--- title: Critical Point Processes in Random Fields url: https://www.emergentmind.com/topics/critical-point-processes type: topic --- # Critical Point Processes in Random Fields to=arxiv_search.search ＿欧美json {"query":"all:(\"critical point processes\" OR \"critical points\" Gaussian random field point process)","max_results":10,"sort_by":"submittedDate","sort_order":"descending"} to=arxiv_search.search 天天爱彩票网站json {"query":"ti:\"Critical Point Processes Obtained from a Gaussian Random Field with a View Towards Statistics\"","max_results":5,"sort_by":"relevance","sort_order":"descending"} to=arxiv_search.search 】【。】【”】【json {"query":"\"Critical Point Processes Obtained from a Gaussian Random Field with a View Towards Statistics\"","max_results":5,"sort_by":"relevance","sort_order":"descending"} to=arxiv_search.search 彩票天天乐json {"query":"ti:\"A note on critical Hawkes processes\"","max_results":5,"sort_by":"relevance","sort_order":"descending"} Critical point processes arise in several distinct but adjacent literatures. In the most specific stochastic-geometric sense, they are spatial point processes formed by the critical points of a latent smooth Gaussian random field, or by subsets of those critical points such as minima, saddle points, and maxima [2507.04753]. In related work, closely connected notions of criticality appear in point-process models at borderline regimes, including rigid determinantal processes at critical density, Hawkes processes with branching ratio \(1\), and critical cluster cascades [1211.2435, 1706.03975, 2208.08383]. The common theme is the analysis of locally finite random configurations whose structure is controlled by a critical balance: geometric stationarity and Morse structure in Gaussian fields, density thresholds in determinantal systems, or exact branching balance in self-exciting and cluster models. ## 1. Gaussian-field construction The most direct construction starts from a centered, stationary, isotropic Gaussian random field \[ X=\{X(t),\, t\in\mathbb{R}^d\}, \] assumed to have unit variance and enough smoothness—at least twice continuously differentiable—so that its gradient and Hessian exist [2507.04753]. A critical point is a location \(t\) at which \[ X'(t)=0. \] For each Hessian index \(\ell\in\{0,\dots,d\}\), the associated index-restricted critical point process is \[ Y_\ell = \{t\in\mathbb{R}^d: X'(t)=0,\ \iota\{X''(t)\}=\ell\}, \] where \(\iota\{X''(t)\}\) counts the number of negative eigenvalues of the Hessian. Thus \(Y_0\) is the set of local minima, \(Y_d\) is the set of local maxima, \(Y_{0:d}\) is the set of all critical points, and intermediate \(Y_\ell\) are saddle-type processes [2507.04753]. A central conceptual feature is that this is not a stochastic intensity-driven process in the Cox-process sense. Once the Gaussian field is realized, the point pattern of critical points is deterministic; randomness enters only through the latent field itself. The resulting process inherits stationarity and isotropy from the field. This shifts the modeling emphasis from intensity specification toward differential geometry of random fields, Kac–Rice theory, and the joint law of field derivatives [2507.04753]. The same source develops the class as a full spatial-statistical model rather than as an isolated geometric object. The analysis relies on geometry of random fields, Sobolev space theory, Kac–Rice formulae, chaos expansions, and multiple Wiener–Itô integrals. This framework is needed because counts of critical points depend simultaneously on the gradient constraint \(X'(t)=0\) and on Hessian functionals such as \(|\det X''(t)|\) and the Morse-index indicator \(\iota_\ell\{X''(t)\}\) [2507.04753]. ## 2. Moment structure and dependence A main contribution of the Gaussian-field theory is the derivation of explicit moment characteristics: the intensity parameter, the pair correlation function, and higher-order intensity functions [2507.04753]. The intensity of \(Y_\ell\) is given by a Kac–Rice expectation, and its dependence on the underlying field enters only through the ratio \[ \frac{\lambda_4}{3\lambda_2}. \] For the two principal examples treated explicitly, this ratio is \[ \frac{\lambda_4}{3\lambda_2}=\frac{1}{\varphi^2}\frac{\nu}{\nu-2} \quad\text{for the Matérn field,} \] and \[ \frac{\lambda_4}{3\lambda_2}=\frac{1}{\varphi^2}\frac{d}{d+2} \quad\text{for the random wave model} \] [2507.04753]. Second-order structure is encoded by the pair correlation function. For \(r>0\), under nondegeneracy of the joint gradient vector \(V(r)=\{X'(0)^\top,X'(re_1)^\top\}^\top\), the paper gives \[ g_{L,L'}(r) = \frac{1}{\rho_L\rho_{L'}\,f_{V(r)}(0,0)} E\!\left[ |\det X''(0)|\,|\det X''(re_1)|\, \iota_L\{X''(0)\}\iota_{L'}\{X''(re_1)\} \mid X'(0)=X'(re_1)=0 \right]. \] More generally, for any \(k\ge1\), \[ \rho_L^{(k)}(t_1,\dots,t_k) = f_V(0)\, E\!\left[ \prod_{j=1}^k |\det X''(t_j)|\,\iota_L\{X''(t_j)\} \mid X'(t_1)=\cdots=X'(t_k)=0 \right], \] again assuming nondegeneracy of the joint gradient vector [2507.04753]. The dependence structure is highly sensitive to dimension and point type. For all critical points \(L=\{0,\dots,d\}\), the short-range asymptotics are reported as \[ g_{0:d}(r)\sim c\,r^{2-d}, \] so that the function tends to a constant in \(d=2\), diverges for \(d>2\), and vanishes for \(d<2\). By contrast, for minima or maxima, \[ g_\ell(r)=O(r^{5-d-\varepsilon}),\qquad \ell=0,d, \] under suitable regularity assumptions, which is consistent with stronger local repulsion. For adjacent indices the paper also states asymptotics such as \[ g_{\ell,\ell+1}(r)\sim c\,r^{2-d}. \] The paper further introduces a repulsiveness index \[ I_L(r) = 1+\rho_L^{-1}\int_{B(0,r)}\{g_L(\|t\|)-1\}\,dt, \] with \(I_L(r)<1\) interpreted as net repulsion up to distance \(r\) [2507.04753]. These results show that “critical point process” is not a synonym for repulsive point process. Attraction or repulsiveness depends on the ambient dimension, the covariance structure of the field, and whether one studies all critical points together or only a fixed Morse class. The paper’s tabulations also show that maxima and minima become increasingly rare among all critical points as dimension increases [2507.04753]. ## 3. Simulation and asymptotic statistics The Gaussian-field theory is explicitly statistical in orientation. Because direct exact simulation of the critical point process is essentially infeasible, the proposed strategy is to simulate an approximate realization of the Gaussian field and then numerically locate its critical points. Two approximation schemes are developed, and both are proved to converge at the level of the induced critical point processes [2507.04753]. The first is smoothing of grid-based simulations: \[ X_n(t)=\sum_{x\in L_n\cap W} n^{-d}k_{\xi_n}(t-x)X(x), \] with bandwidth \(\xi_n\to0\). Under the condition \[ n^{-1}\xi_n^{-d-3}\to0 \] and almost-sure \(C^{2+\varepsilon}\)-smoothness of \(X\), one has \[ X_n \to X \quad\text{in } C^2(W)\ \text{a.s.} \] The second is a spectral approximation built from \[ Z_i(t)=\sqrt{-2\log(W_i)}\cos(U_i+t^\top V_i), \qquad X_n(t)=\frac{1}{\sqrt{n}}\sum_{i=1}^n Z_i(t), \] for which the paper proves \[ X_n \xrightarrow{\mathcal D} X \quad\text{in } C^2(W) \] under stronger smoothness assumptions. A key transfer theorem states that if \(X_n\to X\) in \(C^2\) and \(X\) is an MB function, meaning Morse with no boundary critical points, then the corresponding critical point processes converge [2507.04753]. The statistical asymptotics are developed under the increasing-domain regime \[ W_n=[-n/2,n/2]^d. \] For linear statistics \[ \Phi_{1,n}=\sum_{t\in Y_L\cap W_n}\phi_{1,n}(t), \] the paper derives a Hermite-chaos expansion and variance asymptotics. A basic estimator is the intensity estimator \[ \hat\rho_L = \frac{N_L(W_n)}{|W_n|}, \] for which \[ n^{d/2}(\hat\rho_L-\rho_L) \] satisfies a central limit theorem. For bilinear statistics, the paper defines a modified Ripley \(K\)-function estimator \[ \hat K_{\eta,L}(r)=\Phi_{2,n}\,\frac{\rho_L^2}{\hat\rho_L^2}, \] with a lower cutoff \(\eta>0\), and proves the multivariate central limit theorem \[ n^{d/2} \begin{pmatrix} \hat\rho_L-\rho_L\\ \hat K_{\eta,L}(r_1)-K_{\eta,L}(r_1)\\ \vdots\\ \hat K_{\eta,L}(r_m)-K_{\eta,L}(r_m) \end{pmatrix} \Longrightarrow \mathcal N(0,\Sigma) \] for fixed \(r_1,\dots,r_m\) [2507.04753]. This establishes a full inference program. The same paper explicitly suggests minimum-contrast estimation of field parameters through \[ \big(\hat\rho_L-\rho_L(\theta)\big)^2 + \int_{r_1}^{r_2}\big(\hat K_{\eta,L}(r)-K_{\eta,L}(r;\theta)\big)^2\,dr. \] A plausible implication is that critical point processes can be treated as primary statistical objects, not only as geometric summaries of random fields [2507.04753]. ## 4. Critical-regime point processes A different family of results uses “critical” not for critical points of a field, but for borderline point-process regimes. In determinantal theory, the paper on completeness of random exponentials studies the critical density case for rigid determinantal point processes. If \(\Lambda\) is a realization of the continuum sine kernel process, then \[ \mathcal{E}_{\Lambda}=\{e^{i\lambda x}:\lambda\in\Lambda\} \] spans \(L^2[-\pi,\pi]\) almost surely; similarly, a realization of the Ginibre ensemble yields an almost surely complete system in the Fock–Bargmann space. More generally, for a rigid determinantal process with projection kernel \(K\), the family \(\{K(\cdot,x):x\in \Pi\}\) is almost surely complete in the associated closed subspace \(\mathcal H\) [1211.2435]. The paper identifies rigidity, rather than density alone, as the decisive mechanism in this critical case. Critical Hawkes processes are defined by branching ratio \(m=1\), so that the conditional intensity takes the form \[ \lambda(x)=\int_{(-\infty,x)} f(x-y)\,N(dy). \] The central existence result is a necessary condition: if a critical \((F,\lambda)\)-Hawkes process exists, then the random walk induced by the symmetrized law \(\tilde F\) must be transient. The paper further frames the process as a cluster-invariant point process, proves uniqueness, stationarity, and infinite divisibility of the law when it exists, and develops three constructions: a Poisson embedding, a renewal-immigration representation, and a backward construction yielding a Palm version of the critical Hawkes process. The genealogical structure is proposed to be encoded by Kesten trees [1706.03975]. Critical cluster cascades provide another exact-balance model. At step \(n\), cluster centers have intensity \(c/(n+1)\), while each cluster consists of the particles of a branching random walk up to generation \(n\) generated by a critical point process with mean \(1\). The resulting Poisson cluster process \((\overline{\xi}_n)\) converges weakly, and the limit either equals the void process or has the same intensity \(c\) as the prelimit cascade. Persistence occurs if and only if the Palm version of the outgrown critical branching random walk is locally almost surely finite [2208.08383]. A structural Palm-theoretic complement is given by the superposition theorem for independent point processes. If \(\Phi=\Phi_1+\Phi_2\), then the Palm kernel of \(\Phi\) at \(x\) is an intensity-weighted mixture of \(\Phi_{1x}+\Phi_2\) and \(\Phi_1+\Phi_{2x}\), with weights proportional to \(M_{\Phi_1}(x)\) and \(M_{\Phi_2}(x)\). Extensions to finite superpositions and higher-order Palm distributions are obtained analogously [2409.14753]. This suggests a useful general principle: in composite critical models, local typical-point structure can often be decomposed into Palm laws of simpler components. ## 5. Phase transitions, percolation, and self-organization Several papers connect point-process criticality to percolation thresholds. For Gibbs point processes in \(\mathbb{R}^d\), the Boolean model \[ Z_R(E)=\bigcup_{x\in E} B_R(x) \] exhibits a critical activity phenomenon. If the conditional intensity satisfies condition (P), then for every \(R>r\) there exists a finite threshold \(\beta_+=\beta_+(R,r,\delta)\) such that for every \(\beta>\beta_+\), \[ Z_R(E)\ \text{contains almost surely an infinite cluster.} \] Conversely, if the process is locally stable, then for every \(R>0\) there exists \(\beta_-=\beta_-(R,c^*)>0\) such that for \(\beta<\beta_-\), \[ Z_R(E)\ \text{contains almost surely only finite clusters} \] [1305.0492]. The critical parameter is the activity \(\beta\), not the geometry of critical points of a field. The relation between clustering and percolation is more delicate than monotonic heuristics suggest. In the directionally convex ordering framework, a natural conjecture was that larger clustering should increase the continuum-percolation critical radius \(r_c\). The papers on clustering and percolation show that this fails in full generality: one can construct a Cox point process that is \(dcx\)-larger than a homogeneous Poisson process and nevertheless has degenerate critical radius \[ r_c=0. \] They therefore replace direct monotonicity of \(r_c\) by comparison of two auxiliary critical radii, obtaining sandwich bounds and nontrivial phase transitions for weakly sub-Poisson processes, including determinantal point processes, negatively associated processes, \(k\)-percolation, and SINR-percolation [1105.4293, 1112.2227]. A more dynamical form of criticality appears in the rank-based Markov chain on finite subsets of \([0,1]\) introduced as a self-reinforced point process with self-organized criticality. At each step a uniformly sampled point is added; if it lies to the right of the current minimum, the minimum is removed. The system develops the sharp threshold \[ p_c=1-e^{-1}, \] with theorems showing that particles arriving to the left of \(p_c\) are almost surely eventually removed, while for large enough time particles arriving to the right of \(p_c\) stay in the system forever [1405.3609]. Here the critical phenomenon is internal selection of a threshold by the dynamics itself. ## 6. Terminological extensions and adjacent literatures The same words, “critical point” and “point process,” also appear in nearby literatures where the object of study is different. In QCD, the phrase refers to the endpoint of a first-order transition line. One line of work analyzes the QCD phase diagram in strong-coupling lattice QCD with Polyakov-loop effects, finding that chiral and deconfinement transition boundaries deviate at finite \(\mu\), with a critical point at \[ (\mu_{\mathrm{CP}},T_{\mathrm{CP}})=(0.58,0.19) \] at \(\beta=4\) in the chiral limit, and proposing that prompt black hole formation may dynamically sweep the critical region in isospin-asymmetric matter [1201.6206]. Complementary work on critical dynamics near the QCD critical point identifies thermal diffusion, viscous diffusion, and sound as the hydrodynamic slow modes, finds that bulk viscosity and thermal conductivity diverge strongly at criticality, and derives dynamic critical exponents \[ z_{\rm thermal}\sim 3,\qquad z_{\rm viscous}\sim 2,\qquad z_{\rm sound}\sim -0.8 \] [1201.6408]. In machine-learning applications, neural temporal point processes are used for critical-care event prediction rather than for modeling critical points of a random field. One study formulates adverse-event onset prediction in ICU data as continuous-time event forecasting over event times \(0