Papers

We prove unique ergodicity of the projective process associated with the two dimensional Navier–Stokes equation in vorticity form, with additive diagonal noise acting on every nonzero real Fourier phase and satisfying two sided power law bounds. Consequently, the exact Furstenberg–Khasminskii formula for the top Lyapunov exponent holds. The main new ingredient is a compact dense mechanism for asymptotic generalized coupling in the absence of a Foiaş–Prodi type high-low mode decomposition for the projective dynamics. Using the dense range of the Malliavin derivative and compactness of the state derivative, we construct finite rank perturbations of the Wiener path that compensate, to first order, for perturbations of the initial condition, leaving a residual whose logarithmic growth has negative stationary mean and hence contracts locally at an exponential rate, with a cost controlled by a triangular scheme of blockwise Ramer transformations.

We give a two-player zero-sum repeated game between a learner and nature whose value identity generates Bayesian updating and an exact accounting of exponential-weights regret at once, and supplies the comparator-class variational form that a wide class of concentration phenomena share. The terminal payoff is the most a comparator can gain at fixed relative entropy from the prior, and the one-step constraint is an information budget on nature's move under the learner's mixed action. With the learner's move otherwise unrestricted, Gibbs/Bayes weights emerge as its unique Bellman equalizer – the mixed action that makes the per-round loss independent of which direction nature moves – with log-partition functions playing the role of value functions. The regret decomposes exactly into three parts: a per-round information loss reflecting the variation in observed outcomes, an additive retempering drift that accounts exactly for any change of measurement scale between rounds, and the information the comparator carries relative to the prior. The variance and bounded-range proxies that drive standard regret bounds are looser relaxations of this decomposition, which holds generally and governs them all. Both players' strategies are read off from the decomposition term by term, and repeated play yields an information-theoretic ledger of self-play in place of the usual quadratic-variation surrogate. The same comparator-class geometry accounts for the classical large-deviation bounds, and methods across bandits, posterior sampling, aggregation, and boosting are specializations of the one regret decomposition.

We study the supercritical regime $\beta>\beta_c$ of a large family of real-valued spin models on $\mathbb Z^d$, including the Blume-Capel and $P(\varphi)$ models. We consider their random cluster representations and prove that they are well-behaved, in the sense that local uniqueness of macroscopic clusters occurs with high probability, uniformly in the boundary conditions. This implies, among other things, a surface-order exponential bound for the (lower) large deviations of the empirical magnetisation. These results were previously known only in the cases of the Ising and $\varphi^4$ models.

We study the problem of guessing a realization of an i.i.d. random sequence given element-wise correlated side-information. We use type-counting to provide estimates of the tail probabilities of the number of guesses for the case without side-information, which was shown earlier through large-deviation techniques. We then extend the same counting argument to the conditional setting, obtaining new explicit expressions for the corresponding guesswork exponents as divergences involving conditional tilted distributions. Finally, we provide an application of these exponents to brute-force password guessing with side-information.

For an AR($1$) sequence with coefficient $0 < a < 1$, the probability of staying above zero decays exponentially in the number of steps; at the unit root $a=1$ it decays polynomially, with order $n^{-1/2}$ for symmetric increments. We study this transition for AR($1$) sequences driven by symmetric and asymmetric $\alpha$-stable innovations. We write $\Lambda(a,\alpha)$ for the symmetric exponential persistence rate and $\Lambda(a,\alpha,\rho)$ for the rate of a strictly stable law with positivity parameter $\rho$. The entire chain admits an exact representation through a single stable Lévy process observed on a geometrically expanding time grid. In the symmetric case, comparison with continuous half-line survival gives $\Lambda(a,\alpha) \leq \frac{\alpha}{2}\log{(1/a)}$. For $0 < \alpha < 2$, this bound disproves the stable specialization of a conjecture of Hinrichs, Kolb and Wachtel concerning regularly varying innovation tails. Combining stable closure under subsampling with a monotonicity coupling yields a lower bound of the same near-unit order. This proves $\Lambda(a,\alpha) \asymp \log{(1/a)}$ as $a \uparrow 1$ and shows that $\Lambda(a,\alpha)/\log{(1/a)}$ converges to a limit in $(0,\alpha/2]$ equal to the supremum of that ratio over $0 < a < 1$. A Lamperti transformation reduces identification of this constant to a dense-sampling problem for a stationary stable Ornstein-Uhlenbeck process. Existing Gaussian theory determines the sharp value at $\alpha=2$; for $0 < \alpha < 2$, rescued crossings between observations remain the obstacle. For asymmetric strictly stable innovations with $\alpha \neq 1$ and admissible $\rho\in(0,1)$, the corresponding bounds are $\Lambda(a,\alpha,\rho) \leq \alpha(1-\rho)\log{(1/a)}$ and $\Lambda(a,\alpha,\rho) \asymp \log{(1/a)}$. At $\alpha=2$, only the symmetric case $\rho=1/2$ occurs.

We establish an asymptotic formula for squarefree polynomials over a finite field whose coefficients are in $\{0,1\}$. This resolves a function-field analogue of a conjecture of Erdős, Mauduit and Sárközy. More generally, we estimate the probability of a polynomial $\sum \varepsilon_it^i$ being squarefree, where $\varepsilon_i$ are random variables sampled according to measures on $\mathbb{F}_q$.

Let $\varepsilon_1,\ldots,\varepsilon_n$ be independent Rademacher signs and let $a=(a_1,\ldots,a_n)\in\R^n$ satisfy the normalization below. For the normalized Rademacher sum, we determine how its higher moments depend on the fourth-order mass. Combining a sharp fixed-q moment envelope with a separate argument below the convexity threshold gives the Gaussian stability inequality for the full range $p\geq4$ of this linear-in-q bound. The same fourth-order framework determines the sharp finite dimensional $L_p/L_4$ Khintchine constant for $p\geq5$, with the flat coefficient vector as the extremizer. These results settle the conjectures of Jakimiuk and of Barański, Murawski, Nayar, and Oleszkiewicz stated below. We also prove Jakimiuk's conjectured quadratic stability estimate at $p=3$. The resulting bounds retain information about sparsity and effective dimension, with applications to Rademacher random projections and randomly signed errors; those applications are not developed further here. Their Laplace-transform form also gives coefficient-sensitive tail bounds. The proofs are discovered with substantial assistance from ChatGPT 5.6 Sol.

Kolmogorov's "Grundbegriffe der Wahrscheinlichkeitsrechnung" introduced the standard measure-theoretic formalization of probability but included only a two-page section about the relation of the mathematical theory of probability to the world of experience. This note discusses this brief non-mathematical section concentrating on what I regard as its two most controversial features: having two bridges between the mathematical theory and the world of experience instead of the standard one, and the possibility of observing a prespecified event of probability zero in a sequence of trials.

We study a resource extraction problem of ergodic type, where harvesting quotas are imposed that restrict the harvesting rate by a function of the current biomass. In a setting with diffusive dynamics, we provide conditions under which the optimal harvesting strategy is of threshold type. In contrast with the unrestricted case with singular controls, the optimal threshold may be 0, which corresponds to harvesting at the maximal possible rate at all times. Furthermore, we establish the convergence of the solution to the corresponding singular control problem as the harvesting rate tends to infinity.

Let $\delta>0$ be fixed, let $m = \lceil(1+\delta)n\rceil$, and let $\Gamma$ be an $n\times m$ matrix with independent standard Gaussian entries. For the Gaussian Gluskin polytope $G_m = \Gamma(B_1^m)$ we prove $$ \p\left\{d_{\mathrm{BM}}(G_m,B_1^n) \ge c_\delta n^{5/8}(\log n)^{-1/8}\right\} \ge 1-Ce^{-cn}. $$ Consequently, for all sufficiently large $n$, $$ R_1(n)\ge c n^{5/8}(\log n)^{-1/8}. $$ The proof uses an exact coefficient quantization, a simultaneous Gaussian compression in coefficient quotients, and an inverse-trace estimate for arbitrary inscribed bases.

We characterize bivariate measures of concordance of degree at most two in the sense that the measures evaluated at copula-mixture segments are polynomials of degree at most two. Our main device is the canonical polarization, which converts quadraticity of a functional into separate affinity, thereby allowing an integral representation for affine functionals to be applied sectionwise. Using this technique, we prove that measures of degree at most two are in bijection with copula-indexed measure fields satisfying several properties including square-group equivariance. The field at independence determines the affine linearization, while its displacement determines the purely quadratic part. Moreover, exact degree two is equivalent to field non-constancy. Leveraging our characterization result, we provide a construction of degree-two measures of concordance based on affine copula operators that are square-group equivariant. As a more concrete example, we derive a necessary and sufficient condition for a weighted averaging operator of square-group transformed copulas to generate a measure of concordance of degree two.

It is well known in random matrix literature that the limiting spectral distribution of a Wigner matrix is the semi circular law while the limiting spectral distributions of other patterned matrices like Toeplitz, Hankel, symmetric circulant and reverse circulant matrices have unbounded supports. One fundamental difference between the Wigner matrices and the other matrices mentioned above is that the Wigner matrices have $O(n^2)$ independent random variables while the others have $O(n)$ independent random variables. In this paper, we show that this is not true in general. In particular, we form matrices with $O(n)$ independent random variables whose empirical spectral distributions are arbitrary close to the semi circular law.

In this paper, we consider the connectivity of a random network of N mobile devices in three dimensions (3D), where the location of each device or node has a Gaussian distribution in each dimension. For each pair of nodes, the probability of connectivity is related to the nodes' separation by a Gaussian connectivity function. The fundamental analytical tool for studying such systems is the probability of a given network state, derived and expressed in terms of its graph Laplacian. Leveraging this result, we obtain results for the connectivity of small networks, the probability of a complete network (where all nodes are connected to all other nodes), and the probability of an isolated node, which gives an approximation to the connectivity probability of larger networks. The general results are then simplified for special cases and limiting scenarios.

We address the inverse problem of reactive transport in heterogeneous porous media, where unknown model parameters must be inferred from sparse experimental observations. The problem is complicated by strong nonlinearities, spatial heterogeneity, and limited data availability. We propose a Conditioned-on-Design Bayesian Physics-Informed Neural Network (CoDe-BPINN), which combines a domain-decomposed PINN solver with a Bayesian inference network that learns the conditional distribution of model parameters given experimental design variables. The framework is trained by maximizing a physics-informed Evidence Lower Bound (ELBO), enabling simultaneous reconstruction of spatiotemporal concentration fields, probabilistic parameter estimation, and uncertainty quantification. We demonstrate the approach using laboratory experiments on contaminant transport through a multilayer porous column with an iodinated contrast medium. The model accurately reproduces breakthrough dynamics while revealing systematic parameter dependence on flow conditions. In particular, it identifies a nonlinear decrease in effective sorption capacity with increasing flow rate and porosity, consistent with kinetic limitations and reduced adsorbent mass. The Bayesian formulation also uncovers a strong negative correlation between sorption affinity and sorption capacity, quantifying the intrinsic non-identifiability of the inverse problem. CoDe-BPINN provides a robust framework for parameter inference and uncertainty quantification in data-scarce reactive transport problems.

We derive convergence rates for the approximation of the Fréchet distribution $\mathcal{F}(\alpha)$ with parameter $\alpha > 0$ by sequences of renormalized maxima in the extreme value theorem. Our proofs rely on the application of the infinitesimal generator approach to Stein's method to max-stable distributions, using the family of Markov semi-groups recently introduced in \cite{CostacequePhD, Costaceque24}. We develop two different approaches to compute rates of convergence; the first one relies on the second-order regular variation assumption, while the second one requires the existence of a density function for the base distribution. In particular, with the first approach, our bounds are expressed using the Kolmogorov distance, and the Wasserstein distance when $\alpha > 1$. The second approach allows also rates for a smooth Hölder distance when $\alpha \in (0,1)$. In both cases, we also obtain convergence rates for moments when they exist.

We prove quantitative pathwise stability estimates for one-dimensional stochastic differential equations driven by a common Brownian motion and a common symmetric $\alpha$-stable process, where $\alpha\in(1,2)$. The comparison is made under a synchronous coupling and is measured by $\sup_{0\le t\le T}\mathbb E|X_t-\wt X_t|^{\alpha-1}$. The perturbed drift and Brownian diffusion coefficients are spatially Lipschitz, while the perturbed stable jump coefficient may be Hölder continuous down to the critical exponent $1/\alpha$. The estimate is expressed in terms of the initial error and three law-weighted coefficient errors: the drift error $B$, the Brownian diffusion error $A$, and the stable jump-coefficient error $S$. The Brownian component produces a second-order correction term in Itô's formula. This term is controlled by a second-derivative estimate for a Komatsu-type mollification of $|x|^{\alpha-1}$. For stable jump coefficients with Hölder exponent $\wt\eta>1/\alpha$, the estimate gives explicit power rates in $B,A,S$; at $\wt\eta=1/\alpha$, it gives a logarithmic rate. The same method yields estimates with predictable forcing errors and time-uniform tail bounds via stopped quasi-martingales.

We study exponential-scale fluctuations of the Hardy-Szegő zero process, investigated by Peres and Virág in the disk setting, in its upper-half-plane realization, which reveals a different probabilistic geometry. This conformally invariant determinantal zero process is equivalent to its disk realization, but the upper-half-plane coordinates make real-translation invariance explicit and single out long horizontal windows as natural observables. For the vertical window from height one to height $a>1$, let $N_a(L)$ denote the number of zeros in the corresponding horizontal window of length $L$. We identify the limiting scaled log-moment generating function explicitly. As a consequence, we prove a large deviations principle for $N_a(L)/L$, with rate function given by the Legendre transform of this limit. We also prove a strong Szegő expansion for the log-moment generating function, including an explicit order-one correction, locally uniformly in the natural complex strip. The proof uses the planar determinantal structure before projection, reduces the problem to a one-dimensional Fredholm determinant, and combines fixed-power trace asymptotics with a Wiener-Hopf comparison.

By means of the theory of compound (non-homogenoeus) Poisson processes we define a general framework which includes most of the generalizations of the Poisson processes, both in the Skellam sense and in the space-fractional sense. We prove that Bernstein subordination is the only time-changing leading to a compound Poisson process. We also consider the case of subordination with inverse Bernstein subordinators. Then, we focus on the integrals of compound processes. We give an explicit representation of the fractional integral of compound Poisson process, showing that for fixed t it is distributed as a compound Poisson random variable. We then extend this result to more general integral forms. Furthermore, we obtain some limit results, explicit forms of the iterated integrals and their governing equation. Finally, we study the integral of compound renewal processes in the Fourier-Laplace domain.

We study the persistence probability of fractional Lévy fields, i.e. the analogue of fractional Brownian motion with generalised (multi-dimensional) index sets. First, we compute the persistence exponent of the hyperbolic fractional Lévy field. The result matches the rate obtained in Molchan (1999) for Euclidean space and the one in Aurzada/Helmer (2026) for the sphere. This enables us to study persistence for fractional Lévy fields indexed by a large class of Riemannian manifolds (whenever that process exists) through a local comparison argument with the spherical and hyperbolic case.

We study a Brownian time change on the unit square whose speed measure is constructed from a Dirichlet eigenfunction expansion with independent non-Gaussian coefficients. For $0<\gamma<\sqrt2$, the measure is obtained by a second-moment martingale argument. Finite coefficient translations induce coherent exponential tilts of the speed measure, and conditioning on the complementary coefficients gives positive Lebesgue densities on every finite-dimensional orbit. Unitary transport along these orbits gives a common-domain analytic family and explicit first-order cluster derivatives. A first-order splitting argument proves almost-sure simplicity, while the local eigenfunction-square identity and a Vandermonde argument give joint densities for all finite vectors of ordered positive eigenvalues.

We determine the exact spectral gaps for both the finite-particle and infinite-particle unlabelled Ginibre interacting Brownian motions. In every finite-particle system, the spectral gap is equal to one, whereas for the infinite-particle system it is equal to two. The infinite-particle dynamics therefore has a strictly larger spectral gap than any of its finite-particle counterparts, revealing an enhancement of spectral gaps in the passage from finite to infinite particle systems.

Motivated by a conjecture of Vaikuntanathan and Zamir, we study the pseudo-mixing of Kac's walk on $\mathrm{SO}(n)$: whether short trajectories are indistinguishable from Haar measure by low-complexity tests. We prove that the first $k$ columns mix in Wasserstein distance in $O(n(k+\log n)\log n)$ steps for fixed accuracy, resolving a conjecture of Oliveira. Combining this with a representation-theoretic variance bound, we show that if $T=\omega(nk(k+\log n)\log n)$, then every degree-$k$ polynomial normalized to have unit Haar variance has expectation under the $T$-step law within $o(1)$ of its Haar expectation. As an application, we show that this pseudo-mixing estimate can be used to prove the effectiveness of a fast Johnson–Lindenstrauss transform with the usual target dimension.

We show that, for Weil–Petersson random closed hyperbolic surfaces of large genus, the normalized weighted count of prime geodesics with norm in the interval $(X,X+H]$ is asymptotically Gaussian, provided $H \leq X$ and $H/\log X\to\infty$ as $X\to\infty$. In particular, it applies to intervals which are not necessarily short.

Let $B_1^n$ be the standard cross-polytope in $\mathbb R^n$, let $g_1,\ldots,g_m$ be independent standard Gaussian vectors in $\mathbb R^n$, and set $G_m=\operatorname{conv}{\pm g_1,\ldots,\pm g_m}$. For $m=n^3$ it is proved that $$ \mathbb P\left\{d_{\mathrm{BM}}(G_m,B_1^n)\geqslant c n^{5/8}(\ln n)^{-5/8}\right\}\geqslant 1-\frac2n $$ for a suitable absolute constant $c>0$. This independently improves the polynomial exponent $4/7$ in Friedland's preceding work. Independent concurrent work of Friedland, which appeared after completion of the present manuscript, obtains the same polynomial exponent with the stronger logarithmic factor $(\ln n)^{-1/4}$ by a different argument. The proof uses Friedland's discretization and conditioning argument together with the $K/U$ decomposition. A selected family of $K$ vectors is suppressed and the remaining $K$ vectors are quotiented out. In the resulting quotient simultaneous bounds are proved for every top-dimensional exterior product formed from the suppressed $K$ vectors and the $U$ vectors. A Dvoretzky-Rogers selection after L"owner normalization converts these determinant estimates into a bound for the minimum volume ellipsoid of the whole projected polytope and Maurey's empirical method then gives the required Gaussian measure estimate.

We study centered quadratic variations of fractionally filtered generalized Hermite processes. For every chaos order $q\ge2$, every admissible anisotropic monomial kernel $g_{\boldsymbol\gamma}$, and every fixed $0<h\le1/2$, we prove that $N^{2h+\theta-1}V_N$ converges to a Rosenblatt process with self-similarity exponent $1-\theta$, without subtracting any chaos projection. Thus the visible roughness $h$ determines the normalization, whereas the latent singularity exponent $\theta$ determines the memory of the limit. The proof uses a sharp paired-filter threshold and charged diagram power counting. At $h=0$, the bare kernel has logarithmically divergent energy. Variance-normalized $h$-regularization and a hard endpoint cutoff have the same residue and converge in finite-dimensional distributions to the universal Gaussian field $G_t=(\zeta_t-\zeta_0)/\sqrt2$. Although $G$ has no stochastically continuous modification, the iterated boundary-first quadratic energies converge to $\sqrt3B$. In the isotropic case, simultaneous limits $h_N\downarrow0$ are governed by $\lambda_N=h_NN^{1/2-\theta}$: $\lambda_N\to0$ yields a Brownian limit, $\lambda_N\to\lambda\in(0,\infty)$ yields an independent Brownian-Rosenblatt sum, and $\lambda_N\to\infty$ yields a Rosenblatt limit after division by $\lambda_N$. We also obtain an explicit polynomial rate in fixed Malliavin-Sobolev norms and functional convergence in little-Hölder spaces.

Competitive interactions can maintain diversity, yet coexistence is often fragile in well-mixed populations, where stochastic fluctuations can lead to extinction. This is the case in non-transitive systems, such as rock-paper-scissors dynamics, where no single type dominates globally. While spatial structure can stabilize these systems by providing refuges in space, it remains unclear whether analogous mechanisms can operate in time in well-mixed environments. Here, we develop a population-genetic framework showing that dormancy can act as a temporal refuge, preserving lineages and preventing collapse to fixation under interaction-driven fluctuations. We introduce a discrete-time Wright-Fisher model that combines generalized seed-banks with frequency-dependent interactions, allowing individuals to inherit their type from potential parents sampled across multiple past generations. This construction provides a tractable framework in which dormancy stores and later reintroduces lost types. In the case of either weak or moderate selection, we prove a multidimensional diffusion limit for the resulting type-frequency process and use it to analyze complex selective interactions. In non-transitive systems, dormancy stabilizes trajectories that would otherwise collapse through stochastic extinction, extends fixation times, and sustains coexistence. These effects cannot be explained solely by an increase in effective population size. Our results show that dormancy introduces temporal memory that qualitatively alters competitive dynamics, stabilizing otherwise fragile systems and enabling long-term coexistence.

We prove probabilistic pointwise convergence to the initial datum for the 2D periodic quintic NLS for data in $H^s(\mathbb T^2)$ with $s > 0$. This is an improvement with respect to the deterministic setting, in which convergence is known to fail if $s < 1/3$. The proof is based on a nonlinear maximal characterization for convergence, Bourgain's linear-nonlinear decomposition and the corresponding nonlinear smoothing for which we employ random tensor estimates.

We study the effect of sparse random reinforcement on near-critical Bernoulli site percolation on the first quadrant of the square lattice. The reinforcement is generated by randomly selecting horizontal and vertical lines and increasing the opening probability $p$ only at their intersections, so that the reinforcement is supported on a sparse set of points with long-range dependence. We determine how the geometry of this random set influences the phase transition. We first investigate whether the reinforcement lowers the critical threshold. When the spacings between consecutive selected lines are geometric, we prove that, for every $p$ above the critical threshold, percolation occurs even if the opening probability $q$ outside the reinforcement is strictly below criticality. In contrast, if the spacings in one direction decay slower than exponentially while those in the other are geometric, then no threshold shift occurs: percolation is impossible for $q$ below the critical threshold, regardless of $p$. We next study the critical case $q=p_c$. We show that sufficiently high moments of the spacing distributions imply percolation at criticality, whereas the absence of a fractional moment of order $\alpha\in(0,1)$ rules it out. Both regimes remain unchanged after an additional independent thinning of the reinforced vertices. Thus, reinforcement supported on random intersections may either lower the critical threshold or leave it unchanged while fundamentally altering the nature of the phase transition.

Anticoncentration is central to hardness arguments for approximate sampling. In the independent Gaussian surrogate for collision free Gaussian boson sampling, the moment ratio studied here also determines the averaged ideal linear cross entropy reference value. Let $H_{k,n}=\mathrm{haf}(X^{\mathsf T}X)$, where $X\in\mathbb{C}^{k\times 2n}$ has independent standard circular complex Gaussian entries. We evaluate $\mathbb{E}|H_{k,n}|^2$ and $\mathbb{E}|H_{k,n}|^4$ exactly by reducing four hafnian copies to a rank two Gaussian integral. For $R_{k,n}=(\mathbb{E}|H_{k,n}|^2)^2/\mathbb{E}|H_{k,n}|^4$, we obtain $R_{k,n}=4^{-n}\binom{2n}{n}/F_{k,n}$, where $F_{k,n}={}3F_2(-n,-n,1/2;1,k/2;1)$ is a terminating generalized hypergeometric polynomial. If $k/n^2\to c>0$, then $F{k,n}\to e^{1/c}I_0(1/c)$, where $I_0$ is the modified Bessel function of the first kind of order zero, and consequently $R_{k,n}\sqrt{\pi n}\to[e^{1/c}I_0(1/c)]^{-1}$. Thus $k\asymp n^2$ is a smooth Bessel crossover, whereas the scaling order boundary for inverse polynomial weak anticoncentration is $k\asymp n^2/\log n$. These conclusions concern the Gaussian surrogate moment criterion; finite dimensional Haar moment transfer and high probability small ball anticoncentration remain separate problems.

We prove a shape theorem for a multidimensional periodic branching diffusion, which is a branching particle system where particle trajectories evolve according to a diffusion, drift, branching rate and offspring distribution which all depend periodically on space. More precisely, we show that almost surely as $t\to\infty$, the particle cloud at time $t$, normalized by $t$, converges in Hausdorff distance to a deterministic convex set $\mathcal{W}$. We further establish that the boundary of the asymptotic shape $\mathcal{W}$ is of class $C^2$, strictly convex, and has positive Gaussian curvature bounded away from zero. This work generalizes the approach of arXiv:2507.10515 for $g$-BBM and furthers our understanding of the shape $\mathcal{W}$ in that setting.