Rigorous foundation of the Cline-Politzer-Rey-Wise high-peak asymptotics for Gaussian random fields: asymptoticity of the 1/T expansion, the all-orders multiplicative factorization conjecture, and the exact negative-definite Hessian constraint

Cline, Politzer, Rey and Wise (Commun. Math. Phys. 112, 217 (1987)) developed the classical high-threshold asymptotics for the N-point correlation functions of high peaks - and of high-threshold exceedance points - of Gaussian random fields: formal expansions in the inverse dimensionless threshold T, jointly expanded in the small interpoint correlations, with the negative-definite Hessian constraint on peaks replaced by an extended integration domain whose error is exponentially bounded. Three elements of that theory were explicitly left uncontrolled by its authors: the series are 'presumed, though not proven, to be asymptotic'; the multiplicative factorization of three-point correlations into two-point factors at leading order in the interpoint correlations is verified only to order $T^{-2}$ and conjectured to hold to all orders; and the exact corrections from enforcing the negative-definite Hessian domain are bounded but never computed. The problem is to supply these three missing rigorous or exact ingredients.

Background

Setting. Let $\varepsilon(x)$ be a real Gaussian random field on $D$-dimensional Euclidean space - a random function whose values at any finite set of points are jointly Gaussian - assumed statistically homogeneous and isotropic, so that it is fully characterized by its two-point correlation function $\xi(|x-y|)=\langle\varepsilon(x)\varepsilon(y)\rangle$; in Fourier space the field is described by the spectral density $\tilde{\xi}(k)$, the Fourier transform of $\xi$, which must be positive for the Gaussian probability measure to be normalizable. Write $\xi(0)$ for the variance and $t$ for an excursion threshold; the natural dimensionless threshold is $T=t^2/\xi(0)$. A peak (local maximum) is a point where the gradient $\nabla\varepsilon$ vanishes and the Hessian matrix of second derivatives $\nabla\nabla\varepsilon$ is negative definite. The peaks above threshold form a random point process whose statistics are captured by the number density $n_m(x)$ (a sum of delta functions at the peak positions, weighted by $|\det\nabla\nabla\varepsilon|$) and by $N$-point correlation functions defined through the joint probability $P_N(x_1,\dots,x_N)$ of simultaneously finding high peaks at the given positions via $1+\xi_N=P_N/(P_1)^N$. Although $\varepsilon$ is Gaussian, $n_m$ is a strongly non-Gaussian random measure. Origin of the question. The subject originates in biased galaxy formation: Kaiser (1984) observed that rare high peaks of the cosmological mass-density fluctuation field are more strongly clustered than the field itself (peak bias), and Politzer & Wise (1984) showed that at leading order the $N$-point correlations of high-threshold regions factorize multiplicatively into products of two-point correlations. Bardeen, Bond, Kaiser & Szalay (1986) computed the one-point peak density exactly, at all thresholds, retaining the full negative-definite Hessian domain (building on the exact joint distribution of the field and its Hessian due to Doroshkevich (1970)); Jensen & Szalay (1986) and Otto, Politzer & Wise (1986) developed operator techniques for multi-point correlations in the spirit of Berezin's second quantization (a formalism of creation and annihilation operators, adapted here to classical random fields). Cline, Politzer, Rey & Wise (1987) - the source of this problem - then gave the first systematic high-threshold asymptotics for the $N$-point correlations of both threshold exceedances and peaks, in arbitrary dimension $D$. For exceedances they represent $P_N$ by a compact differential-operator expression built from the normalized correlation matrix $M_{ij}=\xi(|x_i-x_j|)/\xi(0)$, expand it in powers of $1/T$, and obtain explicit two- and three-point correlation functions to $O(T^{-2})$, jointly expanded in the small normalized correlations $a_{ij}=\xi(|x_i-x_j|)/\xi(0)$ that hold at large separations. For peaks, the joint density involves integrals over the peak heights $m_j\ge t$ and over Hessian matrices $\omega_j$ restricted to $\mathscr{D}$, the domain of negative-definite symmetric matrices; because $\mathscr{D}$ is a complicated non-convex domain, they replace it by the space of all symmetric matrices (approximation $P'_N$), representing the determinant $\det\omega$ with Grassmann (anticommuting) variables, and derive $1/T$ expansions for the peak correlations. They prove a theorem bounding the difference $|P_N-P'_N|$ by an exponentially small quantity at large thresholds and large separations (the relative error tends to zero; for $N=1$ minima and saddle points are exponentially suppressed), and they prove positivity of the key block matrices at infinite separation from $\tilde{\xi}(k)>0$, arguing only perturbatively for large finite separations. What was left open. Three elements of this theory were left uncontrolled by the source paper and remain so. First, the expansions are, in the authors' words, 'presumed, though not proven, to be asymptotic' - no remainder control of any kind is given. Second, the multiplicative factorization $1+\xi_3=\prod(1+\xi_2)$ is verified only to $O(T^{-2})$ and conjectured to hold to all orders in $1/T$ provided only the leading order in $a_{ij}$ is kept at each order; the authors state that the situation is 'just the same' for peak correlations. Third, the exact consequence of enforcing the negative-definite Hessian domain is bounded but never computed. Modern context. Modern large-scale-structure research recasts peak bias in the language of renormalized perturbative bias expansions - systematic field-theoretic expansions for the clustering of biased tracers (Desjacques, Jeong & Schmidt 2018). The two-point peak correlation of a Gaussian field is now available (i) as exact finite-dimensional integrals - 14-dimensional for the two-point function in three dimensions (Regős & Szalay 1995) - evaluated by Monte-Carlo methods (Matsubara & Codis 2020; Baldauf, Codis, Desjacques & Pichon 2021, whose implementation is valid at all separations), and (ii) as large-separation perturbative series at arbitrary threshold (Matsubara 1995; Matsubara & Codis 2020). These routes determine the two-point function numerically where the source's asymptotics do not apply, but none of them supplies the missing rigorous foundation of the original $1/T$ asymptotics, settles its factorization conjecture, or computes the exact Hessian-domain corrections.

Problem Statement

Let $\varepsilon(x)$ be a homogeneous, isotropic Gaussian random field on $\mathbb{R}^D$ with correlation function $\xi$, positive spectral density, and $\xi$ four times differentiable at the origin; let $t=(T\xi(0))^{1/2}$ be the threshold. Consider (a) the $N$-point correlation functions of the point process of threshold exceedances (points where $\varepsilon>t$) and (b) the $N$-point correlation functions of the point process of high peaks (points where $\nabla\varepsilon=0$, $\nabla\nabla\varepsilon$ is negative definite, and $\varepsilon>t$), in both cases defined through the joint probabilities via $1+\xi_N=P_N/(P_1)^N$. In the framework of Cline, Politzer, Rey & Wise (1987), these correlations admit formal expansions in inverse powers of $T$, jointly expanded in the small normalized correlations $a_{ij}=\xi(|x_i-x_j|)/\xi(0)$, with the peak Hessian integrals over the negative-definite domain $\mathscr{D}$ approximated by integrals over all symmetric matrices ($P'_N$) and the error $|P_N-P'_N|$ exponentially bounded. The problem is to replace the three uncontrolled elements of this asymptotic theory by rigorous or exact ones.

1. Asymptoticity. Prove or disprove that the $1/T$ series for the exceedance probabilities and correlations (the source's Eqs. (9)-(15)) and for the peak probabilities $P'_N$ and peak correlations (its Eqs. (53)-(60)) are genuine asymptotic expansions - the remainder after truncation being of the order of the first neglected term - for each fixed $N$, in the joint limit $T\to\infty$, $|x_i-x_j|\to\infty$ with $a_{ij}=\xi(r)/\xi(0)\ll 1/T^2$ (i.e. the source's validity regime: high threshold, large separations, and this ordering for the joint $(1/T,a_{ij})$ expansion). (The one-point exceedance series is the classical divergent Mills-type expansion - the inverse-threshold expansion of the Gaussian tail probability - whose asymptoticity is standard; the open content lies in general $N$ and the peak sector.) This includes justifying the hypotheses the derivation rests on: positivity of $J_i=\sum_j(M^{-1})_{ij}$ for each point, and positive definiteness of the height-sector and Hessian-sector block matrices at finite large separations (established in the source only at infinite separation and perturbatively), as well as the joint ordering of the double expansion, which is controlled only in the regime $\xi(r)/\xi(0)\ll 1/T^2$.
2. All-orders factorization conjecture. Prove or disprove the source's conjecture that

$$1+\xi_3(x_1,x_2,x_3)=(1+\xi_2(x_1,x_2))(1+\xi_2(x_2,x_3))(1+\xi_2(x_3,x_1))$$

holds to all orders in the $1/T$ expansion when only the leading order in $a_{ij}$ is retained at each order of $1/T$ (including the exponent). The source verified this to $O(T^{-2})$ for exceedances and states that the situation is 'just the same' for peak correlations; the conjecture concerns the exceedance and peak correlations at orders beyond the verified $O(T^{-2})$, within the conjectured regime $a\ll 1/T^2$. 3. Exact Hessian constraint. Determine the exact consequence of enforcing the negative-definite Hessian domain $\mathscr{D}$, going beyond the exponentially small upper bounds. These three components are the uncontrolled elements of a single asymptotic theory for a single family of objects (the $N$-point correlations of high excursions and high peaks of a Gaussian field), in full generality over $D$, the class of $\xi$, and $N$, within the stated framework. Two further directions raised by the source paper - the behavior of peak correlations at moderate thresholds or separations, and the extension to non-Gaussian fields - have been substantially or partially developed by the later work recorded above and are therefore context rather than part of the core.

References

1. James M. Cline, H. David Politzer, Soo-Jong Rey, Mark B. Wise, 'Correlations of peaks of Gaussian random fields', Communications in Mathematical Physics 112, 217-235 (1987). DOI: 10.1007/BF01217812. https://doi.org/10.1007/BF01217812 (open-access full text at Project Euclid: https://projecteuclid.org/journals/communications-in-mathematical-physics/volume-112/issue-2/Correlations-of-peaks-of-Gaussian-random-fields/cmp/1104159878.full)
2. N. Kaiser, 'On the spatial correlations of Abell clusters', Astrophysical Journal 284, L9 (1984). DOI: 10.1086/184341. https://doi.org/10.1086/184341
3. H. D. Politzer, M. B. Wise, 'Relations between spatial correlations of rich clusters of galaxies', Astrophysical Journal 285, L1 (1984). DOI: 10.1086/184352. https://doi.org/10.1086/184352
4. J. M. Bardeen, J. R. Bond, N. Kaiser, A. S. Szalay, 'The statistics of peaks of Gaussian random fields', Astrophysical Journal 304, 15 (1986). DOI: 10.1086/164143. https://doi.org/10.1086/164143
5. L. G. Jensen, A. S. Szalay, 'N-point correlations for biased galaxy formation', Astrophysical Journal 305, L5 (1986). DOI: 10.1086/184673. https://doi.org/10.1086/184673
6. S. Otto, H. David Politzer, Mark B. Wise, 'Two-Point Correlation for Rich Clusters of Galaxies', Physical Review Letters 56, 1878-1881 (1986). DOI: 10.1103/PhysRevLett.56.1878. https://doi.org/10.1103/PhysRevLett.56.1878
7. Paolo Catelan, Francesco Lucchin, Sabino Matarrese, 'Peak Number Density of Non-Gaussian Random Fields', Physical Review Letters 61, 267-270 (1988). DOI: 10.1103/PhysRevLett.61.267. https://doi.org/10.1103/PhysRevLett.61.267
8. S. L. Lumsden, A. F. Heavens, J. A. Peacock, 'The clustering of peaks in a random Gaussian field', Monthly Notices of the Royal Astronomical Society 238, 293-318 (1989). DOI: 10.1093/mnras/238.2.293. https://doi.org/10.1093/mnras/238.2.293
9. E. Regős, A. S. Szalay, 'Density and velocity correlations of peaks from random Gaussian fluctuations', Monthly Notices of the Royal Astronomical Society 272, 447-461 (1995). DOI: 10.1093/mnras/272.2.447. https://doi.org/10.1093/mnras/272.2.447
10. Takahiko Matsubara, 'Diagrammatic Methods in Statistics and Biasing in the Large-scale Structure of the Universe', Astrophysical Journal Supplement Series 101, 1 (1995). arXiv:astro-ph/9501056. https://arxiv.org/abs/astro-ph/9501056
11. Takahiko Matsubara, 'Stochasticity of Bias and Nonlocality of Galaxy Formation: Linear Scales', Astrophysical Journal 525, 543-553 (1999). DOI: 10.1086/307931. arXiv:astro-ph/9906029. https://arxiv.org/abs/astro-ph/9906029
12. V. Desjacques, 'Baryon acoustic signature in the clustering of density maxima', Physical Review D 78, 103503 (2008). DOI: 10.1103/PhysRevD.78.103503. arXiv:0806.0007. https://arxiv.org/abs/0806.0007
13. C. Gay, C. Pichon, D. Pogosyan, 'Non-Gaussian statistics of critical sets in 2D and 3D: Peaks, voids, saddles, genus, and skeleton', Physical Review D 85, 023011 (2012). DOI: 10.1103/PhysRevD.85.023011. arXiv:1110.0261. https://arxiv.org/abs/1110.0261
14. Vincent Desjacques, Donghui Jeong, Fabian Schmidt, 'Large-Scale Galaxy Bias', Physics Reports 733, 1-193 (2018). DOI: 10.1016/j.physrep.2017.12.002. arXiv:1611.09787. https://arxiv.org/abs/1611.09787
15. Takahiko Matsubara, Sandrine Codis, 'The large-separation expansion of peak clustering in Gaussian random fields', Physical Review D 101, 063504 (2020). DOI: 10.1103/PhysRevD.101.063504. arXiv:1910.09561. https://arxiv.org/abs/1910.09561
16. Tobias Baldauf, Sandrine Codis, Vincent Desjacques, Christophe Pichon, 'Nonperturbative halo clustering from cosmological density peaks', Physical Review D 103, 083530 (2021). DOI: 10.1103/PhysRevD.103.083530. arXiv:2012.14404. https://arxiv.org/abs/2012.14404
17. A. G. Doroshkevich, 'Spatial structure of perturbations and origin of galactic rotation in fluctuation theory', Astrophysics 6, 320-330 (1970). DOI: 10.1007/BF01001625. https://doi.org/10.1007/BF01001625
18. Vladimir I. Piterbarg, 'Asymptotic Methods in the Theory of Gaussian Processes and Fields', American Mathematical Society, Translations of Mathematical Monographs 148 (1996; electronic edition 2012). eBook ISBN 978-1-4704-4565-2. DOI: 10.1090/mmono/148. https://bookstore.ams.org/mmono-148