Papers

We study the particle number distribution of locally interacting bosonic quasi- free states, which arise in various areas of mathematical physics. We show that the particle number decomposes into an infinite sum of independent geometrically distributed random variables, confirming predictions from the physics literature. This representation yields exponential tail bounds for the particle number, a law of small numbers together with a large deviation principle at logarithmic speed when the lattice spacing diverges, and a central limit theorem when it vanishes. As an application, we obtain a detailed description of the statistics of the quantum depletion in Bose–Einstein condensates, with the constants explicit in terms of the scattering length of the interaction potential.

Will an attack on a network affect its structure in a major way? In this work we study this question by investigating a dynamic preferential attachment-type graph in which an attacker interferes with the growth mechanism. We show that our model of an attack exhibits a phase transition. If the skill of the attacker is low, the network preserves its scale-free structure inherited from the underlying preferential attachment structure (even though the tail of the degree distribution becomes lighter due to the attack). When the skill of the attacker increases and crosses a critical boundary, the scale-free property is destroyed. The skill of the attacker is expressed as the probability that, at a given step, the attacker succeeds in preventing adding an edge to the network and, instead, causes a deletion (in a preferential manner) of an edge. In all cases we describe the tail behaviour of the degree distribution.

The goal of this paper is to present a one dimensional heat equation with a singular drift for which solutions of the Cauchy problem with smooth initial data are bounded and are discontinuous at an interior point (common to all solutions).

Consider a one-dimensional branching Brownian motion. Let $W_\infty(\beta)$ denote the limit of the additive martingale in the subcritical regime $\lvert \beta\rvert < \beta_c$ and $Z_\infty$ be the limit of the derivative martingale at criticality. Madaule (Stochastic Process. Appl. 126 (2016), no. 2, 470–502) established the following convergence \[ \frac{W_\infty(\beta)}{\beta_c-\beta}\xrightarrow[\beta\nearrow \beta_c]{\mathbb{P}} 2Z_\infty. \] The goal of this paper is twofold: firstly, we strengthen this result into an almost sure convergence; secondly, we describe the fluctuations occurring in this convergence by proving \[ \frac{1}{\beta_c-\beta}\left( \frac{W_\infty(\beta)}{\beta_c-\beta} - 2 Z_\infty +2(\beta_c-\beta)\log(\beta_c-\beta) Z_\infty\right) \xrightarrow[\beta\nearrow \beta_c]{(d)} S, \] where, conditionally on $Z_\infty$, $S$ follows a spectrally negative 1-stable distribution with scale and shift parameters proportional to $Z_\infty$. Furthermore, these results are extended to the setting of complex additive martingales and the fluctuations to a multi-dimensional convergence.

We prove the Ding–Song–Sun inequality for continuous spin models on finite ferromagnetic graphs whose even single-site potentials have convex derivatives on the positive half of their domain. This class, introduced by Ellis, Monroe and Newman, includes the $\varphi^4$ and sinh-Gordon potentials. The proof uses a rank-one interpolation of the interactions and a maximum principle for a cooperative parabolic system on a half-plane. The boundary conditions follow by induction on the number of vertices and the number of positive coordinates of the field increment. In particular, the truncated two-point function in an arbitrary external field is bounded above by its zero-field value.

Let $X=(X_1,\dots,X_n)\in\{0,1\}^n$ have a multivariate totally positive ($\mathrm{MTP}_2$) law. We prove that $$ \sum_{i<j}\mathbb{E}\left[\left|\mathrm{Cov}(X_i,X_j \mid X_{[n]\setminus\{i,j\}})\right|\right] \le n/2, $$ and more generally a weighted MaxCut inequality for the fully conditioned covariances. As an application, we confirm a conjecture of Allen and O'Donnell on correlation rounding for signed $\mathrm{MTP}_2$ laws.

Motivated by Wasserstein measures of dependence, we study the largest possible 2-Wasserstein distance between a joint distribution and the product of its prescribed marginals. For two uniform marginals, Catalano and Lavenant conjectured that the monotone and antimonotone couplings maximize the distance from the independent coupling. We prove the Gaussian analogue for an arbitrary number $n\geq 2$ of one-dimensional standard Gaussian marginals. More generally, for every probability measure $\mu$ on $\mathbb R$ with finite second moment, we characterize the laws on $\mathbb{R}^n$ with all marginals equal to $\mu$ that are farthest from the $n$-dimensional standard Gaussian.

We consider a broad class of exchange dynamics with arbitrary weighted graph or hypergraph update structures, which includes the Kipnis–Marchioro–Presutti (KMP) model and the energies of Kac's walk on the sphere. These are conservative continuous-spin systems whose reversible measures are Dirichlet distributions. We prove that their spectral gap is always attained by a polynomial of degree at most two in the energy variables. Equivalently, through an intertwining with the associated discrete particle systems, the dominant mode is always represented by either a one-particle or a two-particle observable. We interpret this as a manifestation of an Aldous-type spectral gap phenomenon. In particular, the resulting two-particle spectral gap identity settles a recent conjecture of Alon and Puder. We also characterize sharply when one particle suffices and when a genuinely two-particle mode dominates. Under a common rescaling of the Dirichlet parameters, segment-like geometries are precisely those for which the gap is always of one-particle type, mean-field geometries are precisely those for which it is always of two-particle type, and all other geometries exhibit a nontrivial transition. The proof relies on the analysis of the so-called hidden model, a dual representation of the original process. The mechanism underlying the degree-two reduction is remarkably robust and extends to a much broader class of stochastic exchange models and their associated particle systems, including the harmonic process, the immediate exchange model, averaging-type processes, and nonreversible variants. Finally, we analyze boundary-driven versions of the KMP model, in which the bulk exchange dynamics interacts with reservoirs, generally resulting in nonreversible processes. In contrast to the conservative setting, we prove that the spectral gap of a boundary-driven model is always of one-particle type.

We study nonlinear Schrödinger equations with nonlinear Stratonovich noise \begin{equation*} \mathrm{d} u\,=\, i\bigl[ \Delta u \,+\, \lambda|u|^{p-1}u\bigr] \, \mathrm{d} t \,+\,i|u|^{(q-1)/2}u\circ \mathrm{d} {W}, \end{equation*} in their energy space $H^1(\mathbb R^d;\mathbb C)$. By combining the stochastic Strichartz estimates derived in [Potential Anal. 41 (2014), pp.\ 269–315] with the approach from [Ann.\ Inst.\ H.\ Poincaré Phys.\ Théor.\ 46 (1987), pp.\ 113–129] we obtain local well-posedness for all energy-subcritical nonlinearities $p,q\in [1, 1+4/(d-2)_+)$ together with a corresponding blow-up alternative. For a linear multiplicative noise $q=1$, a real-valued noise $W$ and a defocusing nonlinearity $\lambda\le 0$, we check this blow-up condition using a bound on the energy, resulting in the global well-posedness of the equation. If both nonlinearities are mass-subcritical, i.e., $p,q\in [1, 1+4/d)$, we provide an improved blow-up criterion involving the $L^2(\mathbb R^d;\mathbb C)$-norm. Using the conservation of mass for real-valued $W$, we obtain global well-posedness also in this case. Compared to previous results on stochastic nonlinear Schrödinger equations, we thereby improve the range of exponents $p$ and $q$ and the spatial regularity assumption on the noise.

We characterize equality in the finite free Stam and entropy-power inequalities, proving that Hermite polynomials are the unique extremizers among simple real-rooted inputs, up to independent translations and scalings. The proof turns this classification into a rigidity problem for projective plane curves. Using hyperbolicity and the Helton-Vinnikov theorem, we express the Jacobian defect as an off-diagonal squared norm in a definite symmetric pencil. Together with score transport, this yields a matrix proof of Stam for all real-rooted inputs. For each simple base pair, the directions in which the defect vanishes are the independent translations and common dilation, forming a three-dimensional subspace in every degree whose curves split into $n$ projective lines. At a collision, the leading configurations are again finite free convolutions of normalized derivatives of the velocity polynomials of the colliding input clusters. Combined with incidence counting, this local formula shows that at most one real fiber is singular, every real singularity is an ordinary totally real multiple point, and the ordered collision multiplicities determine the real normalization covering. For $n\geq3$, every non-split curve has at least $2n-2$ non-real projective discriminant zeros, counted with multiplicity, with equality attained by irreducible curves of geometric genus zero through every simple input pair. The leading Fisher-information coefficient is determined by the colliding tangent configurations, while the finite entropy term retains the gaps between clusters. For $n\geq3$, maximal logarithmic entropy divergence along the optimally weighted score direction is equivalent to Stam equality.

We study the large time behavior of a periodic ASEP with fixed period started from the step initial condition. We establish the limit shape theorem and the asymptotic distribution of a single second class particle. Our single-species results extend the previous results of Lam and Ayyer-Linusson about periodic TASEP to the ASEP case. These results can also be interpreted as describing the limit behavior of a random walk on affine Hecke algebra.

We prove a canonical (or fixed-population) local equilibrium theorem for the symmetric simple exclusion process on the discrete torus $\mathbb T_N^D$, $D\ge2$, at particle densities bounded away from $0$ and $1$, uniformly over all deterministic initial configurations with the prescribed particle number. At times \[ t_N(s)=\frac{\log(N^D)+s}{2\gamma_N}, \qquad \gamma_N=2-2\cos\left(\frac{2\pi}{N}\right), \] the Radon–Nikodym density of the process relative to equilibrium converges in $L^2$ to a canonical exponential tilt generated by the unique small mean-zero calibration field whose one-site marginals match the evolving one-particle heat profile. Consequently, whenever the profile coordinate converges, the corresponding total variation profile is a Gaussian shift. More precisely, for a deterministic initial sequence $(S_N)$, if the covariance-normalized squared amplitude $\mathfrak q_N^{S_N}(s)$ converges to a scalar $\mathfrak q$ at a fixed $s$, then the distance to stationarity converges to $ 2\Phi\!\left({\sqrt{\mathfrak q}}/2\right)-1 $, where $\Phi$ is the standard normal distribution function. The profile coordinate is asymptotically determined by the one-particle eigenspace corresponding to the smallest nonzero eigenvalue. The proof combines a calibrated canonical comparison, a fixed-degree comparison between independent and exclusion dynamics, and all-degree control obtained from pair energy estimates and preservation of the Strong–Rayleigh property.

A Hawkes process is a simple point process whose intensity depends on its history; the resulting dynamics are generally non-Markovian. We establish a sample-path moderate deviation principle for a nonlinear Hawkes process in the full moderate regime. Since a Poisson cluster representation is unavailable for nonlinear Hawkes processes, we use the past configuration as a Markov state and construct a potential, or Poisson corrector, for the centered stochastic intensity. A monotone Poisson coupling shows that the add-one increment of the corrector is uniformly bounded. The centered counting process is consequently the sum of a martingale with bounded jumps and an exponentially negligible boundary term. Exponential stabilization of the predictable quadratic variation follows from the process-level large deviation principle for nonlinear Hawkes processes. The martingale moderate deviation theorem then yields the result for every scale between the central-limit and large-deviation scales. The same construction gives a response formula for the asymptotic variance and, in particular, verifies that the variance dominates the stationary mean intensity in the self-exciting case.

We study high-dimensional conditioned dynamics obtained from an arbitrary number of independent copies of a mixing Markov chain. The dynamics is conditioned to avoid a family of holes whose stationary measure vanishes as the dimension grows, and we ask whether the resulting process admits a quasi-stationary measure close to the stationary product measure. Our problem is motivated by conditioning on holes with complicated geometry in high dimension, such as sets arising naturally from large deviations of suitable observables. Our main tool is the ANOVA decomposition, which separates functions according to their dependence on different subsets of coordinates. This allows us to exploit the product structure of the dynamics and obtain contraction estimates that are uniform in the dimension. Combined with a Keller Liverani perturbation argument, these estimates yield the existence of quasi-stationary densities converging to the stationary density as the size of the hole vanishes. Under stronger one-step mixing and regularization assumptions, we obtain sharper convergence in a Sobolev norm, with a square-root dependence on the measure of the hole. The improvement relies on the increasingly strong contraction of higher-order ANOVA components, thereby overcoming the usual loss of control with dimension. Finally, we apply our results to additive-noise Markov chains on the circle with smooth, uniformly positive transition densities

We study additive observables of a non-confined Feller process characterized by a non- normalizable stationary probability density and return times with infinite mean. The process is equivalent, after a deterministic rescaling, to a squared Bessel process, but we focus here on a question: the spatial structure of its local-time field. We show that the local time admits a fac- torization in the long time limit. Its spatial profile is governed by the non-normalizable stationary density, whereas its temporal fluctuations are controlled by a single Mittag-Leffler random ampli- tude. As a consequence, normalized spatial correlations of the local time converge to a universal constant independent of the two observation levels. Occupation times of finite intervals follow as corollaries and display Darling-Kac fluctuations. In contrast, non-integrable power observables exhibit self-similar squared-Bessel-type limits rather than Mittag-Leffler statistics. This spatial- field perspective provides a unifying framework for infinite ergodic theory under state-dependent multiplicative noise.

In planar soft-stick percolation, each vertex of the square lattice independently opens one uniformly chosen outgoing arrow and, with probability $\epsilon$, the opposite arrow as well. Motivated by the question on its critical parameter raised by Bäumler et al., we prove that $\epsilon_{c}\le0.99<1$, establishing percolation below the two-arrow endpoint. More precisely, at $\epsilon=0.99$ the origin has an infinite forward cluster with probability greater than 11/20. The proof uses a Peierls argument in which boundary turns are encoded by a three-state transfer matrix to bound the contour sum.

For some discrete parameters, multivariate (Dunkl-)Bessel processes on Weyl chambers $C$ are projections of Brownian motions on Euclidean spaces $V$; the most prominent examples are Dyson Brownian motions. More generally, the projections of Brownian motions on $V$ with drift are Bessel processes with drifts on $C$, where again the associated transition densities can be described in terms of multivariate Bessel functions. Moreover, Dunkl processes with drift can be defined in an analogous way. In this paper we prove central limit theorems for these Bessel and Dunkl processes with drift for regular drift vectors, and arbitrary root systems and multiplicities. These results improve recent results of the author.

We study the geodesic convex hull of a stationary Poisson point process restricted to a horoball in $d$-dimensional hyperbolic space. The resulting random set is an unbounded hyperbolic polyhedron with a distinguished ideal direction. Projecting its boundary facets to the bounding horosphere yields a stationary Euclidean tessellation of $\mathbb{R}^{d-1}$, which we identify as a dual Poisson–Laguerre tessellation with an explicit height density. We derive an exact formula for its cell intensity and, in the critical regime where the intensity of the Poisson point process is matched with the height of the horoball, we prove local convergence of the projected tessellation to the classical Poisson–Delaunay tessellation in $\mathbb{R}^{d-1}$. As consequences, the typical cell converges in distribution and the intensities of all $k$-dimensional faces converge to their Poisson–Delaunay counterparts. We also study a localised volume functional of the hyperbolic convex hull, determine its limiting expectation in the same regime, and use an Efron-type identity to obtain a second-order asymptotic expansion for the vertex intensity. In addition, we derive an exact formula for the expected localised surface area and determine its critical asymptotics. In dimensions $d\ge3$ the expected unnormalised local surface area converges to a finite limit, whereas in dimension $d=2$ it exhibits logarithmic growth.

Numerically solving McKean-Vlasov stochastic differential equations is computationally challenging due to the compounding costs of discretizing in time and in the distribution of the solution. Multilevel ideas have been proposed to provide speed-ups. In this work, we study the multilevel Monte Carlo method proposed by Ricketson (2015) for equations whose drift and diffusion terms depend on the law of the solution $X_t$ through the expectation $E[R(X_t)]$. The scheme follows the single-ensemble paradigm, where particles interact across levels at each timestep. While cross-level feedback makes this scheme attractive in practice, the correlations it introduces have so far confined its cost-error analysis to a model problem with linear drift, deterministic diffusion, and $R$ the identity. We use additional coarse particles to enforce geometrically decaying coupling errors towards the coarser levels. This allows us to prove our main contribution, an $L^p$-error of $O(\epsilon)$ at cost $O(\epsilon^{-2-\delta})$ for any $p\ge2$ and $\delta>0$ (with a constant that grows as $\delta\to0$), assuming only global Lipschitz bounds on the drift, the diffusion, and $R$. An exploratory experiment is consistent with the derived rates and suggests that in practice the constant does not grow significantly for small $\delta$. Our methodology and proof strategy may also be useful for other single-ensemble multilevel schemes, such as multilevel ensemble Kalman filters.

The contact process on the regular tree $\mathbb{T}_d$ when $d\geq 3$ has the two phase transitions of global and of local survival, found by Pemantle and Liggett at values $\lambda_1$ and $\lambda_2$. We start the system with every vertex infected and let it relax to what is known as the upper invariant infection. In this stationary state, $\lambda_p$ is the critical value beyond which the infected vertices can percolate through $\mathbb{T}_d$, and $\lambda_{p^\complement}$ is the parameter before which the healthy vertices can percolate. We find these are both distinct phase transitions \[0<\lambda_1<\lambda_p<\lambda_2<\lambda_{p^\complement}<+\infty\] on $\mathbb{T}_d$ when $d\geq 7$. The most interesting of these comparisons is $\lambda_1<\lambda_p$, which we find for all $d\geq 3$. This comparison $\lambda_1<\lambda_p$ is a long-standing open question on $\mathbb{Z}^d$ with $d\geq 2$ and was not yet found on any other graphs except where $\lambda_p$ is infinite.

We study fluctuation limit of a slow-fast system driven by $\alpha$-stable Lévy noise with $1<\alpha<2.$ The slow component is generated by an odd polynomial function $f(y):=y^q,$ while in the fast component, the drift is $g(y):=-|y|^p\operatorname{sgn}(y)$ for some $p>0.$ Although the noise is given, the fluctuation limit is either a stable process or a Brownian motion, depending on both $p$ and $\alpha.$ The critical line between stable limit and Brownian limit is $q+1-p=\alpha/2.$

We study comparison inequalities for contact-process-type infection dynamics on $\mathbb Z^d$ with general local transmission mechanisms. The processes considered include both discrete-time oriented-percolation models and continuous-time contact processes in which infection events may transmit to random, possibly correlated sets of neighbouring sites. Our main results apply local-to-global comparison criteria beyond stochastic domination: if one local infection rule is more likely than another to hit every non-empty test set of neighbours, then the corresponding process has larger survival probabilities at all times. We compare classical independent-infection models with exchangeable fixed-budget, all-or-nothing, and burst-type infection mechanisms, deriving orderings for example of finite-time and infinite-time survival probabilities and stopping-time distributions. In continuous time, we further obtain consequences for critical survival thresholds of exchangeable infection laws. The results show that infection processes with identical or comparable mean-field infection intensity can nevertheless be rigorously ordered at the stochastic level once spatial geometry is taken into account. Additionally, it turns out that the classical continuous-time contact process dominates a variety of related continuous-time models, whereas its most natural discrete-time version, Bernoulli oriented percolation, is not dominant with respect to related discrete-time models.

The Portmanteau Theorem gives several equivalent characterizations of weak convergence of probability measures (equivalently, convergence in distribution of random variables). This note presents a unified proof of the theorem in a seven-statement formulation, emphasizing the structure of the implication cycle connecting expectations of bounded continuous functions with probability bounds on open and closed sets. The argument highlights standard approximation techniques and clarifies the role of intermediate function classes such as bounded Lipschitz functions.

We study the quenched pressure of the nearest-neighbor Edwards–Anderson Ising model with free and periodic boundary conditions. First, we prove that the infinite-volume pressure is $\beta d$-Lipschitz in the coupling law for the $1$-Wasserstein distance, yielding quantitative thermodynamic limits for spatially inhomogeneous disorder. Second, for periodic volumes, we prove that almost-sure convergence under every joint coupling of the finite-volume disorder arrays is equivalent to complete convergence of the one-volume pressure laws. Third, we prove that $\mathbb{E}\vert{}J\vert{}^{1+1/d}<\infty$ guarantees this universal-coupling conclusion. We further show that this exponent is optimal among uniform power-moment assumptions: for every $1\leq q<1+1/d$, there is a centered symmetric law with finite $q$-th moment for which canonical nested volumes converge almost surely, whereas independently resampled volumes with the same fixed-volume marginals converge in probability but not almost surely. Finally, in dimension one, we prove that the first-moment condition is also necessary for a finite limiting pressure.

We introduce a left $G$-circulant decomposition for matrices indexed by an arbitrary finite group $G$, extending the diagonal decomposition associated with cyclic groups. We show that, when $A\in M_{|G|}(\mathcal A)$ is free from $M_{|G|}(\mathbb C)$, the components arising from the left $G$-circulant decomposition of $A^t$ form a free family, with the components associated with inverse pairs forming $R$-diagonal pairs. We also describe the distributions of these components in terms of the distribution of $A$. Our results recover the cyclic case and show that different group structures of the same order may lead to different free decompositions of the same matrix.

Let $D=2^n$ and equip $\operatorname{PU}(D)$ with the one-step-cliff Nielsen metric, with quadratic metric coefficients one in Pauli directions of weights one and two and $D^2$ in all higher weights. We prove that the distance from the identity of a Haar-random element, normalized by $D$, converges to $\pi/\sqrt3$ in probability and in $L^p$ for every $1\le p<\infty$. Quantitatively, it lies within $O(D^{-1/8}(\log D)^{1/2})$ of this limit outside a set of Haar measure at most $\exp\{-\Omega(D^{7/4}\log D)\}$. For each fixed $0<x<\pi/\sqrt3$, the ball of radius $xD$ has Haar measure $\exp\{-\Theta_x(D^2)\}$; for $x>\pi/\sqrt3$, its complement has measure at most $e^{-c_xD^2}$ for some $c_x>0$. As $x\uparrow\pi/\sqrt3$, the lower and upper logarithmic rates are both asymptotic to $(\pi^2/3-x^2)^2/(16\zeta(3))$. The small-ball upper bound follows from a comparison of Jacobi determinants, obtained by rescaling the linearized geodesic equations and applying Kato transport. Weyl integration reduces the remaining integral to an Abel-regularized logarithmic-energy estimate on the circle. A centered principal logarithm and concentration of the circular unitary ensemble eigenangle second moment give the distance upper bound.

Neural networks can leverage feature superposition to encode more concepts than dimensions, but cross-feature interference constrains the linear accessibility of simultaneously active features. By framing linear accessibility as a compressed sensing problem, we derive high-probability bounds for fixed supports under subgaussian noise, proving the sufficient dimension scales linearly ($d=O_{\varepsilon}(k \log m)$) rather than prior worst-case quadratic limits. We characterize the asymmetry between active and inactive interference and the trade-off between interference and observation-noise budgets. We then validate these bounds across system parameters through Gaussian-tail approximations. We also introduce IHT-SAE, which uses learned iterative refinement to improve feature recovery beyond the limits of linear availability. These results quantify the geometric constraints of the linear representation hypothesis, providing a framework for evaluating sparse autoencoders, compositional generalization, and neural interpretability.

We study online vector balancing with $N$ random vectors in $\mathbb{R}^M$ revealed sequentially, where each vector must be assigned an irrevocable sign upon arrival. The goal is to minimize the expected $\ell^\infty$ norm of the final signed sum. For i.i.d. entries with mean zero, variance one, and a finite fourth moment, we prove that, as $M/N\to\alpha\in(0,\infty)$, the optimal value divided by $\sqrt N$ converges to a limit $R_\alpha$ independent of the entry distribution. This limit is the stochastic control value identified for Gaussian inputs by Fiedler, Jackson, Lacker, and Niles-Weed. In particular, it determines the exact asymptotic optimum for Rademacher inputs. For every $\kappa>R_\alpha$, we construct a randomized online algorithm whose final signed sum has $\ell^\infty$ norm at most $\kappa\sqrt N$ with high probability; for $\kappa<R_\alpha$, every online algorithm has vanishing success probability. Consequently, the online threshold of the symmetric binary perceptron is universal at every positive margin. The main step is a coupling that transfers Brownian controls to non-Gaussian inputs, while truncation controls rare large entries.

Let $(X _n )_n$ be a sequence of i.i.d. $R^d$-valued random variables uniformly distributed on a compact subset $M$ of $R^d$ . In this work, we study the asymptotic behavior of maximal packings of this compact subset $M$. Under some assumptions on $M$ (in particular, that it is a smooth submanifold of $R^d$ without boundary), we establish a precise extreme value law (of Gumbel type) for the minimal distances from the sample $(X_1 , \cdots, X_ n)$ to the centers of maximal packings. This result leads to explicit asymptotic confidence bounds for this maximal packing and thus to the support $M$. A distinctive feature of our contribution is the explicit derivation of the scaling sequences in the Gumbel convergence, depending only on the geometry of the support M. These formulas provide interpretable confidence bounds for $M$, which represent a novel complement to previous approaches such as those by Fasy et al. (2014). Our approach bridges geometric and combinatorial probability arguments and relies on analytic tools such as the Lambert $W$ function. In addition, we provide examples of main sample spaces in directional and circular statistics (circle, sphere, torus), along with simulations that illustrate and support the theoretical findings.

In this article, we study the second-order fluctuation of the solutions of one-dimensional polynomial stochastic partial differential equations (SPDEs) of the form \begin{equation*} (\partial_t - \Delta) \Phi_{\varepsilon} = -P(\Phi_{\varepsilon}) + \xi_{\varepsilon}, \end{equation*} where $P$ is a polynomial of degree greater than or equal to $2$, $\xi_\varepsilon$ is the white-noise after being convoluted (in space) by the heat kernel $K_\varepsilon = e^{\varepsilon \Delta}$. More precisely, taking advantage of the local solutions of pointwise well-posedness of limits $\Phi= \lim_{\varepsilon \to 0}\Phi_{\varepsilon}$, we characterise the limit of $\Phi^{err}=\lim_{\varepsilon \to 0} \varepsilon^{-1}(\Phi_{\varepsilon}-\Phi)$ as the solution of a more irregular stochastic partial differential equation. This is performed by applying the Da Prato–Debussche decomposition in the non-linear equation and characterising the limit of each of the terms. We also discuss possible characterisations of second-order fluctuations performed for higher-dimensional SPDEs in the weakly-coupled regime.