Papers

We study the rational canonical form of sparse random matrices over a finite field. Suppose $A_n\in \operatorname{Mat}_n(\mathbb{F}_p)$ has independent and $\alpha_n$-balanced entries. We prove that if $$ \liminf_{n\to\infty}\frac{n\alpha_n}{\log n}>1, $$ then, for every fixed collection of distinct monic irreducible polynomials over $\mathbb{F}_p$, the corresponding primary partitions of $A_n$ converge jointly to the same asymptotically independent Cohen-Lenstra distributions as in the uniform model studied by Fulman in his thesis. The sharp sparsity threshold for the full rational canonical form coincides with the threshold previously obtained by Lee for finite-field cokernels and by Jung-Lee-Yu for random matrix models over $\mathbb{Z}_p$. Our proof is based on the surjection moment method over the function field $\mathbb{F}_p[t]$, applied to the finite module $\operatorname{Cok}_{\mathbb{F}_p[t]}(tI_n-A_n)$, whose primary decomposition records the rational canonical form. We also construct degree-$d$ critical sparse obstructions, suggesting a polynomial-dependent threshold $1/d$ for statistics associated with irreducible polynomials of minimal degree $d$.

We introduce and study a new geometric representation for general classical spin models. It consists of two coupled percolation configurations that are the joint FK (random cluster) representation of the Ginibre rotation of two independent copies of the spin model, and can be viewed as an extension of Sheffield's cluster swapping defined in the context of height functions. Our approach combines the advantages of the random current and FK representations of the Ising model: it provides a percolation interpretation for various truncated correlation functions and at the same time satisfies the FKG inequality (for a large subclass of models). We highlight its strength and versatility by establishing two very different results. We first prove the converse of the classical fact that boundary multi-point correlation functions of planar Ising models are given by Pfaffians of the two-point functions. Indeed, we show that if a general spin model on a general graph satisfies the Pfaffian relations, then up to natural local modifications it must actually be an Ising model on a planar graph. In particular, the algebraic Pfaffian relations imply the topological feature of planarity. Our second application is a proof of sharpness of the phase transition for a class of spin models first considered by Ellis, Monroe and Newman, which we do by generalising the celebrated argument of Duminil-Copin and Tassion, replacing the use of the random current with our new representation.

We consider non-Hermitian random band matrices with growing bandwidth and study convergence of their empirical spectral distributions to the circular law. Let $N$ denote the matrix size and $W_N$ the bandwidth. From a universality perspective, it is conjectured that the circular law holds whenever $W_N\to\infty$. Previous results have mainly required $W_N\gg N^{1/2}$, with the principal exception of \cite{Han2511}, which proves the $W_N\to\infty$ circular law for an open-boundary block-tridiagonal model. Here we prove the circular law at this optimal threshold for several periodic models and under the near-optimal condition $W_N\gg\log N$ for a genuinely discrete model. For the periodic hard-indicator profile and its uniform and polynomially tapered generalizations, we prove the circular law under bounded-density and finite-third-moment assumptions whenever $W_N\to\infty$. At the same threshold, we prove the circular law for continuous, integrable profiles locally bounded below on every finite interval, including exponentially and Gaussian decaying profiles, with circular complex Gaussian entries. For the periodic full-block model, we prove the circular law under bounded-density and finite-third-moment assumptions whenever $W_N\to\infty$. The finite-third-moment assumption in the bounded-density results can be weakened to a finite $(2+\alpha)$-moment assumption. For the same full-block model without a density assumption, we prove the circular law for centered variance-one real subgaussian atoms when $W_N\gg\log N$. The proof uses compositions of random transfer operators. An established high-band circular-law input on a short auxiliary ring calibrates the full exterior coefficient norm, and a boundary-uniform local comparison lifts this calibration to target rings even when $N/W_N$ is arbitrarily large.

In the article, a novel framework for rough differential equations on finite-dimensional smooth manifolds driven by branched rough paths is developed utilizing the theory of natural bundles. The role of vector fields is played by sections of certain associated fiber bundles. The solutions are defined in a generalized Davie sense via a local approximation in such a way that they are invariant under changes of coordinates. Existence and uniqueness of the solutions is proved and a necessary and sufficient condition for the invariance of a submanifold for the solution is given. The approach allows the treatment of rough differential equations driven by fully branched rough paths on manifolds without directly relying on a shuffle product formula, bracket extension, or the Connes-Kreimer Hopf algebra and without imposing additional structure on the manifold.

The present research is devoted to the nonparametric estimation of a density-dependent drift coefficient in a multivariate McKean–Vlasov diffusion from independent observations at a common time, as well as the stationary density. Under certain assumptions on the (known) potential, we reduce the problem to the one-dimensional one and construct a sieve maximum-likelihood estimator based on sparse ReQU neural networks subject to structural and Hölder constraints. Using the endpoint-adapted graded approximation, we achieve the rate of $\left(b_n\log n/n\right)^{2(\beta+1)/(2\beta+3)}$ for the Kullback-Leibler divergence between the true and estimated stationary densities, with $b_n$ being at most a logarithmic factor. Similarly, it is shown that the constructed estimator for the drift coefficient converges to the true one at the rate of $\left(b_n\log n/n\right)^{\beta/(2\beta+3)}$ in the $L^2$-metric. A matching Assouad lower bound proves minimax optimality of this bound up to logarithmic factors.

We consider the voter model on the giant component of a hyperbolic random graph, which is a spatial scale-free network, in the sparse and linear-giant regime $\alpha\in(1/2,1)$. We find that the quenched expected consensus time has order $n^{2-1/\alpha}$, as the number of vertices $n\to\infty$, with probability arbitrarily close to one. This is generalised to the voter model where each vertex changes its opinion at rates $q(v)={\rm d}(v)^\varphi$, where we also establish the consensus time orders for all $\varphi\geq 0$. These orders have 3 regimes, with a phase transition at $\varphi=2-2\alpha$. For the upper bounds, our main proof idea is to connect the meeting set to some fixed target vertex of appropriate height in the product chain electrical network, to make rigorous an argument due to Durrett.

We prove an obstruction to quasi-invariance of Gaussian fields under divergence free transport flows on Riemannian manifolds. For the centered Gaussian field with covariance $(1-\Delta_g)^{-\alpha}$, $\alpha>\frac{d}{2}$, quasi-invariance under the transport flow is equivalent to invariance. This happens precisely when the flow acts by isometries, or equivalently when the underlying vector field is Killing. The proof leverages the Feldman–Hájek theorem and utilizes localized pseudo-differential computations. On $\mathbb R^d$ this leaves only rigid motions, while on flat tori this leaves only translations.

Characterizations of pointwise Gaussian upper bounds on graphs with possibly unbounded geometry in terms of localized functional inequalities contain errors depending on the vertex degree. We introduce a new space-time averaged form of the heat semigroup in terms of time-averaged $\ell^{q/(q-1)}-\ell^q$-estimates and obtain Gaussian upper bounds from large-scale Faber-Krahn inequalities which avoid such errors. A main analytic ingredient is an integrated version of Davies' method which yields off-diagonal estimates for this averaged quantity, and which tends to pointwise Gaussian bounds as time tends to infinity. Conversely, on large scales volume doubling and Gaussian bounds on this averaged heat semigroup norms imply relative Faber-Krahn inequalities for subsets of prescribed relative measure. Our proof is based on a lower bound for Dirichlet eigenvalues in terms of these averages. The Faber-Krahn dimension is scale-dependent, but converges to the doubling dimension for increasing radii. This gives a characterization of asymptotic Gaussian heat kernel behavior in terms of Faber-Krahn inequalities.

Uniform-state discrete diffusion models update all tokens in parallel while keeping every position revisable. Even when the commonly used top-$p$ rule leaves only one candidate at a position, that choice affects only the current reverse step and can be revised at the next sampling step. We ask what changes when selected hypotheses instead become persistent context for later predictions. We therefore propose committed reveal sampling (CRS), a training-free sampler that stores selected argmax tokens and inserts them into subsequent model inputs. Our analysis gives a rationale for selecting later and for keeping selected tokens visible. Under the exact forward process, the Bayes error of selecting a clean token cannot increase as noise decreases, while in a simple latent-mode model, keeping the selected token visible helps later parallel predictions agree on the same sequence-level choice. Empirically, paired experiments on Duo-distilled then separate this persistent effect from single-step top-$p$ restriction and scalar temperature scaling. Under the same finalization rule, CRS without top-$p$ truncation reaches lower generative perplexity (GenPPL) than fixed $p=0.95$ and $p=0.9$ baselines across budgets of 8–64 function evaluations (NFE). At 64 NFE, the comparison at matched unigram entropy also gives lower GenPPL for CRS, yielding a more favorable GenPPL–entropy tradeoff. Base Duo shows the same direction in a descriptive comparison, while other diversity and continuation metrics can rank these operating points differently. These results identify support restriction and persistent context as distinct controls of that tradeoff.

Recursive LLM agents can broaden their search by spawning specialists. Some branches later request tools that send data or deploy code. When should a branch receive authority to act? We distinguish sandbox spawning, in which external controls prevent the specified harm, from capability activation, in which a selected branch crosses an irreversible-action boundary. Progressive Risk Vesting (PRV) holds a trajectory-level risk budget in escrow and debits it as branches are activated. We prove an anytime harm bound for adaptively generated trees. Branch outcomes may be dependent, but each local certificate needs to remain valid conditional on the full pre-activation history, including the information used to select the request. When activation gates, branch charges, and compute constraints are held fixed, delayed vesting preserves every policy available under irrevocable spawn charging. Marginal risk estimates can still fail after branch selection. In a stylized branching model, trajectory harm changes as the authority reproduction number $\mathcal{R}_A$ crosses one. As local risk $p$ approaches zero, trajectory harm is proportional to $p$ below criticality, proportional to $\sqrt{p}$ at criticality, and retains a positive floor above it. A finite-type occupancy model yields risk and compute shadow prices. For nested fanout modes with decreasing marginal value per unit risk, these prices produce a threshold rule. Branching calculations and a split-sample experiment illustrate the results. These synthetic studies do not estimate safety in deployed agents. The analysis suggests a design rule: search broadly in the sandbox and grant recursive authority sparingly, with an explicit risk charge.

We study the asymptotics as $n$ goes to infinity of $\mathbb E\left[\mathcal{N}(S_n,[0,2\pi])\right]$, the expected number of zeros in $[0, 2\pi]$ of a random periodic signal $S_n$ of the form \[ S_n(t)=\sum_{k=1}^{n}a_k f(kt), \] where $f$ is a non-analytic $2\pi-$periodic function and the coefficients $(a_k)$ are i.i.d. random variables, centered with unit variance. We show in particular that if $a_1$ admits a finite third moment and if the function $f$ is piecewise polynomials and of class $\mathcal C^{7}$, then we have the following universal asymptotics, independent of the particular law of the coefficients $(a_k)$ \[ \lim_{n \to +\infty}\frac{\Esp\left[\mathcal{N}(S_n,[0,2\pi])\right]}{n}= \frac{2}{\sqrt{3}}\sqrt{\frac{\|f'\|_{L^2([0,2\pi])}}{\|f\|_{L^2([0,2\pi])}}}. \] This result thus extends in expectation and at the scale of the whole period $[0,2\pi]$ the local universality property established {in [Angst-Poly, IMRN, 2019]}, in distribution and in shrinking intervals of size $1/n$. Moreover, it generalizes to a non-analytic context the global universality results obtained in the more classical frameworks of random trigonometric polynomials or random analytic functions. Our approach combines a new almost sure Central Limit Theorem à la Salem–Zygmund for the function $S_n$ when evaluated at a uniform random point in $[0, 2\pi]$, and as well as suitable uniform integrability and anti-concentration estimates.

We prove uniform-in-$N$ partition-function estimates at all positive temperatures for the centered modulated energy of logarithmic, Riesz, and Bessel–Riesz kernels on $\mathbb R^d$ in the locally square-integrable range $0\le s<d/2$. They yield finite-particle exponential integrability, quadratic behavior at small parameter, explicit kernel-dependent growth at large parameter, and uniform entropy control of the associated Gibbs measures. As applications, we obtain uniform Rényi-divergence and relative-entropy bounds for interacting Gibbs equilibria and entropic mean-field convergence near thermal equilibrium; the optimal $N^{-1}$ normalized relative-entropy rate for the three-dimensional Coulomb flow; and an $N^{-1/2}$ Gaussian approximation for fixed-time finite-dimensional fluctuations. Finally, the modulated energy converges to a random variable in the second Wiener chaos, and for every positive parameter the partition functions converge to its Laplace transform, represented by a Carleman–Fredholm determinant. This formula identifies the sharp small-parameter behavior and, when the reference density is bounded below on a ball, the sharp large-parameter growth.

Let $L^x$ be the total local time of one-dimensional super-Brownian motion with $(1+\beta)$-stable branching mechanism, $0<\beta<1$. We prove that $\{x\in\mathbb R:L^x>0\}$ is almost surely a bounded open interval $(\mathsf L,\mathsf R)$ and that \[ h_L(\mathsf L)=h_L(\mathsf R)=1+\frac{2}{\beta} \] almost surely, where $h_L$ denotes the pointwise Hölder exponent. Thus the pointwise $\gamma$-Hölder condition holds at both endpoints for every $\gamma<1+2/\beta$ and fails for every $\gamma>1+2/\beta$. The same conclusions hold under the canonical excursion measure.

Let $U$ be Haar-uniform on $\mathbb S^{p-1}$, let $a_1,\ldots,a_k$ be arbitrary nonzero vectors, and let $w_1,\ldots,w_k$ be simplex weights. Define \[ g(U)=\sum_{j=1}^k w_j\frac{a_j}{a_j^\top U}, \qquad N(U)=\frac{g(U)}{\|g(U)\|}. \] We prove the universal incidence law \[ \{U^\top N(U)\}^2\sim\operatorname{Beta}\!\left(\frac12,\frac{p-1}{2}\right), \] independently of the number, arrangement, rank, or overcompleteness of the directions and of the weights. Thus a deterministic, generally non-Haar function of $U$ has the same squared-cosine law as an independent Haar direction. One proof combines a Herglotz–Cauchy boundary principle, a Haar-random two-plane with one common phase, and an exact Beta–Cauchy tangent-projection equivalence. A second proof specializes the positive-semidefinite Pillai–Meng identity. The planar structure leads to converses: plane-conditional Cauchy laws recover positivity, while for signed measures an exact phase-cancellation deficit equals twice the hidden negative mass. This yields local-to-global rigidity under a phase-norming condition strictly weaker than injectivity and an unconditional exclusion of negative atoms. The law extends to probability measures under almost-sure reciprocal integrability. We characterize this condition by an exact Wiener–Dini belt series, prove finite Shannon entropy to be the sharp universal criterion for countable weights, and give an entropy–geometry extension for clustered measures. Every compact carrier of zero one-dimensional Hausdorff measure is admissible, whereas a nonzero rectifiable arc component forces divergence on a set of positive Haar measure. In orthogonal coordinates, the theorem also gives a weight-free scaled $F$ law for Pearson divergence from a fixed simplex vector to a $\operatorname{Dirichlet}(1/2,\ldots,1/2)$ vector.

We study the phase boundary of the finite-lifespan frog model. For rate-one continuous-time simple random walk on an infinite, connected, locally finite transitive graph of superlinear growth, we prove that the critical-density curve is continuous and strictly decreasing, with $-\log\lambda_c$ locally bi-Lipschitz. The inverse critical-lifespan curve has the corresponding regularity wherever it is finite. Thus the critical-curve conjecture of Angel, de la Riva, Hermon, and Shi reduces to finiteness of the critical lifespan at every positive density. We identify the exact short-lifespan limit $t\lambda_c(t)\to\Delta\log(1/(1-p_c(G)))$, where $\Delta$ is the degree and $p_c(G)$ is the bond-percolation threshold, and extend critical-curve regularity and sharpness to a class of long-range frog models. The main tool is a local-to-global principle for activation processes generated by independent finite rooted ranges: local one-hit comparison implies comparison of global reachability and survival. The comparison allows disconnected ranges and source-dependent laws, and therefore also applies to activation models beyond random walks on a fixed graph.

Bayesian synthesis, which generates synthetic data by sampling from the posterior predictive distribution, is a popular approach for privatizing sensitive personal data. However, how attribute disclosure risk is affected by feature dimensionality, the number of individuals in the original dataset, and the amount of released synthetic information remains poorly understood. We propose a mathematically tractable Bayesian synthesizer for multivariate normal data based on a conjugate normal-inverse-Wishart prior for the mean vector and covariance matrix. The conjugate structure yields closed-form posteriors and enables direct investigation of an adversary's ability to infer records under different data and release settings. We then demonstrate several intuitive properties of synthetic data generation through several simulations. Specifically, we show that disclosure risk decreases with the size of the original dataset, but increases with the dimensionality of the feature space and the amount of synthetic information released, whether through the release of larger synthetic datasets or multiple generator realizations. Finally, the proposed synthesizer was used to generate synthetic versions of a polygenic risk score dataset, with the synthetic data exhibiting distributional properties comparable to those of the original data.

For an endomorphism $\phi$ of a closed Riemannian manifold $M$, we study the pushforward action $\phi_\ast$ on the Wasserstein space $\mathcal{P}(M)$ at a measure $\mu_0$ preserved by $\phi$. We show that if $\phi$ is a $C^2$ covering map and $\mu_0$ has positive $C^1$ density, then $\phi_\ast$ is Gâteaux differentiable at $\mu_0$ along tangent directions, with derivative given by the transfer operator of $\phi$ acting on vector fields, followed by orthogonal projection onto the tangent space. The derivative is the adjoint of the Koopman operator restricted to the tangent space, and its fixed space consists of the directions in which $\mu_0$ can be deformed while preserving invariance to first order. For appropriate pairs of endomorphisms of $\mathbb{T}^d$, we compute the intersection of their fixed spaces, show that it contains an infinite family of linearly independent continuous vector fields, and construct, for every $n$, an embedded $n$-dimensional family of measures that are nearly invariant under both endomorphisms. First-order rigidity therefore fails in every dimension. In particular, rigidity phenomena such as higher-dimensional analogues of the Furstenberg conjecture, if true, are genuinely nonlinear.

Let $X_1,X_2,\ldots$ be i.i.d. finitely supported random variables in a torsion-free abelian group, and write $S_k=X_1+\cdots+X_k$, and $H(S_k)$ is the Shannon entropy $S_k$, for all $k \ge 1$. We prove that, for every fixed $n\geq1$, \[ H(S_{n+1})-H(S_n) \geq \frac12\log\frac{n+1}{n} -o_{H(X_1)\to\infty}(1), \] uniformly over the ambient group and the input law. This proves a conjecture of Tao [29] in 2010.

We study the spatial Maki–Thompson rumor model on infinite, connected graphs of bounded degree. Spreaders transmit the rumor to ignorant neighbors but become stiflers upon contacting non-ignorant neighbors. We prove extinction on Cayley graphs of linear growth, for every \(\lambda,\alpha>0\) and every initial configuration with finitely many non-ignorant vertices, and establish an explicit extinction criterion on arbitrary bounded-degree graphs for processes started from finitely many spreaders. On Cayley graphs of superlinear growth, we prove survival from a single spreader whenever the ratio of the stifling to the transmission rate lies below an explicit threshold depending only on the maximum degree. On Cayley graphs of polynomial growth of degree \(D\ge2\), we further show that the range has positive lower density with positive probability. Under a stronger condition, macroscopic annuli contain a surface-order number of simultaneously active spreaders for a total duration bounded uniformly away from zero. When \(\alpha>0\), every finite region eventually contains no spreaders, so global survival forces the rumor to move continually into new regions. Under either subcriticality or a sufficiently small stifling rate, we prove central limit theorems for the final stifler density and the total spreader occupation time, together with a functional central limit theorem for the empirical survival function. These results follow from a central limit theorem for stationary stabilizing functionals of i.i.d.fields on polynomial-growth Cayley graphs; the functionals may depend on the field outside the observation set.

Dual quasi-mixing and dual mixing of a measure preserving transformation are uniform individual ratio limit properties of the transfer operator which entail the quasi-mixing and mixing properties of Krickeberg (respectively). The real restrictions of normalised, parabolic, non-Möbius inner functions of the upper half plane preserve Lebesgue measure and are dual quasi mixing; such a restriction being dual mixing iff it is exact and singular dual quasi-mixing otherwise.

In 2023, Wyman and Xi asked “Can you hear your location on a manifold?” We pose a related question, asking if probabilistic data can be used to recover location within a manifold. More precisely, we ask if location can be recovered from the distribution of exit times of Brownian sample paths. We show that exit time moments determine location up to symmetry for a variety of two-dimensional domains. Beginning with (convex) domains whose boundaries are conic curves, we show that in elliptic, parabolic, and hyperbolic domains, the first two exit time moments are sufficient to detect location up to symmetry. We establish similar results in unbounded wedge domains and equilateral triangular domains. We show that the full sequence of exit time moments determines location in rectangular domains. We end with two location detection results on general planar domains. The key tool driving our work is a “Poisson hierarchy,” which establishes a connection between exit time moments of Brownian motion and solutions to a family of PDE problems.

We study first-passage percolation on the $\ell$-spread-out one-dimensional cycle of size $n$, where vertices are connected if their graph distance is at most $\ell$. We assign i.i.d.~non-negative random weights from a Weibull distribution $\omega_e \sim \mathrm{Exp}(1)^{1/\theta}$ to the edges for $\theta>0$ fixed. This paper investigates the transition in the asymptotic behavior of the passage time $T_n$ between two typical vertices and the hop-count of the optimal path as the connectivity parameter $\ell$ diverges with $n$. We identify two fundamentally distinct geometric regimes. In the mesoscopic regime ($1 \ll \ell \ll n$), the optimal path locally mimics a spatial branching random walk but remains globally constrained to a one-dimensional geometry. We establish a law of large numbers characterized by the front speed of a Crump–Mode–Jagers branching random walk, prove a central limit theorem with Gaussian fluctuations when $\ell\ll n^{1/4}$, and show that the expected hop-count grows proportionally with the spatial distance. In the macroscopic regime ($\ell \approx \lambda n$ for $\lambda \in (0,1/2)$), the graph becomes a highly connected mean-field network. We prove that the passage time collapses to a $\log n$ scale with constant order non-Gaussian fluctuations, explicitly determining the extreme-value limit driven by the collision of two independent non-spatial CMJ processes. We establish a law of large numbers for the hop-count. Finally, we rigorously trace the transition in the order of the mean of $T_n$ between these two regimes, demonstrating an order transition for the passage time across the critical connectivity threshold $\ell \asymp n/\log n$. Our results provide a comprehensive deterministic-range interpolation from spatial Gaussian fluctuations to mean-field extreme-value fluctuations.

In this paper, we initiate the study of colored exponential random graph models (ERGMs), a class of exponential-family models for networks with multiple types of edge relations. Using the framework of probability graphons, we first derive a variational representation for the limiting free energy, whose maximizers determine the asymptotic structure of typical samples from the model. Then we identify several general families of colored ERGMs exhibiting replica symmetry, where the variational problem has constant maximizers and the model asymptotically concentrates on product colorings with independent edges. For general colored ERGMs, we derive Euler-Lagrange fixed-point equations for the variational maximizers, which in turn yield a general high-temperature uniqueness criterion. In the complementary zero-temperature regime, we establish a two-level selection principle: the leading energy term determines the ground states, while the lower-order energy terms, combined with entropy, act as a tie-breaker to determine the asymptotic zero-temperature structure of the model. We illustrate this principle through the induced wedge and rainbow triangle ERGMs. Both models have natural interpretations in multitype networks, and their zero-temperature limits exhibit interesting structures that connect to well-known results in extremal combinatorics. We further establish finite-temperature symmetry breaking for both these models and complement the rigorous results with numerical experiments.

The motivation for this paper is the investigation of the trade-offs implicit in probabilistic models used in machine learning. Models are often used to make predictions in the form of conditional probabilities. However, a pair of conditional distributions p(x|y) and p(y|x) may not be compatible with any joint distribution p(x,y). Given two such conditionals, determining if there exists a compatible joint is known as the compatibility problem. For discrete random variables, when the conditionals are encoded as probability tables, the compatibility problem has a known solution, which is computationally tractable. In this paper, we formalise and study a succinct version of the problem, encoding conditional distributions as arithmetic circuits. This is applicable to practical applications of probabilistic modelling in high-dimensional settings, including neural network models. We show that, for succinct circuit representations of conditionals, the compatibility problem is intractable. In the case that all probabilities are non-zero, the problem is co-NP-complete. In the case that probabilities can be zero, we give examples to demonstrate that several notions of compatibility can be distinguished, and we prove that multiple versions of the problem are PSPACE-complete. Furthermore, we show that, assuming the polynomial hierarchy does not collapse, there exist compatible succinct conditionals whose joint cannot be expressed succinctly. Implications of these results for probabilistic modelling and machine learning are discussed.

We investigate the fake stationarity properties of solutions to forward Stochastic Volterra Integral Equations (SVIEs) with affine drift and long-memory (regular) kernels, both on finite horizons and in the long-run regime. By either deriving explicit closed-form specifications for the deterministic initial condition $\phi$ and the mean-reversion function $\mu$ appearing in the drift, or by introducing a deterministic stabilizing factor $\varsigma$ in the diffusion coefficient associated with the kernel while keeping $\mu$ fully flexible, we show that it is possible to induce a fake stationary regime, in the sense that all marginal distributions share the same mean and variance. Afterwards, using a refined asymptotic analysis, we further establish that, in both frameworks, the time-shifted solutions of these long-memory SVIEs converge weakly, in the functional sense, toward a family of $L^2$-stationary processes sharing the same covariance structure, for suitable classes of diffusion coefficients. These results are applied to a class of exponential-fractional Stochastic Volterra Integral Equations driven by an $\alpha$-gamma fractional integration kernel, in the particular regime \(\alpha \geq 1\), which regularizes diffusion paths and invoke textit{ long-term memory}, persistence or long range dependence.

We prove a correspondence between the Polchinski flow and the Connes-Kreimer approaches to perturbative renormalisation, in the sense that the first yields the same renormalisation as the latter. More precisely, we show that an ansatz based on decorated graphs (Feynman diagrams),with a combinatorial renormalisation procedure solves the Polchinski equation. This result holds for very general Euclidean quantum field theories. We are able to derive from this the form of the renormalised potential which is, to the best of our knowledge, the most general one in the literature. The main ingredients of the proof are multiple morphism properties with respect to the renormalisation, as well as a novel duality formula for forests of decorated graphs.

Let $r\geq3$ be fixed, and let $\mathbf{G}_n$ be the set of all simple graphs with vertex set $[n]=\{1,\ldots,n\}$. We consider an exponential random graph model which gives higher probability to $G \in \mathbf{G}_n$ than to $H \in \mathbf{G}_n$ if $G$ has fewer $r$-cliques than $H$. But all graphs in $\mathbf{G}_n$ have positive probability. The degree to which graphs with fewer $r$-cliques are given higher probability is determined by a positive weight $w$. We prove that, asymptotically almost surely as $n \to \infty$, a random graph from $\mathbf{G}_n$ has a vertex partition into $r-1$ parts of roughly equal size, the density of edges between the parts is close to $1/2$, and for every $\varepsilon > 0$ the density of edges within any part is less than $\varepsilon$. The asymptotic structural properties are independent of the weight $w$ as long as it is positive. We also extend the result to the context of several clique sizes, each one with its own weight.

Let $\{Y_n; n\ge 1\}$ be a sequence of independent and identically distributed random variables with mean zero in Peng's framework of the sub-linear expectation space $(\Omega,\mathscr{H},\widehat{\mathbb E})$, and $S_n=\sum_{i=1}^nY_i$. In this paper, we establish a limit law of \begin{align*}\lim_{n\to \infty}\max_{k\le n}\frac{S_k}{\sqrt{2k \log\log n}}. \end{align*} Different from the result obtained by Chen (2015) in which the limit is a constant, it is shown that under the upper capacity the limit may be prescribed as a given function of $Y_1,Y_2,\ldots$, taking values in the standard deviation interval. As a result, it is also shown that the set of limit points in the compact law of the iterated logarithm can be a symmetric random interval. This paper (Chinese version) has been submitted to Special Issue of Science in China-Mathematics in Celebration of Professor Peng Shige's 80th Birthday. In Theorem 2.2 of the original paper, an additional condition (2.6) is needed.

We study the two-dimensional directed polymer model in a Gaussian environment which is independent in time and spatially correlated, with covariances $h(x)$ either summable or with a critical decay, satisfying $h(x) \sim (\log |x|)^a/|x|^2$ as $|x|\to\infty$ for some $a>-1$. We determine the precise high-temperature asymptotics of the free energy, confirming a conjecture of Lacoin (Ann. Probab. 2011), later refined by Cosco, Cottini and Donadini (2025). We also establish a phase transition for the diffusively rescaled partition functions: below some critical point they converge to the Lebesgue measure, while above it they converge to zero. A key feature of our approach is that both results are obtained using only second-moment estimates and are based on a simplified change-of-measure argument in the supercritical regime, that may prove useful for other disordered models.

Let the center of a $d$-dimensional Euclidean ball be uniformly distributed modulo $\mathbb Z^d$. We study the resulting number of integer lattice points as the dimension and radius tend to infinity. The critical scale is $4\pi^2R_d^2/(d+2)=\log d$. In a logarithmic window around this scale, the logarithm of the lattice-point count divided by the volume satisfies a central limit theorem. At a fixed offset from the critical scale this yields a genuine lognormal limit. Above the critical scale, the count-to-volume ratio converges to $1$ in measure; its normalized error has a standard normal limit, and its variance satisfies an asymptotic formula.