arXiv:2606.30927v2 Announce Type: replace-cross Abstract: We introduce a discrete-energy Sobolev space $\mathcal{W}^{1,p}_{\mathscr V}(T)$ on Ahlfors regular snowtrees, a class of metric trees where every arc is a snowflake of the same type. Our main result shows that, for every partition $\mathscr V$ and every $1

<\infty$, this discrete space coincides quantitatively with the Korevaar--Schoen space on $T$. This fact and the independence of the space on the particular partition used to define $\mathcal{W}^{1,p}_{\mathscr V}(T)$ are both novel even for the class of geodesic trees. We also determine the critical Korevaar-Schoen exponent for Ahlfors regular snowtrees and prove capacity attainment and upper estimates, which reveal the appropriate walk dimension needed for the corresponding probabilistic profile on these trees.

arXiv:2604.22670v2 Announce Type: replace-cross Abstract: We study the inverse optimal transport problem of recovering the ground cost from an optimal transport plan. In discrete settings, this problem reduces to inverse linear programming and is intrinsically ill-posed, exhibiting non-identifiability and flat directions. We show that in the continuous setting, the regularity of the marginals fundamentally alters the structure of the inverse problem. Assuming smooth positive densities for the source and target measures, we characterize the second variation of the optimal transport functional with respect to the ground cost in H\"older spaces. In particular, we show that it is non-degenerate modulo the natural transport invariances, yielding a strict curvature property that is absent in discrete transport. As a consequence, we obtain local identifiability and stability results for inverse optimal transport. For the structured family of bilinear costs (i.e. Mahalanobis parametrizations), the ground cost can be uniquely recovered up to the intrinsic invariances from a single optimal coupling under a natural spanning condition. We further show that this identifiability property is generic under arbitrarily small perturbations of the marginals, while settings where the optimal transport map is affine (for instance Gaussian or elliptical marginals) remain degenerate. Finally, we establish precise bounds on the bias and statistical variance of inverse optimal transport under entropic regularization. These results reveal a structural parallel between forward and inverse optimal transport: regularity of the marginals ensures smooth optimal maps in the forward problem, while non-degeneracy of the induced transport plan yields curvature and local invertibility in the inverse problem.

arXiv:2604.20612v2 Announce Type: replace-cross Abstract: We develop e-values and e-processes testing the null hypothesis that a distribution over nonnegative integers is monotone, and that a distribution over integers is unimodal given a certain mode. Our e-processes lead to tests of power one under any non-null distribution with a sequence of i.i.d. observations, and consistent set-valued mode estimators that eventually equal the true set of modes. Additionally, we characterize the set of all e-values, and therefore the set of all valid tests, with one monotone and unimodal observation, as well as the most powerful e-value for a fixed alternative. We then show that many of our results can be generalized to continuous random variables, relating them to the existing results in the shape-constrained inference literature.

arXiv:2602.18396v2 Announce Type: replace-cross Abstract: We propose PRISM-FCP (Partial shaRing and robust calIbration with Statistical Margins for Federated Conformal Prediction), a communication-efficient Byzantine-robust federated conformal prediction framework that uses partial model sharing to mitigate stochastic model-poisoning attacks during training and histogram-based filtering to mitigate adversarial calibration submissions. Existing approaches address adversarial behavior only in the calibration stage, leaving the learned model susceptible to poisoned updates. In contrast, PRISM-FCP mitigates attacks end-to-end. During training, clients partially share updates by transmitting only $M$ of $D$ parameters per round. This attenuates the expected energy of an adversary's perturbation in the aggregated update by a factor of $M/D$, yielding lower mean-square error (MSE) and tighter prediction intervals. During calibration, clients convert nonconformity scores into characterization vectors, compute distance-based maliciousness scores, and downweight or filter suspected Byzantine contributions before estimating the conformal quantile. Extensive experiments on both synthetic data and the UCI Superconductivity dataset demonstrate that PRISM-FCP maintains near-nominal empirical coverage in the studied Byzantine settings while avoiding the interval inflation observed in standard FCP, with reduced communication. These results support PRISM-FCP as a robust and communication-efficient approach to federated uncertainty quantification.

arXiv:2511.01537v2 Announce Type: replace-cross Abstract: We study explosive connectivity and mechanical rigidity in three-dimensional cubic lattice structures under Achlioptas-type product-rule dynamics. Our work combines extensive numerical simulation with a theoretical framework based on rigorous finite-size scaling. Using massive-scale simulations up to $L=192$ ($N \approx 7 \times 10^6$) with 20,000 independent realizations, we demonstrate that for $k \ge 8$, the peak susceptibility scales with an exponent of $\gamma = 1.000$, and the maximum single-step jump stabilizes at a macroscopic fraction. This confirms that while the transition is continuous in the infinite thermodynamic limit, it exhibits the exact finite-size scaling signatures of a first-order discontinuity in finite physical systems. For rigidity, we discover numerically that for richly-connected hosts, increasing the number of choices $k$ optimally enhances the efficiency of rigidification. To explain this phenomenon, we propose a theoretical model centered on a conditional progress function that links an edge's local product-rule score to its global mechanical utility. We show that while local rigidification efficiency monotonically increases, the global rigidity gap exhibits a ``Goldilocks'' minimum at intermediate $k$ due to the emergence of maximally floppy, tree-like components at large $k$. Altogether, our work provides new insights into the relationship between local dynamics and global connectivity and rigidity in cubic lattice structures via both theory and computation.

arXiv:2411.02721v2 Announce Type: replace-cross Abstract: In this work, we study probability functions associated with Gaussian mixture models. Our primary focus is on extending the use of spherical radial decomposition for multivariate Gaussian random vectors to the context of Gaussian mixture models, which are not inherently spherical, but conditionally so. Specifically, the conditional probability distribution, given a random parameter of the random vector, follows a Gaussian distribution, which allows us to rewrite the probability function as a tractable integrated Gaussian mixture. This assumption, together with spherical radial decomposition for Gaussian random vectors, enables us to represent the probability function as an integral over the Euclidean sphere. Using this representation, we establish sufficient conditions to ensure the differentiability of the probability function and provide an integral representation of its gradient. Furthermore, we approximate the probability function using random sampling over the parameter space and the Euclidean sphere. Finally, we present a numerical example that illustrates the advantages of this approach over classical approximations based on random vector sampling.

arXiv:1406.7675v3 Announce Type: replace-cross Abstract: (Due to the limit on the number of characters for an abstract set by arXiv, the full abstract can not be displayed here. See the abstract in the paper.) We study dispersive equations with a time non-homogeneous modulation acting on the linear dispersion term. As primary models, we consider the Korteweg-de Vries equation (KdV) and related equations such as the Benjamin-Ono equation (BO) and the intermediate long wave equation (ILW), imposing certain irregularity conditions on the time non-homogeneous modulation. In this work, we establish phenomena called regularization by noise in three-folds: (i) When the modulation is sufficiently irregular, we show that the modulated KdV on both the circle and the real line is locally well-posed in the regime where the (unmodulated) KdV equation is known to be ill-posed. In particular, given any $s \in \mathbb R$, we show that the modulated KdV on the circle with a sufficiently irregular modulation is locally well-posed in $H^s(\mathbb T)$. Moreover, by adapting the $I$-method to the current modulated setting, we prove global well-posedness of the modulated KdV in negative Sobolev spaces. (ii) It is known that certain (semilinear) dispersive equations such as BO and ILW exhibit quasilinear nature. We show that sufficiently irregular modulations make the modulated versions of these equations semilinear by establishing their local well-posedness by a contraction argument, providing local Lipschitz continuity of the solution map. (iii) We also prove nonlinear smoothing for these modulated equations, where we show that a gain of regularity of the nonlinear part becomes (arbitrarily) larger for more irregular modulations. As applications of our approach, we also include further examples.

arXiv:2606.25204v2 Announce Type: replace Abstract: Fix $b\in(0,1)$, let $1\leq k\leq n$, and let $A=(A_{ij})$ be an $n\times n$ random matrix with independent real entries satisfying $$ \sup_{x\in\mathbb{R}}\mathbb{P}\{A_{ij}=x\}\leq b<1 \qquad (1\leq i,j\leq n). $$ We show that there exists $c>0$ such that $$ \mathbb{P}\{\operatorname{rank} A\leq n-k\}\leq \exp(-cnk), \qquad 1\leq k\leq n. $$

arXiv:2606.16018v2 Announce Type: replace Abstract: In this note, we prove a non-asymptotic version of a theorem by R\'acz and Richey, showing that a Wishart matrix is close in total variation to an affine transformation of a GOE matrix. The proof mirrors a proof in a paper by Bubeck, Ding, Eldan, and R\'acz, with some changes made to make it non-asymptotic.

arXiv:2601.17175v2 Announce Type: replace Abstract: This paper establishes a conservation identity for mean-zero martingales stopped by extended-valued stopping times. For any mean-zero martingale $\{M_n\}$ and any extended-valued stopping time $T$ satisfying $E|M_T|I(T<\infty)<\infty$, the quantity $L\equiv E[M_T I(T<\infty)]$ exists and equals $\lim_n E[-M_n I(T>n)]$, a limit which always exists. The optional sampling theorem for stopping times and uniformly integrable martingales -- and Wald's equation for mean-zero random variables, as its i.i.d.\ specialization -- is recovered with a little extra effort, in which case the limit also vanishes. The identity itself remains in force whether or not $L=0$, and whether or not $P(T<\infty)=1$. Two corollaries and an application derived from this identity provide information on the rate of decay of the tail probability of the stopping time. Moreover, a necessary and sufficient condition is presented to characterize when $E|M_T|I(T<\infty)$ is finite. The characterization applies more generally whenever $|M_n|$ is a sequence of random variables, each having finite expectation. A third theorem provides sufficient conditions ensuring that certain exceedance-level, potentially extended-valued, stopping times are finite with probability one. It further implies that $\limsup M_n=\infty$ almost surely. We demonstrate these results through examples and explore their implications for different families of martingales. Our findings extend classical results in martingale theory and provide new insights into the behavior of stopped martingales, especially when the expected value of the stopped martingale on the set where the extended-valued stopping time $T$ is finite differs from the expected value of the martingale at time 1.

arXiv:2504.07672v2 Announce Type: replace Abstract: We study a general non-homogeneous Skellam-type process with jumps of arbitrary fixed sizes. We express this process in terms of a linear combination of Poisson processes and study several properties, including the summation of independent processes of the same family, some possible decompositions (which present particularly interesting characteristics) and the limit behaviors. A compound Poisson representation and a discrete approximation are also presented. Then, we study the fractional integral of the process as well as the iterated integral of the running average. Finally, we consider some time-changed versions related to L\'{e}vy subordinators, connected to the Bernstein functions, and to the inverses of stable subordinators.

arXiv:2410.18554v2 Announce Type: replace Abstract: We study the asymptotic behaviour of stationary densities of one-dimensional random diffeomorphisms, at the boundaries of their support, which correspond to deterministic fixed points of extremal diffeomorphisms. In particular, we show how this stationary density at a boundary depends on the underlying noise distribution, as well as the linearisation of the extremal diffeomorphism at the boundary point (in case the corresponding fixed point is hyperbolic), or the leading nonlinear term of the extremal diffeomorphism (in case the corresponding fixed point is not hyperbolic).

arXiv:2408.16649v2 Announce Type: replace Abstract: Let $(A_v)_{v\in \mathcal{T}}$ be the balanced Gaussian Branching Random Walk on a $d$-ary tree $\mathcal{T}$ and let $M^A$ be the multiplicative chaos with parameter $\gamma \in (0, \sqrt{2\log d})$ constructed from $A$. In this work we establish the precise first order asymptotics of negative exponential moment of $M^A$, i.e.\ we prove that for $t_k = \lambda p^k$ with $\lambda>0$ and $p$ an explicit constant depending only on $\gamma$, we have as $k \to \infty$, \begin{equation} -\frac{1}{d^k} \log \mathbb{E}[e^{-\lambda p^k M^A } ] \to h(\lambda), \end{equation} where $h\colon (0,\infty)\to \mathbb{R}$ is a non-explicit positive continuous function. This result allows us to study the law of $A$ tilted by $e^{-t_k M^A}$ for particular values of $\lambda$, with $k\to \infty$. In this setting we prove that the normalized $L^1$ norm of $A$ in generation $k-a$ is bounded and converges to $0$ when first $k\to \infty$ and then $a\to 0$. As an application we prove that in this setting, under the tilt $e^{-t_k M^A}$ and with $k\to \infty$, the Branching Random Walk $A$ exhibits a weak decay of correlations, which is not present in the non-tilted model. Our methods also apply to the usual Branching Random Walk $(S_v)_{v\in \mathcal{T}}$ and with $M^A$ replaced by $\frac{1}{2}(M^+ + M^- )$, where $M^+$ and $M^-$ are the multiplicative chaoses with parameter $\gamma \in (0, \sqrt{2\log d})$ constructed from $S$ and $-S$. In that case we prove that, as $k\to \infty$, \begin{equation} -\frac{1}{d^k} \log \mathbb{E}[e^{- \frac{\lambda p^k}{2}( M^+ + M^-) }] \to \tilde h(\lambda), \end{equation} where $\tilde h\colon (0,\infty)\to \mathbb{R}$ is again a non-explicit positive continuous function.

arXiv:2403.06821v2 Announce Type: replace Abstract: We consider three kinds of discrete-time arrival processes: transient, intermediate and recurrent, characterized by a finite, possibly finite and infinite number of events, respectively. In this framework, we study renewal processes which are externally stopped at an independent stopping time which may be defective or non-defective. For defective stopping time, the resulting arrival process is of intermediate nature. For non-defective stopping time, the resulting arrival process is transient, i.e. stopped almost surely. For these processes we obtain finite time and asymptotic properties. Particular attention is devoted to the class of transient renewal processes, that is, renewal processes with defective interarrival times. Among these, we consider two examples: The "Defective Bernoulli Process" and the "Defective Sibuya Process". We validate some analytical results using Monte Carlo simulations. We apply these results to biased and unbiased random walks on the $d$-dimensional infinite lattice and as a special case on the two-dimensional triangular lattice. We study the spatial propagator of the walker and its large time asymptotics. In particular, we observe the emergence of a superdiffusive (ballistic) behavior in the case of biased walks. For geometrically distributed stopping times, the propagator converges to a stationary non-equilibrium steady state (NESS), which is universal in the sense that it is independent of the stopped process. In dimension one, for both light- and heavy-tailed step distributions, the NESS has an integral representation involving alpha-stable distributions.

arXiv:2305.04716v3 Announce Type: replace Abstract: Consider a finite inhomogeneous random graph evolving in continuous time, where each vertex is assigned a mass, and an edge between any pair of vertices appears at a rate proportional to the product of their masses. The process tracking the evolution of component sizes evolves according to the multiplicative coalescent dynamic and can be encoded using the simultaneous breadth-first walk introduced by Limic (2019). We extend this encoding to incorporate surplus edge data within each connected component. Two distinct graph-based representations of the multiplicative coalescent, each with its own advantages and limitations, are analyzed in detail. In particular, a canonical multigraph introduced by Bhamidi, Budhiraja and Wang (2014), which is naturally connected to the augmented multiplicative coalescent, emerges from our framework. We demonstrate that a transformation of the simultaneous breadth-first walk, supplemented with an additional and independent source of randomness, encodes the full dynamics of the augmented multiplicative coalescent.

arXiv:2302.09847v4 Announce Type: replace Abstract: This paper is divided into two parts. The first part is devoted to the study of a class of Approximate Message Passing (AMP) algorithms which are widely used in the fields of statistical physics, machine learning, or communication theory. The AMP algorithms studied in this part are those where the measurement matrix has independent elements, up to the symmetry constraint when this matrix is symmetric, with a variance profile that can be sparse. The AMP problem is solved by adapting the approach of Bayati, Lelarge, and Montanari (2015) to this matrix model. \\ The Lotka-Volterra (LV) model is the standard model for studying the dynamical behavior of large dimensional ecological food chains. The second part of this paper is focused on the study of the statistical distribution of the globally stable equilibrium vector of a LV system in the situation where the random symmetric interaction matrix among the living species is sparse, and in the regime of large dimensions. This equilibrium vector is the solution of a Linear Complementarity Problem, which distribution is shown to be characterized through the AMP approach developed in the first part. In the large dimensional regime, this distribution is close to a mixture of a large number of truncated Gaussians.

arXiv:2607.08757v1 Announce Type: cross Abstract: Score matching controls average error under the forward marginals, but a discretized reverse-time sampler evaluates the learned score along its own trajectory. We show that small forward-marginal error does not guarantee numerical stability. We construct a single smooth score field with arbitrarily small forward-marginal $L^2$ error. The learned reverse-time process is nonexplosive, has moments of every order, and can be arbitrarily close to the exact reverse-time process in path-space total variation. Yet its Euler--Maruyama discretizations converge in probability while every positive moment diverges. Thus weak convergence can hold even though every Wasserstein distance $W_p$, $p\ge1$, diverges. The same failure can occur within one fixed finite neural architecture. We construct a family of bounded, globally Lipschitz denoisers for which both the forward-marginal error and the path-space total variation distance tend to zero, while their Euler--Maruyama endpoints diverge in every $W_p$. For compactly supported data, we also give a simple positive result. Projecting the learned denoiser onto a known bounded closed convex set containing the support preserves pointwise accuracy, gives grid-uniform moment bounds, and yields Wasserstein convergence under mild local regularity. Experiments with a small fixed DiT-style network show large growth along rare numerical trajectories and its suppression by denoiser projection, while overall trajectory errors remain small.

arXiv:2607.08677v1 Announce Type: cross Abstract: We construct a measure that exhibits two aspects of a new type of universality and dramatically simplifies the integration of tensors $T_{a_1,a_2,\ldots,a_D} \in \mathbb{C}$ ($a_1,\ldots,a_D=1,\ldots,N$) at large $N$. In contrast to matrix integration, in which matrix traces canonically yield the integrand, tensors need additional information (equivalent to a $D$-coloured graph $B$) to contract their indices and form a tensor trace $B(T)$. We show that, whenever each $B_1,\ldots, B_n$ can be obtained by a recursive construction known as melonicity, then the leading order in $N$ of the integral of $ {B_1}(T) {B_2}(T) \cdots {B_n}(T) $ is independent of the -- often intricate -- combinatorics of the traces $B_i$, but also, to our surprise, independent of $D$ as far as $D\geq 3$. Instead, at large $N$, these integrals are some functions (indexed by $n$) of the number of vertices $2p_i$ of $B_i$ which we call melonic polynomials. Melonic traces cumulants with respect to any ('interacting') measure \[ \exp\Big\{-N^{D-1} \sum_{i=1}^m g_i {B_i}(T)\Big\} \mathrm{d}\mu_0(T) \quad (g_1,\ldots,g_m \in \mathbb{R}, \mathrm{d}\mu_0(T) =\text{the tensor Gaussian}) \] with each $B_i$ melonic, can be computed with our universal measure that replaces each $B_i$ by a canonical trace depending only on $p_i$. We prove that any two melonic tensor models are indistinguishable at large-$N$, independently of the number of tensor indices (first universality aspect), and of the fine-grainedness of their interactions (second universality), being a sufficient condition that the couplings (the parameters $g_i$ above) agree and their respective traces are monomials with the same degree in $T$.

arXiv:2607.08670v1 Announce Type: cross Abstract: We study stochastic barycentric estimators for proximal points and metric projections obtained by exponentially reweighting Gaussian perturbations. Our main result is an abstract comparison theorem for probability measures with densities proportional to an exponential weight, under a radial dominance condition relative to a prescribed profile. This yields an explicit upper bound for the norm of the associated barycenter in terms of a one-dimensional comparison measure. We also provide tractable sufficient conditions for radial dominance, including strong convexity, addition of nonnegative convex terms, and star-shaped constraints. As a consequence, we obtain a refined convergence rate for stochastic proximal estimators of weakly convex functions, together with asymptotic sharpness of the constant. The same framework yields a corresponding rate for stochastic projection estimators onto closed convex sets. We further establish basic structural properties of the barycentric approximation operator, such as smoothness and cocoercivity. Numerical experiments illustrate the predicted rate, the dimensional scaling of the constant, and its asymptotic sharpness.

arXiv:2607.08607v1 Announce Type: cross Abstract: Recently, there has been significant interest in applying the method of moments developed by Wood and others to study distributions of finite abelian groups that arise in number theory and combinatorics. When the moments do not grow too fast, they determine a unique distribution. We construct large families of distributions that have the same moments. These families include several distributions that arise naturally in the study of sandpile groups of families of random graphs. Wood determined the distribution of Sylow $p$-subgroups of sandpile groups of Erd\H{o}s--R\'enyi random graphs. This was extended by M\'esz\'aros to sandpile groups of random $d$-regular graphs, who observed an interesting special case when $d$ is even and $p = 2$. We study Sylow $p$-subgroups of sandpile groups of random bipartite graphs and similarly find a special case for $p =2$. Although this distribution differs from that of M\'esz\'aros, we show that they have the same moments and fit into our broader construction. To compute the moments of the distributions we study, we apply combinatorial tools from the theory of Hall--Littlewood functions.

arXiv:2607.08592v1 Announce Type: cross Abstract: We study a class of random inner functions $\varphi$ whose Clark measure at $1$ is the weighted sum of point masses supported on independent uniformly distributed points of $\mathbb T$. Our first result shows that $\varphi$ is almost surely a Blaschke product. We then investigate when $\varphi$ admits angular derivative almost surely and we provide a $0 - 1$ law. These conditions have a direct interpretation in terms of the other Clark measures associated with $\varphi$. Finally, we obtain quantitative estimates for the zeros of $\varphi$, proving that, in suitable regimes, their distribution satisfies summability conditions stronger than the classical Blaschke condition.

arXiv:2607.08531v1 Announce Type: cross Abstract: Assuming that the asset price $X$ follows a constant elasticity of variance process, this paper studies the optimal prediction problem $\inf_{0\leq \tau\leq T}\mathbb{E}|X_\tau-\ell|$, where the infimum is taken over stopping times $\tau$ of $X$ and $\ell$ is a hidden aspiration level independent of $X$. Adopting the aspiration level hypothesis, we show that a class of admissible laws of $\ell$ leads to optimal trading boundaries which are located relative to the median interval of $\ell$ and serve as predictors of the resistance and support levels. The existence of these boundaries is proved and nonlinear integral equations are derived to characterise them uniquely. In the positive drift case the stopping set is bounded by two curves, while in the negative drift case the stopping set is described by a single boundary.

arXiv:2607.08472v1 Announce Type: cross Abstract: We address a long-standing open problem posed by Hansen and Pitts (2006) on nonparametric inference for the service-time distribution in an M/G/1 workload model. We consider an M/G/1 queue with unknown arrival rate $\lambda>0$ and service-time distribution $B(\cdot)$, without assuming stability or stationarity. A statistician observes the workload process at discrete times $t=0,1,\ldots,n$ and aims to estimate $B(w)$ at a fixed point $w>0$. We propose an estimator $B_n(w)$ based solely on the observed workload trajectory. The construction relies on a screening mechanism that extracts conditionally i.i.d. compound Poisson increments from the workload process, thereby reducing the dependent-data problem to a Laplace-transform inversion framework. Under mild regularity assumptions on $B(\cdot)$, i.e., continuous differentiability on $[0,\infty)$, twice differentiability at $w$, and a finite second moment, we establish the bound \[ \mathbb{E}\bigl|B_n(w)-B(w)\bigr| =\mathcal{O}\!\left(\frac{\log n}{\sqrt{n}}\right), \qquad n\to\infty. \]This provides the first solution to the Hansen-Pitts problem achieving a parametric $L^1$-risk rate (up to a logarithmic factor), without requiring stationarity, stability, or knowledge of the arrival rate.

arXiv:2607.08435v1 Announce Type: cross Abstract: This article concerns interacting particle systems with singular kernels, driven either by degenerate deterministic controls or by degenerate decomposable noise. In the deterministic setting, we establish global exact controllability and a topologically robust property called solid controllability. Moreover, we prove a result that guarantees global approximate controllability with prescribed trajectories. For stochastic dynamics, we obtain ergodicity and exponential mixing by utilizing coupling and recurrence mechanisms based on controllability. Our approach exploits the singularity and applies to a broad class of models, including Biot-Savart, Coulomb, Riesz, and Yukawa interactions, as well as heterogeneous multi-species systems.

arXiv:2607.08415v1 Announce Type: cross Abstract: Let $X=(X_1,\ldots,X_n)$ be independent nonnegative random variables, not necessarily identically distributed. Let $D=(D_0,D_1,\ldots,D_n)\sim\operatorname{Dir}(1,\ldots,1)$ be independent of $X$, and define $K(x)=\mathbb{P}\{\sum_{i=1}^n x_iD_i\le1\}$. We prove that, for every $n\ge1$, whenever $\mathbb{E} X_i\le1$ for every $i$, $\mathbb{P}\{K(X)\le\alpha\}\le\alpha$ for all $0\le\alpha\le1$. Thus $K(X)$ is a finite-sample, distribution-free $p$-value for testing the null hypothesis $\mathbb{E}X_i \le 1$ for all $i$. This proves a conjecture of Gaffke (2005).

arXiv:2607.08385v1 Announce Type: cross Abstract: We investigate pathwise well-posedness of the stochastic modulated Korteweg-de Vries equation (KdV) on the circle with a multiplicative noise, where a time non-homogeneous modulation acts on the linear dispersion term. (i) In the Young case (= fractional-in-time case with Hurst parameter greater than $\frac 12$), we establish a new regularization-by-noise phenomenon on the stochastic convolution in a pathwise manner, where a gain of spatial regularity becomes (arbitrarily) larger for more irregular modulations. We then prove that, given any $s \in \mathbb R$ and any multiplicative Young noise, however rough it is in space, the stochastic modulated KdV is pathwise locally well-posed in $H^s(\mathbb T)$, provided that the modulation is sufficiently irregular. (ii) In the rough case (= white-in-time case), irregularity of the modulation does not induce any smoothing on the stochastic convolution, and in fact, there is a slight loss in the spatial regularity. In this case, by slightly regularizing the multiplicative noise term, we prove pathwise local well-posedness in $H^s(\mathbb T)$ for any given $s \in \mathbb R$, provided that the noise is sufficiently smooth in space. We achieve these goals by combining (i) the sewing lemma approach to the nonlinear Young integration theory, introduced by Chouk and the second author (2014), and (ii) the pathwise construction of stochastic convolutions as Young or rough integrals via the random tensor estimate and the sewing lemma, introduced by the first, fourth, and fifth authors (2026). In the appendix, we also present an example of regularization by noise for a stochastic modulated Schr\"odinger equation with a multiplicative Young noise.

arXiv:2607.08157v1 Announce Type: cross Abstract: In this paper, we study an age-dependent branching process. In the simplest setting, the population is divided into two age groups, namely juveniles and adults. Our objective is to estimate the model parameters using observations of the total population size only (i.e., juveniles plus adults). Focusing on the ergodic regime of the model, we introduce a method-of-moments estimator and establish its asymptotic normality. Several extensions are discussed, including models with more than two age groups.

arXiv:2607.07841v1 Announce Type: cross Abstract: We investigate quadratic bilinear systems by developing novel tree-based representations of their solutions. The proposed framework decomposes the solution into a sequence of coupled bilinear subsystems whose components admit explicit expansions indexed by full binary trees. These representations yield sufficient conditions for the existence of global solutions and lead to new output bounds in terms of reachability Gramians. Motivated by these estimates, we introduce time-limited and infinite-horizon reachability and observability Gramians, establish sufficient conditions for their existence, and characterize them through nonlinear matrix equations. The associated Gramians are employed to identify dominant state-spaces and to derive exact reduced-order models obtained by removing Gramian kernels. Building on these results, we develop a balanced truncation method for quadratic bilinear systems and prove an error bound for the reduced-order approximation. The proposed framework provides a unified connection between tree-based solution representations, nonlinear Gramian theory, and balanced truncation for quadratic bilinear systems, closing several theoretical gaps in the analysis of Gramian-based model reduction for this class of systems.

arXiv:2607.07778v1 Announce Type: cross Abstract: Bubeck, Li and Nagaraj conjectured that, for generic data, any two-layer neural network with $m$ neurons that fits $n$ noisy labels must have Lipschitz constant at least of order $\sqrt{n/m}$, with no restriction on the size of the weights. Bubeck and Sellke proved a universal version of this law for Lipschitz-parameterized classes, but under a polynomial bound on the parameters; at depth three that boundedness hypothesis is genuinely necessary. The two-layer unbounded-weight case requires a different argument. We prove the conjectured law, up to one logarithmic factor, for every continuous piecewise-linear activation, in particular for ReLU networks. For data drawn uniformly from $\mathbb{S}^{d-1}$, $d\ge3$, or from $N(0,I_d/d)$, labels in $[-1,1]$ with noise level $\sigma^2>0$, and any width-$m$ two-layer network with arbitrary real weights, biases and affine skip connection, fitting the data $\varepsilon$ below the noise floor forces $\mathrm{Lip}(f)\ge c\,\varepsilon\sqrt{n/(\bar m\log(C\bar m nd/\varepsilon))}$, $\bar m=(K-1)m+1$, with high probability. A realized-kink-count version holds on the same event: every realized two-layer piecewise-linear function with $k(f)\le n$ distinct kink hyperplanes obeys the bound with $\bar m$ replaced by $k(f)+1$, irrespective of how many redundant hidden units parameterize it. The proof replaces parameter-space covering, impossible for unbounded weights, by a function-space covering. The central deterministic ingredient is a rigidity lemma: on $B_2$, and on $\mathbb{S}^{d-1}$ for $d\ge3$, the coefficient of each canonical kink is controlled by the Lipschitz constant of the realized function, because kinks on distinct hyperplanes cannot cancel at generic points. Rigidity genuinely fails at $d=2$, and an explicit two-layer ReLU interpolant with $O(1)$ Lipschitz constant at width $2n$ matches the law at the overparameterized endpoint.

arXiv:2607.08696v1 Announce Type: new Abstract: We study the spectrum of the adjacency matrix $A_n$ of directed inhomogeneous random graphs on $n$ vertices. We assume that $A_n$ has independent entries and diverging average degree scale $s_n$. This framework includes, as special cases, the directed Chung--Lu random graph and directed stochastic block models. Assuming boundedness of the variance profile and that $s_n$ diverges faster than a suitable logarithmic function of $n$, we show that the rank-one Chung--Lu model satisfies a non-homogeneous version of the circular law, which in some situations allows for an explicit expression. Moreover, under mild conditions, we identify the asymptotic singular value distribution using tools from free probability. Finally, for finite-rank directed models, we prove the existence of eigenvalues outside the bulk and establish their joint Gaussian fluctuations at the scale $\sqrt{s_n/n}$, with an explicit covariance matrix. These results extend the theory of spectral outliers and their fluctuations to directed inhomogeneous random graphs.

arXiv:2607.08632v1 Announce Type: new Abstract: We construct an uncountable family of extremal Gibbs states of the low temperature Ising model on hyperbolic lattices embedded in the hyperbolic plane $\mathbb{H}_2$ whose interfaces are complete geodesics of $\mathbb{H}_2$. These states are extracted from the states constructed by D'Achille, Coquille and Le Ny in arXiv:2504.19553v2 by considering path in the dual lattice at close enough distance from geodesics of $\mathbb{H}_2$ thanks to the Morse--Mostow lemma.

arXiv:2607.08551v1 Announce Type: new Abstract: We study a weighted-threshold version of the coupon collector problem in continuous time. Each type $i$ is discovered at rate $\lambda p_i$ and, once discovered, contributes weight $w_i$, where $p$ and $w$ are probability vectors. The stopping time when the total weight of the discovered types first exceeds a fixed threshold $\theta\in (0,1)$ is called the quorum time. We first prove concentration estimates and compare the quorum time with the corresponding deterministic threshold time obtained from the mean discovered weight. When all discovery rates are equal and the largest individual weight tends to zero, the first-order asymptotics are universal and do not depend on the weight vector. We then analyze the aligned Zipf family $p_i = w_i \propto i^{-s}$. This model has three regimes: a deterministic linear scale for $0\le s < 1$, a critical scale $H_NN^\theta$ at $s=1$, with an explicit leading constant, and a non-degenerate random hitting-time limit for $s>1$. Finally, we show that the expected quorum time need not be monotone in the Zipf exponent.

arXiv:2607.08536v1 Announce Type: new Abstract: We show that the lower edge of the appropriately scaled size $n$ Laguerre beta-ensemble with parameter $a=a_n$ converges to the $\operatorname{Airy}_{\beta}$ process as $n\to \infty$ when $a_n\to \infty$ and $\tfrac{a_n}{n}\to 0$. This completes the picture of the possible edge scaling limits of the Laguerre beta-ensemble with a fixed $\beta>0$. When $a_n\gg (\log \log n)^3$ our proof establishes operator level convergence of the inverse of the scaled Dumitriu-Edelman tridiagonal matrix to the inverse of the stochastic Airy operator. Our methods allow us to prove similar operator level limits for the known soft edge scaling limits of the Laguerre and Gaussian beta-ensembles. For $a_n\le (\log n)^{1/2}$ we give a different argument that relies on coupling and a result of Dumaz-Li-Valko for the transition between the hard and soft edge limits of the Laguerre beta-ensemble.

arXiv:2607.08330v1 Announce Type: new Abstract: We consider the zero-temperature stochastic Ising process describing $\pm 1$ spin-flip dynamics on an infinite one-dimensional quasi-transitive graph $G=(V,E)$ with finite interaction range $K$. We prove that the zero-temperature limit of the Glauber dynamics for this Ising model exhibits a Type $\mathcal{I}$ behavior (infinite fluctuations of all vertices) if and only if the graph possesses the so-called \emph{shrink property}. For graphs lacking this property, we introduce an algorithmic framework based on an auxiliary spatial automaton to distinguish, in finite time, between Type $\mathcal{F}$ behavior (almost sure local fixation) and Type $\mathcal{M}$ behavior (a mixed regime characterized by the presence of blinkers). We prove that the classification among these three regimes is algorithmically decidable. Furthermore, we provide a constructive example of a graph supporting blinkers of arbitrarily large size.

arXiv:2607.08314v1 Announce Type: new Abstract: We construct a new family of random fields on the Heisenberg group $\mathbb{H}$, the sub-Riemannian analog of $\mathbb{R}^{n}$. These fields are generalized random eigenfunctions of the sub-Laplacian on $\mathbb{H}$, and can be viewed as the sub-Riemannian counterpart to the Berry random wave model in $\mathbb{R}^{n}$. The construction of such waves relies on the representation theory of $\mathbb{H}$, and differs from the Euclidean case because of the presence of infinite-dimensional unitary irreducible representations. This work represents a first step towards studying random waves and their geometry in sub-Riemannian spaces.

arXiv:2607.08242v1 Announce Type: new Abstract: Random sums of independent random variables have been extensively studied in classical probability theory. We consider random sums of self-adjoint variables from a non-commutative probability space, and establish several $*$-convergence results. In particular, we show that the joint $*$-convergence of the standardized random sum of identically distributed self-adjoint variables and the standardized stopping random variable (rv) is equivalent to the convergence of all moments of the stopping rv together with the convergence of the ratio of its mean to its variance. We obtain central limit theorems for the random sums of free, independent and half independent self-adjoint variables with both deterministic and random scaling. Furthermore, we derive some scaling $*$-convergence limits for randomly indexed self-adjoint variables.

arXiv:2607.08187v1 Announce Type: new Abstract: In this paper, we study the asymptotic behaviours of a critical branching random walk in $\mathbb{R}^d$ under the assumption that the offspring distribution belongs to the domain of attraction of an $\alpha$-stable law with $\alpha\in(1,2]$, and that the jump distribution has a finite $\frac{2\alpha}{\alpha-1}$-th moment. First, we establish the precise decay rate for the tail probability of the all-time maximal displacement $M^d$. Next, we investigate the maximal displacement $M_n^d$ at generation $n$ and prove a conditional limit theorem for the distribution of $M_n^d$ given that the process survives up to generation $n$. These results extend the corresponding 1-dimensional results of Lalley and Shao (2015) to the case $d\ge2$. Finally, we study the asymptotic behaviour of the total progeny $\zeta$. In particular, we show that, conditioned on the event $\{M^d\ge x\}$, $\zeta$ converges in distribution under an appropriate normalization. This result reveals a quantitative relationship between the maximal displacement and the total progeny size.

arXiv:2607.08160v1 Announce Type: new Abstract: We develop a new method for proving a weak functional inequality by first proving it for a sufficiently regular sequence of distributions approximating the stochastic localization (SL) process, and then transferring it to the desired distribution via regularity of the SL process and conductance arguments. We use this strategy to prove a weak Poincar\'e inequality (WPI) holds for the Gibbs measure of the Sherrington-Kirkpatrick model when $\beta < \frac{1}{2}$. A prior result of the authors [arXiv:2605.03718, 2026] proves the ASL process for the Sherrington-Kirkpatrick model satisfies the required regularity conditions. A consequence of the WPI is that a much simpler algorithm -- Glauber dynamics with a warm-start -- efficiently samples the Gibbs measure of the SK model at $\beta < \frac{1}{2}$. This is a significant structural step towards resolution of the conjecture that Glauber dynamics mixes fast in the replica-symmetric regime for the Sherrington-Kirkpatrick model [arXiv:2504.20539, Open-Problem 15, 2025].

arXiv:2607.08129v1 Announce Type: new Abstract: Let $A_n$ be a subset of $\{1,2,\ldots,n\}$ obtained by retaining each integer independently with fixed probability $\theta\in(0,1)$, and let $L_n$ be the least common multiple of the integers in $A_n$. We prove a functional large deviation principle, a functional moderate deviation principle, and a Strassen-type functional law of the iterated logarithm for the process $(\log L_{\lfloor{nt}\rfloor})_{0\le t\le1}$. The large deviation rate function is given by an entropy contraction for geometric marks, while the moderate deviation rate function and LIL cluster set are described by the reproducing kernel Hilbert space associated with the Gaussian limit process.

arXiv:2607.07712v1 Announce Type: new Abstract: We reinterpret the classical Hermite generating function as a Gaussian density ratio: relative to the unit Gaussian reference, it is the density ratio of a Gaussian with shifted mean and unchanged covariance. Applying the heat semigroup in the mean-parameter variable to this generating function produces the corresponding temperature variation. Thus the heat-semigroup time variable is reinterpreted as the temperature variation of the Gaussian density ratio. This parameter-space formulation also gives a parabolic control principle for Hermite approximation errors. Since Hermite projections act in the velocity variable and the heat flow acts in the mean variable, Hermite block energies and truncation tails are subsolutions of the same parameter-space heat equation. This remains useful for heat-evolved non-Gaussian perturbations where no usable closed coefficient formula is available. For Gaussian density ratios with general covariance, the Hermite coefficients satisfy a weighted homogeneity in the mean and covariance-defect parameters. This yields Ornstein--Uhlenbeck covariance, an exact generating function for total-degree Hermite block energies, and the sharp geometric Hermite truncation rate, equal to the square root of the largest absolute covariance defect. We also derive precise isotropic block and tail asymptotics and interpret the estimates for near-Gaussian kinetic distributions.

arXiv:2603.23196v2 Announce Type: replace-cross Abstract: In this work, we investigate Gaussian Mixture Models ({\it abbrv} GMM) and the related problem of non parametric maximum likelihood estimation ({\it abbrv} NPMLE) from the perspective of statistical mechanics. In particular, we establish stability guarantees for the NPMLE procedure that extend well beyond the state of the art. Crucially, we obtain guarantees on the Kullback-Leibler divergence between NPMLE estimators and the ground truth, a type of result which has been known to be challenging in the literature on this problem. In particular, we provide high probability upper bounds on the KL divergence between the NPMLE and the true density that are of the order of $\min\big\{\frac{(\log n)^{d+2}}{n} , \frac{\log n}{\sqrt n}\big\}$, which cover a wide range of scenarios for the comparative sizes of $n$ and $d$. We obtain similar guarantees for approximate solutions to the NPMLE problem, addressing realistic situations wherein optimization algorithms need to be stopped in finite time, allowing access only to approximations to the true NPMLE. A cornerstone of our approach is an analysis of the function class complexity of logarithms of gaussian mixture densities, which is able to handle their unboundedness, and could be of wider interest. Our methods lead to novel confidence-interval guarantees for entropy estimation in Gaussian mixtures, demonstrating their wider impact. We also establish correspondences between stability phenomena in the NPMLE problem and concepts such as chaos and multiple valleys in random energy landscapes of statistical mechanics models. While these correspondences are largely of a conceptual nature at this point, we believe that these connections, especially those with concentration phenomena and Langevin dynamics, may be developed into a toolbox for studying a wide variety of random optimization problems in statistics and machine learning.

arXiv:2603.13953v2 Announce Type: replace-cross Abstract: We introduce the notion of a bivariate random discrete copula on an equidistant mesh and explore its stochastic properties. A random discrete copula is a discrete random field, hence, its value at a given point on the mesh is a random variable. We determine the distribution of this random variable and calculate its expected value and variance. We also consider bilinear extension of a random discrete copula to a random field over the whole unit square.

arXiv:2602.24193v3 Announce Type: replace-cross Abstract: We establish the \emph{hole phenomenon} for the Gaussian analytic function \[ F_{\beta}(z)=\sum_{n=0}^{\infty}\frac{\xi_{n}}{\sqrt{\Gamma\bigl(\frac{2}{\beta}(n+1)\bigr)}}\,z^{n}, \] associated with the power-exponential weight $e^{-|z|^{\beta}}$ on $\mathbb{C}$, where $\beta>0$. Under the condition that $F_{\beta}(z)$ has no zeros in $D(0,r)$, the scaled zero counting measure converges to a limiting measure $\mu_{0}^{\beta}$ vaguely in distribution. This limit exhibits a \emph{forbidden region} \[ \bigl\{1<|z|

arXiv:2512.19854v2 Announce Type: replace-cross Abstract: In this work we address the stability and robustness of uniformly discrete point sets in Euclidean spaces. Firstly, we prove that if a sequence of point configurations contained in $\mathbb{R}^d$ is uniformly diffractive, converges rapidly enough to a discrete set $X$ in $\mathbb{R}^d$, and their diffraction measures $\widehat{\gamma_{X_n}}$ are asymptotically orthogonal with respect to the Lebesgue measure in $\mathbb{R}^d$, then $X$ is necessarily a quasicrystal. The convergence is addressed for a distance that quantifies the statistical closeness between two uniformly discrete point sets in $\mathbb{R}^d$. Secondly, motivated by their applications in the diffraction theory of quasicrystals, we establish the continuity of the Fourier Transform of quasicrystals in this topology. This continuity result, in turn, allows us to rigorously demonstrate that well-known robustness properties of quasicrystals against random errors remain stable under the statistical convergence considered. Some applications for rapidly-solidified quasicrystals are highlighted.

arXiv:2502.05906v2 Announce Type: replace-cross Abstract: We consider a strategic M/M/1 queueing model under a first-come-first-served regime, where customers are split into two classes and class $A$ has priority over class $B$. Customers can decide whether to join the queue or balk, and, in case they have joined the queue, whether and when to renege. We study the equilibrium strategies and compare the equilibrium outcome and the social optimum in the two cases where the social optimum is or is not constrained by priority.

arXiv:2412.03405v3 Announce Type: replace-cross Abstract: Motivated by dynamic risk measures and conditional $g$-expectations, in this work we propose a numerical method to approximate the solution operator given by a Backward Stochastic Differential Equation (BSDE). The main ingredients for this are the Wiener chaos decomposition and the classical Euler scheme for BSDEs. We show convergence of this scheme under very mild assumptions, and provide a rate of convergence in more restrictive cases. We then implement it using neural networks, and we present several numerical examples where we can check the accuracy of the method.

arXiv:2401.13812v4 Announce Type: replace-cross Abstract: We consider an M/M/s queueing model in which customers strategically decide, based on the service reward and waiting cost, whether to join upon arrival or balk and, at any time, whether to remain in the queue or renege. Rational strategic behavior yields an equilibrium whose outcome may be socially efficient or inefficient, depending on the queueing regime. Some regimes yield an efficient equilibrium only under precise calibration to the model parameters. Others are universally optimal, meaning that their equilibrium outcome is efficient for all parameter values. Universal optimality is therefore an appealing property for a planner choosing a queueing regime. We characterize the class of universally optimal queueing regimes. A by-product of our characterization is that preemption plays an unavoidable role in universally optimal regimes.

arXiv:2303.07956v2 Announce Type: replace-cross Abstract: It is proved that the number of subsets of $[n]^d$ that tile $\mathbb{Z}^d$ is $\left(3^{\frac{1}{3}}\right)^{n^d \pm o(n^d)}$.

arXiv:2607.05384v2 Announce Type: replace Abstract: A random $n\times k$ matrix $S$ is an \emph{$(r,\alpha)$-oblivious subspace injection} (OSI) if $\mathbb{E}\|S^\top x\|_2^2=\|x\|_2^2$ for every $x\in\mathbb{R}^n$, and for every fixed $r$-dimensional subspace $V\subset\mathbb{R}^n$, with probability close to one, one has $\alpha\|x\|_2^2\le\|S^\top x\|_2^2$ for all $x\in V$. In this work, we show that in the regime $r=\Omega(k)$ and $\alpha=\Omega(1)$, and under a mild additional structural assumption, no constant-row-sparsity matrix $S$ is OSI, thereby answering, in a strong form, a question raised by Cama\~no, Epperly, Meyer, and Tropp. We show that the failure of the OSI property for sparse random matrices stems from a general deterministic phenomenon, thereby reducing a probabilistic problem to a non-probabilistic one. This phenomenon is related to the restricted invertibility principle introduced in the seminal work of Bourgain--Tzafriri. Let $(n_k)_{k\in\mathbb{N}}$ be a sequence of integers satisfying $\frac{n_k}{k}\to\infty$. For each $k$, let $S^{(k)}$ be a $n_k\times k$ non-random matrix with $O(1)$ nonzero entries per row, whose nonzero entries have average magnitude $O(1)$, and such that the total number of pairs of rows with supports overlapping at two or more indices is $o({n_k}^2/k)$. We prove that for every constant $\varepsilon>0$, as $k\to\infty$, the overwhelming majority of $k\times \lfloor\varepsilon k\rfloor$ submatrices of $(S^{(k)})^\top$ have the smallest singular value $o(1)$. Thus, the well-invertible submatrices whose existence is guaranteed by the Bourgain--Tzafriri theorem are rare. The proof is itself based on probabilistic tools.

arXiv:2607.02172v2 Announce Type: replace Abstract: We consider the Sherrington-Kirkpatrick spin glass model at the critical inverse temperature $\beta = 1$ with zero external field. We prove that the free energy $F_N = F_{N,\beta=1}$ of this model has variance \[ \mathrm{Var}(F_N) = \frac16 \log N + O(1)\,, \] confirming a physics prediction of Aspelmeier \cite{aspelmeier2008free}, and that the centered and scaled $F_N$ satisfies a Gaussian CLT. We also identify the critical two-replica overlap scale, proving \[ \mathbb{E} \langle R_{1,2}^2\rangle \asymp N^{-2/3}\,, \] as conjectured by Talagrand \cite{talagrand2011mean2}, together with a uniform exponential moment bound for $N^{1/3} |R_{1,2}|$. The key input is a critical reweighted moment method, in the spirit of the ``small subgraph conditioning'' technique from probabilistic combinatorics, but capable of capturing diverging fluctuations. Through this reweighting, we relate the critical SK model to the BBP critical edge, which determines the overlap and fluctuation scales.