Papers

Let $\mu_0$ and $\mu_1$ be $\kappa_0$- and $\kappa_1$-strongly log-concave probability measures on $\R^n$, and let $(\mu_t)_{t\in[0,1]}$ be their quadratic Wasserstein geodesic. We prove the sharp Poincaré constant estimate \[ \sqrt{C_P(\mu_t)} \leq \frac{1-t}{\sqrt{\kappa_0}} + \frac{t}{\sqrt{\kappa_1}}. \] The coefficient is optimal for every $t,\kappa_0,\kappa_1$, and the result remains valid for extended-valued potentials without symmetry assumptions. Equality at an interior time holds exactly when the two endpoints have Gaussian factors in the same direction, with variances $\kappa_0^{-1}$ and $\kappa_1^{-1}$. All such directions form a maximal linear subspace. This gives an affirmative answer, without symmetry or parity assumptions, to a question of Aishwarya–Rotem. The proof develops a coupled Bochner method for the two endpoints. We solve a weighted Poisson equation involving the Hessian of the Brenier potential. From its solution we construct a Bochner couple, and separate estimates for its two fields are combined along the interpolation. No curvature bound is needed for the intermediate measures. Applications include centered Gaussian relative entropy, Gaussian Brunn–Minkowski inequalities with barycenter terms, and centered HWI, logarithmic Sobolev, and Talagrand estimates.

We study the global fluctuations of the height function associated with the Airy line ensemble. Using its determinantal structure and a steepest-descent analysis of the extended Airy kernel, we prove that, after a suitable rescaling, the centered height function converges to an explicit pullback of the Gaussian free field. The convergence holds in the sense of joint moments of linear statistics against compactly supported continuous test functions.

It is well known that the McKean-Vlasov stochastic differential equation with a symmetric double-well potential exhibits a continuous phase transition. In contrast, for an asymmetric double-well potential, the system undergoes a discontinuous phase transition, in which an invariant measure abruptly appears in the shallower well and subsequently splits into two as the temperature decreases. In this work, we systematically investigate the types of phase transitions driven by non-convex confining potentials and cooperative interactions in $\mathbb{R}^n$. Our main results consist of a pitchfork bifurcation theorem characterizing continuous phase transitions and a saddle-node bifurcation theorem governing discontinuous phase transitions. In addition, despite the dimension-dependent nature of phase transitions, we are able to extend Dawson's criterion for phase transition points – originally formulated for quadratic interactions in $1$-dimensional space – to $n$-dimensional space. This extension directly relates the critical parameter and bifurcation direction to the eigenvalue and eigenvector of the critical covariance matrix, respectively. Finally, we apply our theoretical results to the aforementioned double-well models, to a class of four-well models in two-dimensional Euclidean space that exhibit multiple phase transitions, and to an attractive Gaussian interaction model.

We study the geometry of reset distributions for absorbed Markov processes with geometric resetting, working at an abstract level that isolates the structural mechanism underlying reset-neutral invariance. We show that the spectral duality endows the simplex of reset distributions with a non-trivial spectral response geometry: the simplex $\Delta_{m-1}$ carries a foliation by level sets of the coupling functional $C$, organized around a critical manifold $\Sigma$. Under four structural conditions (S1)-(S4) on the coupling functional, we establish the existence and explicit characterization of the separatrix $\Sigma$, derive the invariant value $C^* = 1/(1+\sqrt{K})$, identify a projective structure in the spectral coefficients, and prove a global sign principle in the two-site case. In the canonical realization, the biased random walk with multi-site geometric resetting, numerical evidence shows that $\mathrm{sgn}(\partial_\gamma C) = \mathrm{sgn}\langle\pi-\pi^*,\psi(\gamma)\rangle$, where $\psi(\gamma)$ is the linear response functional at the separatrix; this global orientation does not, however, follow from (S1)-(S4) alone, and is dynamical rather than algebraic information. This is the third paper in a program connecting stochastic resetting with spectral theory and information geometry.

We introduce a class of piecewise deterministic Markov processes with resetting, motivated by self-exciting models of cyber attacks. Under assumptions reminiscent of ruin theory, we prove the existence and uniqueness of an invariant distribution and derive its density explicitly, thereby establishing ergodicity of the process. We then formulate an associated long-run average control problem in which resetting acts as the intervention mechanism. For exponentially distributed jump sizes, the model becomes analytically tractable, allowing an explicit characterization of the invariant distribution and of the optimal intervention policy.

Let $X_1,X_2,\ldots$ be independent $\mathrm{Bernoulli}(\theta)$ random variables, and let $\bar X_n = n^{-1}(X_1 + \cdots + X_n)$. We prove that, for every real $p \geq 1$, the sequence $\{\mathsf{E}(\bar X_n^p)\}_{n \geq 1}$ is log-convex. This proves the Bernoulli case of a conjecture of Lamkin and Tkocz [Canad. Math. Bull., 65(2):271-278, 2022]. The proof conditions on the total number of successes among $2n$ trials and reduces the desired inequality to a convex-order comparison between two normalized quadratic functions of hypergeometric random variables. All the log-convexity inequalities are strict for $p > 1$ and $0 < \theta < 1$.

We study the nonisotropic fractional analogs of Laplace, Helmholtz and Yukawa equations. We provide a Duffin correspondence for the Yukawa equation and a Feynman–Kac reconstruction for the Helmholtz equation. The foundation of our analysis is the fact that the nonisotropic fractional Laplace equation is related to a symmetric $\alpha$-stable Lévy process with independent identically distributed components. By using this relation we provide a Walk-on-Cubes algorithm that simulates the solutions of Helmholtz and Yukawa equations.

In this paper, we study the strong uniqueness problem for the mutually catalytic super-Markov chain, which is a two-dimensional degenerate stochastic differential equation with Hölder continuous coefficients. The key step is to find a process which is a function of two coupled processes and satisfies an autonomous one-dimensional stochastic differential equation; uniqueness for this equation follows from a Yamada-Watanabe argument. A large deviation principle is then established, in the irreducible two-state case, by applying the weak-convergence approach of Budhiraja, Dupuis and Maroulas to the controlled equations.

We prove the existence of a nematic phase in a model of $l\times w$ hard rectangles on the square lattice with two allowed orientations and a large aspect ratio $k:=l/w$. The proof is based on a two-scale cluster expansion method developed previously by Disertori–Giuliani for hard rods in 2D and Disertori–Giuliani–Jauslin for hard plates in 3D. Our main contributions lie in explicitly tracking the constants and parameters and completing the arguments left implicit in these works. Hence, the proof produces a sufficient set of quantitative conditions from which estimates for the required aspect ratio can be extracted. A non-optimized evaluation of these conditions yields the bound $k\ge 10^{72}$. Although it vastly overshoots the numerical prediction, $k_{\min}=7$, our result appears to be the first rigorous estimate of the aspect ratio required for the formation of a nematic phase in hard rectangle systems.

This book is about the mathematical foundations of data science. 1. Introduction 2. Curses, Blessings, and Surprises in High Dimensions 3. Singular Value Decomposition and Principal Component Analysis 4. Linear Regression and Regularization 5. Graphs, Networks, and Clustering 6. Nonlinear Dimension Reduction and Diffusion Maps 7. Linear Dimension Reduction via Random Projections 8. Optimization for Data Science 9. Classification 10. A Mathematical Introduction to Deep Learning 11. Large Sample Limit of Graph Laplacians 12. Community 13. Concentration of Measure and Gaussian Analysis 14. Matrix Concentration Inequalities 15. Compressive Sensing and Sparsity 16. Low-Rank Matrix Recovery

We consider a diffuse interface model for the mixture of two incompressible fluids driven by transport noise. Under suitable abstract assumptions, we prove the existence of a global weak martingale solution as well as the pathwise uniqueness of a global strong solution in the two dimensional case. This is the first result addressing a unified framework derived from the works by Brzeźniak & Motyl and Mikulevicius & Rozovskii, on the study of stochastic partial differential equations.

We provide a definitive analysis of the number of correct guesses in the complete-feedback card guessing game after an asymmetric single-shelf shuffle with parameter $p\in (0, 1)$. We explicitly describe the optimal strategy that maximizes the expected number of correct guesses. We study the number of correct guesses, under an optimal strategy, using methods from analytic combinatorics. In addition, we find the explicit distribution for the number of correct guesses, and thus find the mean and the variance for the number of correct guesses. We show that the distribution of the number of correct guesses is log-concave. We also study the limiting behaviour of the number of correct guesses as the number of cards goes to infinity. In particular, we prove a (local) central limit theorem and a large deviation principle with an explicit rate function. We also prove phase transitions for the number of correct guesses near $p=0$ and $p=1$. Prior to this work, only the optimal strategy and the expected number of correct guesses under the optimal strategy were known for the special case of $p=1/2$.

Let $X_1,\ldots,X_N$ be independent random vectors in $\mathbb{R}^n$ with common isotropic log-concave distribution $\mu$ and set $P_{N,n}^{\mu}:=\operatorname{conv}\{\pm X_i:1\leqslant i\leqslant N\}$. Assume that $N/n=\gamma\geqslant \gamma_0$ where $\gamma_0>1$ is an absolute constant. We prove that with probability at least $1-C\gamma\exp(-c n^{1/4})$ every $k$-dimensional subspace $E$ of $(\mathbb{R}^n,\|\cdot\|_{P_{N,n}^{\mu}})$ satisfies $d_{\mathrm{BM}} (E,\ell_\infty^k) \geqslant c\gamma^{-C}k^\alpha$ for every $1\leqslant k\leqslant n$ where $c,C,\alpha>0$ are absolute constants. Consequently, with the same probability, $(\mathbb{R}^n,\|\cdot\|_{P_{N,n}^{\mu}})$ has cotype $q(\gamma)<\infty$ with cotype constant depending only on $\gamma$, in particular the cotype exponent and the cotype constant are independent of $n$ and of $\mu$. The proof adapts the deterministic coefficient scheme of Huang-Tikhomirov replacing the Gaussian estimates in their argument by estimates for isotropic log-concave random matrices. As an application, using the log-concave extension of Gluskin's theorem, we obtain a separable Banach space of finite cotype for which the Banach-Mazur diameter of its $k$-dimensional subspaces is of order $k$ and whose finite-dimensional building blocks are generated by isotropic log-concave random polytopes.

The thrust of this article is to show that on all Banach spaces, stochastic approximation holds when the noise sequence is an i.i.d. sequence with mean 0, without imposing any condition on the geometry of the space. Also, the same is true when the noise is a sequence of independent random variables under appropriate conditions on the moment. In this case, we need to require that the noise sequence is tight.

We study the weakly coupled elliptic Anderson model with spatial white noise on the four-dimensional torus, which provides a basic example of a critical SPDE requiring renormalization at arbitrarily high orders. With coupling $\lambda |\log\varepsilon|^{-\frac12}$ where $\lambda>0$ is sufficiently small, we prove that the Green's function of the corresponding random Schrödinger operator, suitably centered and rescaled, converges to a centered Gaussian random field with explicit covariance. The main difficulty is that, for such critical models, one must expand up to order $|\log\varepsilon|$, while the perturbative expansion contains factorially many pairings and a growing number of renormalization terms. To overcome this, we construct a truncated renormalized parametrix and prove sharp high-order bounds for its remainder. A central ingredient is a multiscale analysis based on a new version of Hepp trees, combined with new estimates for summations over permutations. These estimates reveal a precise balance between logarithmic losses from scale summation and factorial gains from the structure of primitive pairings. The methods developed here are intended as a first step toward a general theory for critical SPDEs with weak couplings.

We study a class of diffusion processes on $\mathbb{R}^n$ interacting through a symmetric matrix $X\in\mathbb{R}^{n\times n}$. When eigenvectors of $X$ are Haar-uniform on the orthogonal group, we derive a dynamical mean-field limit for the empirical law of sample paths, extending the classical Sompolinsky–Zippelius characterization for $X\sim\mathrm{GOE}$. The limit takes the form of a generalized Langevin equation with correlated Gaussian noise and memory, whose correlation and response kernels relate to those of the original dynamics through convolution equations involving the free cumulants of the eigenvalue distribution of $X$. For the overdamped Langevin diffusion associated with $\mu(\boldsymbol{\theta})\propto \exp\!\big(\frac12\boldsymbol{\theta}^{\top}X\boldsymbol{\theta}\big)\prod_{i=1}^n\nu(\mathrm{d}\theta_i)$, we analyze the mean-field limit under a rapid-mixing assumption. The correlation and response kernels admit time-translation-invariant approximants satisfying a fluctuation-dissipation relation. The generalized Langevin equation admits a Markovian approximation coupled to an auxiliary multivariate OU process and converges to a replica-symmetric prediction for the empirical coordinate law under $\mu$. This auxiliary correlation structure is characterized through the infinitesimal generator of a Markov semigroup for the lifted path-history process. Consequently, the free energy converges to a replica-symmetric limit under an explicit high-temperature condition, which for an Ising model is $\|X\|_{\mathrm{op}}<1/2$. By recent dynamical universality results, the same free-energy characterization holds for deterministic models without random disorder when $X$ satisfies a set of deterministic delocalization conditions.

Fix a prime $p$ and a constant $\frac{1}{p}<\alpha\leq 1$. Consider the random Erdős–Rényi bipartite graph $G_{\alpha}(n,u)$ with bipartition $(V_1,V_2)$ of sizes $|V_1|=n$ and $|V_2|=\lceil\alpha n\rceil$, and edge probability $0<u<1$. The authors of [1] and [8] conjectured a limiting distribution for the $p$-Sylow subgroup of the sandpile group of $G_{\alpha}(n,u)$ as $n\to\infty$. We prove this conjecture for odd primes $p$. Similar results have previously been proved by computing the expected number of surjections from the random abelian $p$-group to $H$, for each finite abelian $p$-group $H$. However, in our setting, these surjective moments often diverge to infinity, despite the conjectured limiting distribution having finite moments. We resolve this issue by discarding the graphs for which too many vertices have degrees divisible by $p$. Once we remove the contribution of this rare set of graphs, then the surjective moments converge to the expected values. When $p$ is odd, applying Wood's universality theorem yields the desired convergence in distribution. For $p=2$, our computed moments (after excluding the rare set of graphs) match those of the conjectured distribution. However, these moments do not uniquely determine a distribution.

We prove a full Adam theorem for spectral heavy-tail onset in a closed Gaussian Stein-Hermite teacher-student state-evolution model. The theorem begins with the actual full-batch Adam recurrences, derives the population gradient by Stein-Hermite calculus, proves finite-width covariance concentration, converts multi-step Adam momentum into an exact non-centered Gaussian sign kernel, controls the diagonal Adam denominator by a basis-homogenization theorem, derives a regularly varying projected update response from a Hermite edge-transfer theorem, pushes the response through the exact Gram update, and proves approximate-target KL contraction with matching upper and lower hitting bounds. The final law is (\tau_\varepsilon=\Theta(\Delta_1^{-\gamma}d^\rho\log(\Psi_0/\varepsilon))), where (\Delta_1) is the first spike-bulk spectral gap. The result is full in the following precise sense: every step from Adam's momentum and denominator to the spectral hitting law is formalized inside the closed state-evolution model. We also prove that a stronger arbitrary-gradient Adam theorem is impossible, and that exact two-step linear-network loss dynamics do not identify factor spectra or heavy-tail hitting times.

This paper provides a finite-dimensional spectral characterisation for when the stationary covariance matrix of a finite-state Markov-switching affine recurrence exists. For stochastic recurrences of the form \[ Z_{n+1}=A_{\Theta_{n+1}}Z_n+\zeta_{n+1}, \] strict stationarity is typically governed by negativity of the top Lyapunov exponent of the random matrix products. This condition, however, does not ensure that the stationary law has finite variances, covariances, and Pearson correlations. We show that these second-order objects are governed instead by a Markov-switching Kronecker operator. If \[ T_{ji}:=p_{ij}(A_j\otimes A_j), \] then, under a natural Perron-excitation condition on the innovation covariance, \(\rho(T)<1\) is necessary and sufficient for the stationary solution to be square-integrable. When this condition holds, the regime-weighted second moments solve a finite-dimensional linear system, which yields the covariance matrix explicitly. Hence, this paper translates the mean-square spectral condition into an exact covariance-existence criterion for the stationary distribution. Examples illustrate the separation: the Lyapunov exponent may be negative while the second-order spectral radius exceeds one, so strict stationarity can persist even though variances, covariances, and Pearson correlations are not finite objects.

We study an infinite-horizon optimal stopping problem for a two-dimensional normally reflected diffusion in the quadrant with payoff \(G(x_1,x_2)=x_1\vee \alpha x_2\). The problem combines three features that complicate the usual free-boundary analysis: reflection on the coordinate axes, a genuinely two-dimensional stopping region, and a nonsmooth max-type reward. We formulate the associated reflected obstacle problem, prove a verification theorem under explicit Itô–Krylov–Tanaka admissibility and measure-superharmonicity assumptions, and derive a conditional epigraph structure for the stopping set. The main technical point is that the stopping-gain object \(\Gamma=c+rG-\mathcal LG\) is a signed measure rather than a function. Its diagonal component is $\Gamma^\Delta(dx) = -\frac{n^\top a(x)n}{2\sqrt{1+\alpha^2}}\sigma_\Delta(dx)$, $n=(1,-\alpha)$, which shows that pointwise stopping-gain sign conditions must be interpreted with care. We also prove that the correct potential representation is the killed-resolvent formula $V(x)=G(x)-R_r^{\mathcal C}\Gamma(x)$, rather than the unrestricted reflected resolvent. A constant-coefficient reflected Brownian example illustrates the diagonal singular term explicitly.

We investigate the probabilistic counterpart of Yau's conjecture on the nodal volume of Laplace eigenfunctions on compact manifolds, by studying the high-frequency variance asymptotics of Riemannian random waves. We establish (Theorem A) a quantitative bound for the fluctuations of their nodal volumes, depending on different regimes of spectral windows, including the monochromatic one, of spectral size 1. Notably, our bounds improve, by more than a power 2, the existing results in the literature, cf. Canzani and Hanin (2020), in the case of manifolds without conjugate points, in particular negatively curved ones. As a corollary, we prove that Berry's cancellation phenomenon occurs for monochromatic Riemannian Random Waves on such chaotic manifolds. Our proofs rely on a local and global analysis combining the Kac-Rice formula, the new Wiener-Itô chaos decomposition of Stecconi and Todino (2025), and a sharp analysis of the error in the pointwise Weyl law associated to an arbitrary spectral window (Theorem B). We introduce a general machinery (Theorem C), which ensures variance decay under broad geometric conditions, subject to correlation decay assumptions.

We consider the three-dimensional stochastic Navier–Stokes equations with additive stochastic forcing. We study temporal discretizations based on a semi-implicit Euler as well as a Crank-Nicolson scheme. We prove that locally in time the former converges with rate 1 while the latter converges with rate 3/2. These theoretical results are confirmed by extensive numerical simulations. To the best of our knowledge this is the first time that numerical simulations for the three-dimensional stochastic Navier–Stokes equations have been performed.

Membership problems in the completely monotone–Stieltjes–Bernstein hierarchy often become explicit after passage to a determinate representation. We organize that passage as a witness calculus. For the Laplace, Stieltjes, and Hausdorff transforms used here, W f denotes the unique admissible signed representing datum; a certificate either proves its positivity or exhibits an obstruction. A concrete dictionary for completely monotone, logarithmically completely monotone, Bernstein, complete Bernstein, and Stieltjes functions and for Hausdorff moment sequences is combined with transport rules for products, shifts, sampling, and Bernstein increments. The applications are consequences of this common calculation. A triple-product identity for a digamma bridge controls ordinary Bernstein and logarithmic complete monotonicity ranges for gamma ratios and gives the strict power cutoff in Szabo's problem. For integer gamma quotients, the inverse Laplace density is a damped Jacobi polynomial, so classical zero geometry decides broad parameter regions. Boundary signs distinguish Bernstein from complete Bernstein behavior and separate completely monotone functions from the Stieltjes class. On the discrete side, signed atoms and finite differences settle interpolation and coefficient questions. The Ramanujan integral and a fractional Volterra symbol provide positive recognition examples. Across the applications, the workflow is the same: normalize the class question, compute or transport its witness, and decide one explicit sign problem. Every exclusion is accompanied by an exact certificate.

We study the mixing time of the systematic scan dynamics for the $q$-state ferromagnetic Potts model on the $n$-vertex complete graph, known as the mean-field model. This Markov chain updates vertices sequentially according to a fixed predetermined order, in contrast to the Glauber dynamics which updates a uniformly random vertex at each step. Systematic scan dynamics are attractive in practice as they often demonstrate strong empirical performance. However, their theoretical analysis remains far less developed than that of the Glauber dynamics. We take a step toward addressing this imbalance by showing that for every $q\ge 2$ and $\beta<\beta_s$, where $\beta_s$ is the metastability threshold associated with the onset of slow mixing for the Glauber dynamics, the systematic scan dynamics for the ferromagnetic mean-field Potts model mixes in $\Theta(\log n)$ scans or, equivalently, in $\Theta(n\log n)$ single site updates. We in fact prove a sharper result; namely, that there exists a constant $c(\beta,q) > 0$ such that the mixing time is $c(\beta,q)\log n + \Theta(1),$ which implies that the Markov chain exhibits the cutoff phenomenon, with the total variation distance to the stationary distribution dropping abruptly from nearly 1 to nearly 0 within a narrow $\Theta(1)$ time window. This result is tight in $\beta$ as well since the dynamics mixes exponentially slowly for $\beta > \beta_s$. To the best of our knowledge, this is the first general cutoff result for the systematic scan dynamics in the context of spin systems. The result may also be of independent interest in the theory of Markov chains, since the systematic scan dynamics is both global and non-reversible, two settings in which cutoff remains poorly understood.

In this note, we establish various probabilistic estimates for an interacting particle system that describes the evolution of the membrane potentials in a network of excitatory integrate-and-fire neurons, which are subject to both idiosyncratic and common noise. These estimates serve to support a separate work that studies the large population limit and the corresponding stochastic Fokker–Planck equation for the membrane potential density.

High-dimensional interpolation is common in modern machine learning, but its tail risk is less understood than its expected prediction risk. Existing theory shows that interpolating models can perform well in expectation, yet such guarantees do not determine the probability of rare, severe errors. In operations research and stochastic decision-making applications, rare estimation errors can have disproportionate downstream effects, so tail behavior matters alongside average performance. We study the fragility of high-dimensional linear interpolators using large-deviation methods. We focus on ridgeless regression and compare it with ridge-regularized estimators. We first show that the risk of ridgeless regression can exhibit heavy-tailed behavior: although its expected risk may remain well controlled, its upper tail can decay much more slowly than that of regularized alternatives. We then quantify this phenomenon at the level of large-deviation rates. In the regime we study, ridge regularization suppresses fixed right-tail deviations at the $n^2$ scale, whereas ridgeless regression has only $n\log n$-scale decay, where $n$ is the sample size. This gap shows that interpolation can be statistically fragile even when it is accurate on average. Thus regularization affects the frequency of rare, high-impact risk events in addition to the usual bias-variance tradeoff.

We study log-convexity and log-concavity of densities obtained from sums and differences of two independent noncentral gamma random variables. We give a complete classification of one-sided log-convexity for noncentral gamma differences, a complete log-convexity classification for sums of two independent central gamma random variables, and sharp log-concavity criteria for central differences and for common-scale sums. As special cases, we deduce a log-convexity classification for the density of the product of two correlated normal random variables with arbitrary means and variances, and log-convexity and log-concavity classifications for the densities of the variance-gamma and McKay Type I distributions.

We adopt a modern approach to extend a comparison inequality of Chevet for Gaussian processes to a broader class of functions. Furthermore, we consider centered Gaussian random vectors associated with a family of $n+1$ vectors on the unit sphere $S^{n-1}$ and investigate configurations of the vertices for which these new functionals could be maximum for the regular simplex.

We study a $(1+1)$-dimensional directed polymer in a spatially correlated random environment generated by power-law tail variables: $\omega(i,x)=\sum_{y\in\mathbb Z}\psi_{y-x}\xi(i,y), \psi_y\sim \lambda_r |y|^{-r}, r\in(1/2,1)$, where the variables $\xi(i,y)$ are i.i.d. and have a regularly varying right tail with exponent $\alpha>2$. The spatial covariance of the environment has long-range decay with Hurst parameter $H=\frac32-r\in(1/2,1)$. We identify the limiting fluctuations of the log-partition function in the intermediate disorder regime and show that the critical tail exponent is $\alpha_c=\frac{3}{H}=\frac{6}{3-2r}$. When $\alpha>\alpha_c$, the model has the same scaling limits as the corresponding Gaussian spatially correlated polymer: if $\beta_NN^{H/2}\to\beta\in(0,\infty)$, the centered log-partition function converges to the logarithm of the solution of the stochastic heat equation driven by fractional spatial noise; if $\beta_NN^{H/2}\to0$, its normalized fluctuation converges to a centered Gaussian law. In the regime $2<\alpha\le\alpha_c$, at the scale $\beta_NN^{H/2}=\beta N^{H/2}/l(N^{3/2})$, the log-partition function still satisfies Gaussian fluctuation. The main ingredient is a truncation comparison argument adapted to long-range moving-average environments, together with an invariance principle for polynomial chaos. Due to the non-locality of the environments, we perform a far-near field analysis, as well as multiscale analysis, to prove that the truncated version does not change the log-partition function at the corresponding scales.

This paper establishes a central limit theorem (CLT) for functionals of $M$-bounded persistence diagrams arising from germ-grain random set models. Building on stabilisation methods for marked point processes, we show that, under certain conditions, these topological summaries exhibit asymptotic normality as the observation window increases, particularly for models with exponential decay of correlations. These results are applied in goodness-of-fit tests designed to detect spatial interactions such as clustering or repulsion. Using test statistics derived from rectangular partitions of persistence diagrams and functional summaries (e.g., the APF or the support function of the lift zonoid), the study distinguishes between different models. Finally, the methodology is applied to histological images of breast tissue.