and experimental studies (6–10).In this paper, we introduce the key physical concepts and

mathematical techniques necessary to study the stochastic ge-ometry of signals that can be described as a Gaussian randomfield plus a perturbation that we wish to track. We first show howto treat non-Gaussianities within a local approximation andcalculate how the statistics of extrema change when a nonlineartransformation FNLðHGÞ is applied locally to a Gaussian fieldHG. Then we consider the case of fields that cannot be probeddirectly, by calculating the statistics of umbilical points, which aretopological defects of the lines of principal curvature (11). Finally,we turn to a class of nonlinear diffusion (Eq. 2) and go beyond thelocal approximation by considering the effects of spatial gradientsthat couple values of the field at different locations. As an illus-tration, we solve explicitly for the nonlocal non-Gaussianitiesgenerated dynamically by the deterministic Kardar-Parisi-Zhang(KPZ) equation, which models surface growth (2).

Critical PointsTo gain some insights into the physical mechanisms that gener-ate non-Gaussianities, consider first how an isotropic Gaussianfield HGð~rÞ arises from the random superposition of waves (orequivalently, Fourier modes):

HGð~r Þ=X~k

AðkÞcos�~k ·~r+ϕ~k

�; [1]

with an amplitude spectrum AðkÞ that depends only on the mag-nitude of the wave vectors, k= j~kj. [The power spectrum AðkÞ2 isthe Fourier transform of the two-point correlation function.] Thephases ϕ~k are uncorrelated random variables uniformly distrib-uted in the range ½0; 2π�. The statistical properties of the Gauss-ian field HGð~rÞ are entirely encoded by the function AðkÞ or,equivalently, its moments K2n =

P~kk

2n12AðkÞ2.

The most basic difference between Gaussian and non-Gaussianvariables is that Gaussian ones are always symmetric about theirmean. As a consequence, irrespective of its power spectrum,

Author contributions: A.M.T. and V.V. designed research; T.H.B., A.M.T., and V.V. per-formed research; and T.H.B., A.M.T., and V.V. wrote the paper.

The authors declare no conflict of interest.

This article is a PNAS Direct Submission.1To whom correspondence should be addressed. E-mail: vitelli@lorentz.leidenuniv.nl.

This article contains supporting information online at www.pnas.org/lookup/suppl/doi:10.1073/pnas.1212028109/-/DCSupplemental.

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a Gaussian field has equal densities of maxima and minima.Hence, a nonvanishing imbalance, Δn, between these two types ofextrema serves as a probe to detect and quantify the non-Gaussiancomponent of a signal, provided that it can be measured directly.For example, the statistics of peaks and troughs have been used totest the Gaussianity of the temperature fluctuations in the cosmicmicrowave background (12, 13).Consider the primordial curvature perturbation field,Φ, a nearly

Gaussian field of central interest to modern cosmological studies(1). Within a local approximation, the primordial field is obtainedfrom a Gaussian field ΦG via a nonlinear relation Φ=ΦG +fnlΦ2

G + gnlΦ3G. Determining the parameters fnl and gnl is one of the

central tasks in the study of cosmological non-Gaussianities. As weshall see, the quadratic coefficient can be determined from theimbalance Δn between maxima and minima of Φ.The imbalance can be derived in the more general context of

a non-Gaussian field h that is obtained from a Gaussian field HGvia any nonlinear deformation h=FNLðHGÞ. If FNL is a mono-tonic function, the maxima and minima do not change—onlya nonmonotonic behavior of FNL can alter this balance.The critical points of h are given by ~∇h=F′NLðHGÞ~∇HG = 0,

where the prime indicates the derivative of FNLðHGÞ with respectto HG. This condition shows that h and HG have the same criticalpoints. Note however that, if F′NLðHGð~r0ÞÞ< 0 at a critical point~r0, then a maximum (minimum) at HGð~r0Þ will be turned intoa minimum (maximum) at hð~r0Þ. The number of saddle pointsdoes not change—it must be equal to the sum of maxima andminima according to Morse theory (14).If the transformation has a bias toward converting minima into

maxima, then h will have more maxima than minima; for example,

h=HG + εH2G reverses its slope at sufficiently negative values of

HG, which are most likely to be minima. Following this logic, thefirst step toward calculating the imbalance between maxima andminima is to determine the probability gðzÞ that HGð~r0Þ= z fora minimum~r0 of HG. The symmetry properties of HG imply thatthe analogous probability distribution for maxima is gð−zÞ.The fraction of minima of HG that become maxima of h is

obtained by integrating gðzÞ over the range of z for whichF′NLðzÞ< 0. Likewise, the fraction of maxima of HG that areturned into minima is given by the integral of gð−zÞ over thesame range. The overall imbalance in the densities of the max-ima and minima of h can be readily obtained by adding theseopposite contributions. The result reads

Δn ≡nmax − nmin

nmax + nmin

=Zz:F′NLðzÞ<0

dzðgðzÞ− gð−zÞÞ: [2]

For two dimensions, the exact analytical expression for gðzÞ isexplicitly derived in Materials and Methods—it depends only onthe moments K0, K2, and K4. Rescaling hð~rÞ→ hða~rÞ does notaffect the function gðzÞ, because it increases the density ofmaxima with any value of HG by the same proportion. Hence,only K0 = hH2

Gi (which sets the scale of the distribution) and the

dimensionless parameter λ≡ K22

K0K4can enter in the expression for

gðzÞ (see the plot in Fig. 2, Inset).As an illustration, apply Eq. 2 to the perturbed Gaussian field

h=HG + εH2G. Fig. 2 shows our theoretical formula as a contin-

uous line, validated by numerical data (dots) obtained fromcomputer-generated random surfaces with amplitude spectrumAðkÞ∼ θðkD − kÞ, having λ= 3

4. The imbalance between maximaand minima is particularly useful to track large deviations fromGaussianity, and this can be seen explicitly for the quadraticperturbation considered above. Because HG itself has an equalnumber of maxima and minima, the imbalance Δn in h is onlycreated when one of these critical points is inverted. Whether HGhas a high likelihood of having a negative F′NLðHGÞ= 1+ 2εHG is

controlled by εffiffiffiffiffiffiffiffiffiffiffihH2

Giq

. The probability of j2εHGj exceeding 1 is

exponentially small. Thus, Δn rises roughly as e−c=ðε2hH2

GiÞ, where

L

M

S

Fig. 1. The principal curvature of a surface. The major and minor axes of thesmall ellipses represent the directionand relative sizeofmaximumandminimumcurvature at the center, equivalent to the direction of maximum and minimumpolarization for an optical field. The curvature lines are always tangent to thedirectionofmaximalcurvature.At somepoints thecurvature is the samealongalldirections (the equivalent polarization is circular); these are called umbilicalpoints, of which there are three types, all shown in this image: (Upper Left)Lemon; (Upper Right) star; (Lower Right) monstar. The circles demonstrate theirtopological indices: −1=2 for a star, + 1=2 for the other two. The lemon has one(locally) straight curvature line terminating at it (indicated with a thick line), theother twohavethree. (Inset) Computer-generatedGaussiansurfacewithperiodicboundary conditions, a small square ofwhich served as the sourceof this picture.

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

ε

Δn

−4 −2 0 20

0.2

0.4

0.6

z

g(z)

Fig. 2. The relative difference Δn between the densities of maxima andminima of h=HG + εH2

G, where HG is a Gaussian field with λ= 3=4, asa function of ε. The data points are results from computer-generated fields;the solid curve is the theoretical result (Eq. 2). (Inset) Corresponding distri-bution of minima gðzÞ, which forms the basis of our theoretical result. Boththe theoretical curve and a histogram of data gathered from computer-generated fields.

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c≈ 18. Note that our approach in deriving Eq. 2 and gðzÞ is non-

perturbative, as demonstrated by the agreement between ourformula and numerics over the entire ε-range probed in Fig. 2.

Umbilical PointsThe near-Gaussian fields under investigation are not always di-rectly accessible experimentally. For example, the mass distribu-tion along the line of sight responsible for weak gravitationallensing is believed to be mostly composed of dark matter, andhence it cannot be detected directly (15). If the projected gravi-tational potential over a flat patch of the sky is taken to be theheight of a 2D surface (16), the measurable shear field is given bythe lines of principal curvature (11), as shown in Fig. 1. At somespecial points called umbilics, the curvature is equal in all direc-tions, so the shear field cannot be defined and it must vanish.More precisely, a point~r= fx; yg on a surface with height functionhð~rÞ is an umbilic if the second derivatives satisfy the two con-ditions hxxð~rÞ= hyyð~rÞ and hxyð~rÞ= 0. The ratio between differenttypes of umbilical points (which is a universal number for anisotropic Gaussian field) serves as an indicator of non-Gaus-sianities in lieu of the extrema, which cannot be detected. Asimilar reasoning can be applied to study polarization singulari-ties in the cosmic microwave background (17–20) and topologicaldefects in a nematic (21, 22) or superfluid near criticality (23, 24).Inspection of Fig. 1 reveals that there are three types of umbilics:

lemons, monstars, and stars. Note that these umbilics are topo-logical defects in the curvature-line field. The topological index ofany umbilic is equal to ±1/2, if the curvature-line field rotatescounterclockwise (clockwise) by an angle π along any closed pathencircling only that umbilic in the counterclockwise direction. Astar has three curvature lines terminating at it and a topologicalindex of −1

2. A lemon has only one line and index + 12. A monstar

has index + 12, like a lemon, but three lines terminating at it, like

a star. A striking feature of isotropic Gaussian fields is that themonstar fraction, the relative density of monstars with respect toall umbilics, equals αM = 1

2−1ffiffi5

p = 0:053; this is a universal numberindependent of the power spectrum (8, 10). Any deviation fromthis special value is therefore a sure sign of non-Gaussian effects.Consider a height field h=HG + ε f ðHGÞ, with f a nonlinear

perturbation and ε � 1 a small parameter controlling the size ofthe nonlinearity. We can express αM in terms of the joint prob-ability distribution p of the second and third derivatives of h. Tocalculate p, we use the property that a probability distribution isdetermined by its moments of all orders (if the distribution iswell-behaved). This task is most easily accomplished by calcu-lating the generating function of the distribution, χ; its logarithmcan be directly expressed in terms of the cumulants Cn

log χðλ1; . . . ; λnÞ=X∞m=1

im

m!

Xj1 ;...;jm

Cm

�ξj1 ; . . . ; ξjm

�λj1 . . . λjm ; [3]

where ξ1; . . . ; ξn are the stochastic variables given by the spatialderivatives of h. The probability distribution can then be obtainedfrom the generating function by taking the inverse Fourier trans-form with respect to λ1; . . . ; λn. The cumulants can be written interms of expectation values, e.g., C2ðξ1; ξ2Þ= hξ1ξ2i− hξ1ihξ2i. Inthis context, these expectation values are called moments, whichare not to be confused with the moments K2n defined previously.For Gaussian variables, only the second-order cumulants are

nonzero, which gives rise to a generating function of the form

log χ = −12

Xij

C2�ξi; ξj

�λiλj: [4]

One can easily check that the inverse Fourier transform of χ inEq. 4 precisely yields the probability distribution for a set of

correlated Gaussian variables (assuming hξii= 0; Eq. 12). Moregenerally, by determining all of the cumulants, one can constructthe generating function and from that the probability distribu-tion. We shall derive the monstar fraction up to first order in theperturbation ε f ðHGÞ only; consistently we need to determine allof the cumulants up to first order only.Themonstar fraction (even at order ε) could in principle depend

on f in a complicated way if f is an arbitrary nonlinear function. Infact, quadratic terms in the function f produce degree 3 cumulantsin the distribution function of the field h (i.e., skewness), cubicterms produce kurtosis (degree 4 cumulants), and in general de-gree n terms in f produce degree n+ 1 cumulants. However, themonstar fraction can be determined from just the distribution ofa few derivatives of h, whose cumulants vanish beyond the fourthorder due to symmetry, as shown in SI Materials and Methods.Consequently, the final result for the monstar fraction depends

only on a single parameter, hf ‴ðHGÞi=R∞−∞ f ‴ðuÞe−u

22K0 duffiffiffiffiffiffiffiffi

2πK0p , where

the primes indicate derivatives with respect to HG.The calculation can be briefly summarized as follows. The

monstar fraction is related to the distribution function of some ofthe second and third derivatives of h, which we write in terms ofcomplex coordinates z= x+ iy and z* : hzz, hzzz, and hzzz*. Thedefinition of an umbilic point becomes hzz = 0, where hzz is nowcomplex. All of the cumulants of these variables and their con-jugates may now be calculated (up to order ε). The complexcoordinates allow for optimal use of rotational and translationalsymmetry. Only a few of the cumulants are nonzero, and theseare evaluated in Table 1. With the aid of these cumulants, thegenerating function can be constructed to first order using Eq. 3.Taking the Fourier transform leads to the probability distribu-tion pðhzz; hzzz; hzzz*Þ, which takes the form of a Gaussian per-turbed by cubic and quartic terms in h and its derivatives (25). Toobtain αM , we set hzz = 0 and integrate over hzzz and hzzz*, in-cluding the appropriate Jacobian factor. Integration over all C2

gives the total density of umbilical points, whereas the density ofmonstars is obtained by integrating over a specific range of C2

(SI Materials and Methods). The monstar fraction is then theratio of these two densities. The resulting deviation fromαM = 0:053 is

ΔαM = 0:429 μ�f ‴ðHGÞ

�ε; [5]

where μ≡K34=K

26 . When applied to the local model of the pri-

mordial field Φ described previously, ΔαM in Eq. 5 depends onlyon the cubic coefficient gnl and not on fnl. Hence, the leading-order perturbation that alters the monstar fraction is f ðHGÞ=H3

G.The perturbation εH3

G, like any odd and/or monotonic function ofHG, does not have an effect on the density of maxima and min-ima. Note that ΔαM = :107μκ, where κ= <h4 >

<h2>2 − 3 is the kurtosis ofthe signal. [For a more general type of non-Gaussianity therewould be additional terms in the expression, depending on corre-lation functions of partial derivatives of hðx; yÞ.]Fig. 3 shows αM , as determined by Eq. 5 (continuous line),

together with data from computer simulations (symbols), per-formed using the ring spectrum AðkÞ∼ θðkD − kÞ, for which μ= 16

27.

Table 1. All nonzero cumulants

Cumulant Value

C2ðhzz;hz∗z∗Þ σð1+ 2hf ′ðHÞiÞC2ðhzzz;hz∗z∗z∗Þ τð1+ h2f ′ðHÞiÞC2ðhzzz∗;hzz∗z∗Þ τð1+2hf ′ðHÞiÞC3ðhzz;hzzz∗;hz∗z∗z∗Þ & conj. −3σ2hf ″ðHÞiC4ðhzzz∗;hzzz∗;hzz∗z∗;hzz∗z∗Þ −8σ3hf ‴ðHÞiC4ðhzzz;hzz∗z∗;hzz∗z∗;hzz∗z∗Þ & conj. −6σ3hf ‴ðHÞi

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The agreement between theory and numerics is very good in thelinear regime. The monstar fraction is very sensitive to smallnon-Gaussian perturbations: αM changes by 20% when ε is just0.01. For larger values of ε, nonlinear effects become importantand prevent αM from becoming negative. In this regime, ourperturbative result does not hold.

Nonlocal Model and Evolution EquationsIn the first section, we presented an exact expression for the im-balance between maxima and minima for a local perturbation ofa Gaussian random field. This simple class of models may describethe local evolution of a system that starts out with a Gaussian dis-tribution, such as the growth of a population of cells that are initiallydistributed on a dish and then divide without any significant mi-gration from one region to another. However, many dynamicalsystems evolve in a nonlocal, nonlinear way. Non-Gaussianities aregenerated dynamically from the nonlinear equations ofmotion thatthe field hð~r; tÞ obeys, even if the initial condition hð~r; 0Þ=HGð~rÞis Gaussian.A broad class of nonlinear diffusion equations describes the

necessary mixing between regions. Examples include severalmodels of structure formation in both condensed matter (26) andcosmology (1), the Cahn–Hilliard equation for the developmentof order after a phase transition (27), and simplified models ofsurface growth (2). We illustrate our approach in the context ofthe deterministic KPZ Eq. 2, which models the evolution of theheight of a substrate, hð~r; tÞ, as atoms accumulate on it:

∂hð~r; tÞ∂t

= ν∇2hð~r; tÞ+ λ

2ð∇hð~r; tÞÞ2: [6]

To first order, the surface grows at a constant rate that is simplysubtracted out of Eq. 6: the two terms on the right-hand sidecapture additional effects. The first term describes the diffusionof particles along the surface, and the second nonlinear termdescribes approximately how the growth rate varies with the localslope. The surface is assumed to grow at a constant rate perpen-dicular to itself, but because the height is measured vertically, _hdepends on the slope; this gives rise to the term quadratic in ∇h.An alternative interpretation of Eq. 6 is obtained by taking the

gradient on both sides; one thus derives a Burger’s equation thatdescribes the velocity field~vð~r; tÞ= ~∇ hð~r; tÞ. The saddles, maxima,and minima of h correspond to stagnation points, sources, andsinks of the potential flow field vð~r; tÞ. If the term quadratic in thegradient in Eq. 6 is substituted by a term quadratic in the field,−h2, one obtains the Fisher equation, which describes the growth

and saturation of a population. In all of these cases, we can studythe time evolution of an initially Gaussian height profile. Uponsetting the coefficient of the nonlinear term λ equal to zero, wealways retrieve the heat equation, which preserves the Gaus-sianity of h for all later times. However, for λ≠ 0, h attains a non-Gaussian statistics.For concreteness, concentrate on how an imbalance between

maxima and minima is generated by nonlocal non-Gaussianities inEq. 6. The nonlinear term breaks the symmetry between positiveand negative values of h, which is a necessary condition to generatean imbalance. Note however that in the case of a local evolution,this imbalance grows exponentially slowly as e−α=ε

2, where ε is the

coefficient in front of the nonlinear term and α a constant. Localevolution cannot create new extrema; it can only convert a maxi-mum into a minimum whenever h happens to have a sufficientlylarge fluctuation. It is the presence of the diffusion term that is ableto create new maxima and minima, even though, on its own, itwould not be able to generate any imbalance, because of thesymmetry h→− h. The two terms on the right-hand side of Eq. 6conspire together to change the number of maxima and minimaasymmetrically. This mechanism causes the imbalance betweenmaxima and minima, Δn, to exhibit a power law increase in λ thatcan be calculated within perturbation theory.We use the same approach adopted for the umbilical points

and express the joint distribution of hz, hzz, and hzz* in terms ofthe relevant cumulants, listed in Table 2 up to third order in λ.The resulting expression for Δn reads (28)

Δn=ffiffiffiffiffiffi6πα

r 43β

σ+49δ

α−1027

γ

α

: [7]

Eq. 7 is a general result. To apply it to the KPZ equation, we firstchange variables to

uð~r; tÞ =2νλ

�expλ

2νhð~r; tÞ

− 1�

≈ hð~r; tÞ+ λ

4νhð~r; tÞ2

: [8]

Note that u is a monotonic function of h, so u has the sameprofile of maxima and minima as h. This field satisfies the heatequation whose general solution is

uð~r; tÞ≈Z

d2r′Gð~r;~r′; tÞhð~r′; 0Þ+ λ

4νhð~r′; 0Þ2

; [9]

where Gð~r;~r′; tÞ denotes the Green’s function.The correlations listed in Table 2 can now be determined from

the distribution of hð~r; 0Þ, leading to an expression for ΔnðtÞ thatis valid over an arbitrary time span, provided that λ is small.Analytical results can be obtained for a few convenient choices ofthe power spectrum of hð~r; 0Þ. For example, if we take a Gauss-ian spectrum, AðkÞ2 ∼ expð−k2=2k20Þ, we find

Δn=λ

ν

16τ3ð1+ 4τÞ7=2ffiffiffiffiffi3π

p ð1+ 2τÞ3ð1+ 6τÞ4; [10]

−0.02 −0.01 0 0.01 0.020.03

0.04

0.05

0.06

0.07

ε

αM

Fig. 3. The fraction of monstars (Inset) αM of h=HG + εH3G, where HG is

a Gaussian field with μ= 16=27, as a function of ε. The data points are resultsfrom computer-generated fields; the solid line is the theoretical first-orderresult (Eq. 5). At ε= 0 we retrieve the universal fraction αM = 1=2−1=

ffiffiffi5

p= 0:053, valid for any isotropic Gaussian field.

Table 2. All second- and third-order cumulants

Second order Third order

σ = hjhzj2i β= hjh2z jhzz∗i

α= hjhzzj2i γ = hh3zz∗i

δ= hjhzzj2hzz∗i

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where τ ≡ k20νt. The validity of this equation is illustrated in Fig.4, which shows an excellent agreement between theory andnumerics. The imbalance starts out at zero because the initialchoice for h is Gaussian. However, after long times, this expres-sion decays back to zero. The reason for the decay is that, at longtimes, uð~r; tÞ involves an average over a larger and larger window,so by the central limit theorem, it starts to acquire Gaussianstatistics characterized by a vanishingly small imbalance betweenmaxima and minima.For early times, we can make an expansion in t, valid for an

arbitrary power spectrum, which gives

Δn=λ

ν

19

ffiffiffi6π

r1

K2ffiffiffiffiffiffiK4

p �2K2K6 − 3K2

4

�ðνtÞ2 +O�t3�: [11]

The second-order termOðt2Þ in Eq. 11 happens to vanish for theGaussian power spectrum featured in Eq. 10, but not in general.The agreement over the entire range of times in Fig. 4 is

a peculiar feature of the KPZ equation. The perturbative for-mula in Eq. 11 is not expected to hold at sufficiently late times,because the nonlinearities eventually grow exponentially. Bycontrast, the Cole–Hopf transformation, which exactly maps theKPZ equation to a linear diffusion equation, guarantees that thenonlinearities remain bounded in this case.To sum up, we have illustrated a geometric approach to track

the non-Gaussian component of a field that relies on the statisticsof topological defects and extrema rather than multiple-pointcorrelation functions. Because these measures are topological innature, we expect them to be less sensitive to noise than standardcorrelation-function methods. The nonperturbative formula forthe imbalance betweenmaxima andminima is particularly suitablefor detecting local non-Gaussianities in the limit of large pertur-bations. Applications may range from the analysis of images ofcerebral activation (3) to peaks and troughs statistics in cosmo-logical structure formation (29).The statistics of umbilics are a sensitive probe to measure very

weak deviations from Gaussianity. It would be interesting to

apply our analysis of umbilics to the polarization field of thecosmic microwave background (1, 17–20) or weak gravitationallensing shear maps (15, 16). More controlled experiments can berealized by studying the polarization of a light beam propagatingthrough a nonlinear medium.Finally, we generalized our study of static local non-Gaussian-

ities to fields that obey nonlinear diffusion equations that allow forsome fieldmixing between different spatial locations. In contrast tolocal models, the imbalance between peaks and troughs provesa suitable measure of nonlocal non-Gaussian perturbations also inthe limit of weak deviations from Gaussianity. The most obviousapplication of our dynamical analysis would be to analyze the dis-tribution of peaks and valleys in surface growth experiments, aswell as more detailed studies of the peak statistics in the temper-ature fluctuation field of the cosmic microwave background (12,13), which is affected nonlocally by the mixing of sound waves (1).

Materials and MethodsThe probability distribution gðzÞ can be derived using a method similar tothe one outlined in Longuet-Higgins (9). Consider a fixed point~r. We wish toknow the probability density that at this point we have HG = z (to avoidconfusion with the derivatives of HG, we shall write H from now on) giventhat it is a minimum. The conditions for this can be written in terms ofderivatives of H, namely, Hx =Hy = 0 defines a critical point, whereasHxxHyy −H2

xy > 0 and Hxx +Hyy > 0 distinguishes a local minimum from a sad-dle or maximum. First, we determine the joint distribution of these six var-iables (H and its derivatives), which form a set of correlated Gaussianvariables. The joint probability distribution pðξ1; . . . ; ξnÞ for any such set iscompletely determined by the correlations between the variables

pðξ1; . . . ; ξnÞ= ð2πdetCÞ−n=2exp −Xij

C −1ij ξiξj

!; [12]

where Cij = Æξiξjæ is the matrix of correlations between the variables.Correlations between H and its first and second derivatives can be

expressed in terms of the first three moments (K0;K2;K4) of its amplitudespectrum. By differentiating the Fourier expansion of H we find thatÆH2

x æ= ÆH2y æ= 1

2K2, and likewise that the variances of the second derivativesare proportional to K4. The only variables among the six that are correlatedto one another turn out to be H, Hxx , and Hyy , with ÆHHxx æ= ÆHHyy æ= −K2=2and ÆHxxHyy æ=K4=8. After retrieving the probability distribution, we setH= z, Hx =Hy = 0 and integrate out Hxx , Hyy , and Hxy over the domain de-fining a minimum. The Jacobian determinant jHxxHyy −H2

xy j must be added(9, 30). The probability density thus calculated reflects the chance that Hx

and Hy are close to zero at the point~r0 (there is a vanishing chance that theyare exactly zero). To determine the distribution of extrema in the plane, weneed the probability of the reverse situation—namely, that Hx =Hy = 0 ex-actly at a point within a small range of~r0. The ratio of the two probabilitiesis given by the Jacobian determinant. The final answer reads

gðzÞ=ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

32πð3−2λÞ

se−

3z22ð3−2λÞerfc

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiλ

2ð1− λÞð3− 2λÞr

z

−ffiffiffiffiffiffi32π

rλ�1− z2

�e−

12z

2erfc

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiλ

2ð1− λÞr

z

−1π

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi3λð1− λÞ

pz e−

z22ð1−λÞ

; [13]

where λ= K22

K0K4.

ACKNOWLEDGMENTS. We thank T. Lubensky, R.D. Kamien, B. Jain, A.Boyarsky, L. Mahadevan, B. Chen and W. van Saarloos for stimulatingdiscussions. This work was supported by the Dutch Foundation for Funda-mental Research on Matter and the European Research Council.

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0 2 4 6 80

0.05

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)

Fig. 4. The imbalance between maxima and minimaΔn, as a function of time,for an initially Gaussian field evolving according to the deterministic KPZ equa-tion (Eq. 6). (Left Inset) At t =0, the Gaussian field was taken to have a Gaussianpower spectrum ðAðkÞ2 ∼ expð−k2=2k2

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