Papers

Consider a random geometric graph $G_M(n;r)$ on a compact Riemannian manifold $M$, whose vertices are a cloud of $n$ independently sampled points, and whose edges connect vertices at distance $\le r$. We show that, in the thermodynamic (i.e. bounded expected average degree) regime, if $G_M(n;r)$ is supercritical in the sense of continuum percolation, then the first homology group $H_1(M)$ of $M$ can be correctly inferred from $G_M(n;r)$ with high probability as $n \to \infty$. Specifically, one can obtain $H_1(M)$ by taking the cycle space of $G_M(n;r)$ and quotienting out all the cycles of metric diameter $O(r|\log r|)$ (or of graph diameter $O(|\log r|)$). Our method of estimating $H_1(M)$ exploits a coarse-topological fact about supercritical percolation, as opposed to usual methods, which examine the topology of neighborhoods of the point cloud. Whereas previous methods use combinatorial models which require $O(n \log n)$ edges, our method only requires $O(n)$ edges. We also show that, in all phases of the thermodynamic regime, if one instead takes the quotient by cycles of metric diameter $o(r|\log r|)$, with high probability, one will not recover $H_1(M)$. Thus $\Theta(r|\log r|)$ is the “right scale.” On the way, we show that an arbitrary compact $d$-dimensional Riemannian manifold has a first homological percolation threshold in the sense of Bobrowski and Skraba \cite{BS2020} which coincides with the continuum percolation threshold on $\R^d$, a result previously only known for the flat torus. This strongly suggests that our results are optimal, in the sense that $H_1(M)$ cannot be inferred from $G_M(n;r)$ in the subcritical thermodynamic regime. All results hold for homology with arbitrary coefficients.

Hessian-free high-resolution (HFHR) dynamics augments underdamped Langevin dynamics (ULD) with reversible position diffusion for sampling problems that arise in machine learning. We establish an explicit quantitative contraction rate for HFHR dynamics under a position Poincaré inequality, weighted Hessian and Laplacian bounds, and a compact Sobolev embedding, where the potential function is not necessarily convex. An adapted time-augmented Poincaré inequality yields an explicit rate that improves upon the contraction rate of the underdamped Langevin dynamics. We also give a weak-solution construction and a self-contained spectral proof of the divergence lemma underlying the argument. For HFHR Monte Carlo (HFHRMC) algorithm, which is based on a discretization scheme of HFHR dynamics, we use a path-space Girsanov argument to obtain a non-asymptotic convergence bound and an explicit iteration complexity in total variation distance. The bounds hold for every $\alpha\geq0$ and $\gamma>0$ and remain regular at the ULD endpoint. Optimizing the iteration complexity bound yields a positive, accuracy-dependent position-diffusion parameter at finite accuracy, while its leading high-accuracy order coincides with that of the optimized ULD endpoint. Our iteration complexity bound improves upon the existing work on HFHR algorithms. Numerical experiments including Bayesian learning problems on real data are provided to illustrate the effect of positive $\alpha$ and its benefit.

We study the Ising model on $\mathbb{Z}^d$ with $d\geq 4$ and derive near-critical bounds on the truncated two-point function $\langle\sigma_0;\sigma_x\rangle_{\beta,h} := \langle\sigma_0\sigma_x\rangle_{\beta,h} - \langle\sigma_0\rangle_{\beta,h}\langle\sigma_x\rangle_{\beta,h}$ at parameters $\beta\leq\beta_c$ and $h\geq 0$. As a corollary, we obtain that the associated mass (or exponential decay rate) is equal to \begin{equation*} \max\bigl((\beta_c-\beta)^{1/2},h^{1/3}\bigr)^{1+o(1)}, \end{equation*} where $o(1)$ tends to $0$ as $(\beta,h)$ tends to $(\beta_c,0)$. The proof combines the corresponding result at $h=0$, recently established by Duminil-Copin and Panis, with an interpolation argument inspired by Aizenman and Fernández and carried out via the random current representation of the model.

For $N\ge2$ and $\beta>0$, consider the canonical 2D Coulomb gas ensemble \begin{equation} \mathrm{d}\mathbb{P}_{N,\beta}(Z) =\mathsf{Z}_{N,\beta}^{-1}e^{-\beta|Z|^2/2} \prod_{i<j}|z_i-z_j|^{\beta/N}\,\mathrm{d}Z \qquad\text{on }\mathbb{C}^N. \end{equation} The factor $N^{-1}$ in the pair exponent places the ensemble at the diffusive, or high-temperature, scale. We prove that, for every fixed inverse temperature $\beta>0$, this Gibbs measure satisfies a logarithmic Sobolev inequality (LSI) on the closed Dirichlet-form domain generated by collision-free compactly supported smooth functions, with a positive constant independent of $N$. The LSI constant may be chosen uniformly when $\beta$ ranges over a compact subset of $(0,\infty)$. A second main result establishes Poincaré and logarithmic Sobolev inequalities for planar Gaussian measures weighted by finite products of positive powers of distances to points, with constants depending only on the total variance of the underlying Gaussian measure and the total exponent, not on the number or locations of the points. To prove the full labeled Poincaré inequality, we decompose variance into permutation-invariant and label-dependent parts, controlled respectively by a weighted $\bar\partial$ estimate and by relative-coordinate conditioning with random transpositions; separately, one-site logarithmic Sobolev inequalities, a conditional entropy inequality, and a repulsive partition estimate give a defective logarithmic Sobolev inequality. Rothaus tightening combines the resulting full labeled Poincaré inequality with the defective logarithmic Sobolev inequality. No uniformity as $\beta\to0$ or $\beta\to\infty$ is asserted.

Two competitors choose starting vertices for independent, constant-speed random walks, and each site is acquired by its first visitor. We study the spatial geometry of the resulting first-passage location game. On every finite path the optimal strategies are exactly the distributions supported on the central vertices. The proof combines reflecting-boundary harmonic barriers with a parameter-uniform aggregate estimate for product-chain exit probabilities. After diffusive rescaling, the complete two-start payoff landscape converges uniformly to the game between independent reflected Brownian motions. Its unique equilibrium concentrates at the midpoint, with explicit cubic stability. Beyond paths, attaching two leaves to every vertex of a clique of order k produces a 3k-vertex graph on which every exact optimal strategy randomizes over all k clique vertices; for k=2, this is a six-vertex tree with no pure equilibrium. The continuum best response to an endpoint is uniquely determined. A first-passage random-ranking representation relates the finite game to maximal lotteries without identifying it with nonstrategic painting, deterministic Voronoi allocation, or absorbing-trap placement. Parameter-uniform statements follow from analytic arguments or symbolic polynomial identities; identified finite exceptions and numerical enclosures have reproducible certificates.

We consider the SIS contact process on a Poisson system of independent Brownian motions in $\mathbb{R}^d$ of intensity $\eta$. Two particles are in contact when their distance is at most $2r$. An infected particle transmits the infection to a susceptible particle in contact with it at rate $\lambda\in(0,\infty]$, and recovers at rate $\mu\in(0,\infty]$, after which it is susceptible again. When $\lambda=\infty$, recovery acts only while a particle is isolated. Let $\eta_c^{\mathrm{B}}$ be the critical intensity of the static Boolean model, below which the contact graph has only finite components at every fixed time. We show that for $d\ge2$, with motion, at every intensity strictly below $\eta_c^{\mathrm{B}}$, the infection survives and spreads at positive speed once the recovery rate $\mu$ is small enough, depending on the intensity; that is, started from a single infected particle, at time $t$ there is an infected particle at distance of order $t$ from the origin. In particular there exists $\mu_\dagger>0$ such that $\eta_c(\infty,\mu)<\eta_c^{\mathrm{B}}$ for every $\mu<\mu_\dagger$, and as $\mu\downarrow0$ the set of intensities at which the infection survives extends to all of $(0,\eta_c^{\mathrm{B}})$. We also show that the critical density is positive at every recovery rate, with a lower bound that does not depend on the infection rate, and that for $d\ge2$, at every intensity and every finite infection rate, the infection survives and spreads at positive speed once $\mu$ is small enough.

Motivated by the recent construction of fractional Gaussian fields on the Sierpinski gasket, we study those Gaussian fields whose covariance function exhibits a logarithmic behavior. We introduce a parametric family of random measures associated with these fields, which can be regarded as the analogue to Liouville quantum gravity. The construction is based on Kahane's approach via Gaussian multiplicate chaos. We establish reflection invariance and scaling self-similarity for both the fields and the measures, and in addition construct the corresponding analogue to Liouville Brownian motion.

We prove Breiman's conjecture under the first-moment assumption. Let $Y_1,Y_2,\ldots$ be iid nonnegative random variables with $\mathbb P\{Y_1>0\}>0$, normalized by their sum. If the resulting randomly weighted sum converges to a nondegenerate law for one fixed integrable, nonconstant mark distribution, then the tail of $Y_1$ is regularly varying. More generally, any full-sequence limit for one such mark, including a constant limit, determines the asymptotic regime of the ranked weights: one big jump, a Poisson$\unicode{x2013}$Dirichlet partition, or dust. It consequently determines the limit for every integrable mark, with convergence in the $1$-Wasserstein metric, and the limits of independently marked empirical measures. The inverse step is based on a countable power-sum theorem: signed Fourier$\unicode{x2013}$Mellin identities extract a positive limiting expected power sum from one nondegenerate marked limit, without a moment of order greater than one. A ratio$\unicode{x2013}$Tauberian argument then recovers the tail index. The same method classifies ratios formed from the marked jumps of a nonzero, unkilled, driftless subordinator at zero and at infinity, assuming infinite activity at zero. A nondegenerate limit is equivalent to regular variation of the Lévy tail with index in $(-1,0]$. A constant limit is equivalent to disappearance of the largest normalized jump, or, analytically, to slow variation of the integrated Lévy tail. The latter condition need not imply regular variation of the Lévy tail with index $-1$. A Cauchy-mark example shows that the conclusion can fail without integrability of the mark.

Collision counts in a dilute gas of hard particles reflect the dependence between successive encounters. We study whether these fluctuations can be represented by a positive linear Hawkes process, in which each event adds a nonnegative contribution to the rate of future events. We construct such a model for particle pairs, require invariance under particle relabeling, and derive its stationary mean and covariance. We prove that, for a stationary positive Hawkes process with finitely many event types and finite expected total family sizes, the long-time growth rate of the third centered moment of any count formed by selecting event types is at least that of its variance. More generally, the long-time growth rates are nondecreasing with cumulant order. These predicted growth rates therefore cannot be matched simultaneously to the ones found in literature in approximations that follow velocity and collision count for a particle in an equilibrium gas. Simulations support the covariance predictions and illustrate the dependence of the physical comparison on the observation window. This restriction motivates models that retain the state left by a collision or allow past events to reduce future rates.

This paper studies the longest increasing subsequence (LIS) problem for sequences generated by dyadic-type chaotic interval maps. Starting from a single point $x\in[0,1)$ chosen uniformly at random, we form the order pattern of the first $N$ points of its orbit, with the doubling map as the basic model. Let $\lambda_1^{(N)}$ be the LIS length, equivalently the length of the first row of the Young diagram obtained by Schensted's insertion. We show that $\mathbb E[\lambda_1^{(N)}]/\sqrt N\to 2$, matching the leading asymptotics in the classical Ulam–Hammersley problem for uniform random permutations.

We study second-order asymptotics for optimal thresholds in secretary-type problems. Under a unimodality assumption and a local affine expansion of the discrete payoff increments, we establish a general principle showing that sufficiently accurate rational approximations to the leading threshold proportion yield exact optimal integer thresholds. Continued-fraction convergents provide an immediate application. We also derive an asymptotic rounding law and apply the results to several variants of the secretary problem with fixed and random horizons.

It is shown in this manuscript that a random graph $G$ drawn from the Erdős–Rényi model $\mathcal{G}(n,p)$ with \[ p=p(n)\leq 1/2, \qquad \lim_{n\to+\infty}(np-\log n-\log\log n)=+\infty, \] is a homomorphic core, i.e., every homomorphism from $G$ to itself is an automorphism. This implies tight ETH-based lower bounds of the subgraph isomorphism problem for almost all $k$-vertex patterns with polynomial average degree.

We study a population model with non-overlapping generations, where individuals have a type and retain the memory of the types of all their ancestors. We assume that for each individual, the reproduction law depends on the empirical distribution of ancestral types, unlike for a classical multi-type branching process where it hinges on the parental type only. We further assume the mean reproduction matrix has rank one. Building on techniques from previous works with Bastien Mallein, which themselves draw on foundational contributions by Philippe Flajolet and collaborators in the context of analytic urns, we analyze the population at large generations. In particular, we determine the Malthusian parameter and estimate the expected size of the sub-population whose empirical distribution of ancestral types lies within a given set.

We compute the second mixed moment of elementary symmetric polynomials in the eigenvalues of the random matrix model $AUBU^*$, where $A$ and $B$ are positive, and $U$ is a Haar-distributed random matrix, associated with finite free multiplicative convolution.

We prove absolute continuity of the running maximum $X^{\ast}_T=\sup_{0\leq s\leq T}X_s$ of one-dimensional time-inhomogeneous Lévy–Itô diffusions driven by a Brownian motion and an independent non-truncated pure-jump Lévy process. Using Bismut's directional Malliavin calculus on the Wiener–Poisson space together with the running-maximum criteria of Song–Xie and Nakagawa–Suzuki, we reduce the problem to constructing an admissible direction $\Theta$. The key is to ensure that the directional derivative $D_\Theta X_t$ is strictly positive for all $t\in(0,T]$. We give explicit directions in two regimes. In the uniformly elliptic case with time-inhomogeneous coefficients, a purely Brownian perturbation yields an explicit positive integral representation for $D_\Theta X_t$, and hence $X^{\ast}_T$ admits a density without truncating the jump component. In a Brownian-degenerate pure-jump model with a bounded deterministic time-dependent jump weight $\kappa(t)$ that may vanish on subintervals, we prove absolute continuity under the minimal nondegeneracy-in-time condition $\int_0^t \kappa(s)^2\,ds>0$ for every $t>0$ and infinite activity of the Lévy measure. A weighted Poisson positivity lemma is the key new input. Finally, we show that silent initial intervals can create atoms and we derive an explicit atom–density decomposition.

We study boundary free energies and half-space responses in nearest-neighbor disordered Ising models. First, we prove the existence of fixed-depth tangential pressure and identify its derivative almost everywhere with the limiting boundary response. Second, under quenched exponential boundary-response mixing, we prove Gibbs-state selection independence and exponential convergence of finite-depth pressures. We further show that uniform one-spin mixing yields DLR uniqueness and Gaussian normal localization, and verify the required mixing conditions whenever $(2d-1)\mathbb E\tanh(\beta\vert{}J\vert{})<1$. Finally, we establish an all-face surface limit in the bounded Dobrushin regime and provide counterexamples that delimit general surface and stiffness claims.

We prove an optimal Hardy inequality for every aperiodic, symmetric random walk in $\mathbb{Z}^2$ with finite variance. In particular, we verify null-criticality, and thus, optimality of the underlying Hardy weight. Under suitable moment conditions, we use fine asymptotics of the potential kernel due to Fukai and Uchiyama in order to derive the asymptotics of the weight. For the standard Laplacian, we recover the expected first order term in the asymptotics but also show that next order term is negative. Thus, our result shows that the constant in the Hardy inequality proven by Kapitanski and Laptev cannot be larger than $1/4$, which is the optimal constant in the continuum. The proof of null-criticality rests on a new criterion for general graphs beyond the locally finite case. We also recover the situation of $\mathbb{Z}^d$ with $d \geq 3$ which can also be also treated by our new method.

Let $A_1,\ldots,A_T$ be independent uniformly selected subsets of a finite population of size $n$, with prescribed cardinalities $m_1,\ldots,m_T$. For each population element, define its occupancy level as the number of selected subsets containing it. Let $x_t$ and $x_{\geq t}$ denote the numbers of elements with occupancy exactly $t$ and at least $t$, respectively. The 2025 work introduced a general multi-set allocation occupancy (MAO) representation for higher-order occupancy moments. The present paper establishes its joint-probabilistic interpretation and provides a rigorous unified derivation for arbitrary joint occupancy categories and moment orders. Specifically, for arbitrary $B_1,\ldots,B_\ell\subseteq\{0,1,\ldots,T\}$, we prove the exact representation $F_\ell(B_1,\ldots,B_\ell)=G_T(B_1,\ldots,B_\ell)/(n)_\ell^{T-1}$, where $G_T(B_1,\ldots,B_\ell)$ is the corresponding generalized MAO transversal sum. This identity unifies the joint factorial moments of arbitrary occupancy categories within a single exact finite-population framework. In particular, writing $B_{\geq t}=\{t,t+1,\ldots,T\}$, the factorial moments of exact- and threshold-occupancy counts are obtained as the specializations $\mathbb{E}[(x_t)_\ell]=F_\ell(\{t\},\ldots,\{t\})$ and $\mathbb{E}[(x_{\geq t})_\ell]=F_\ell(B_{\geq t},\ldots,B_{\geq t})$. Mixed factorial moments, raw moments, variances, and covariances follow from the same representation through standard transformations. The formulas are verified by exhaustive enumeration over feasible parameter ranges and by Monte Carlo simulation. The resulting theory provides a rigorous and unified finite-population foundation for exact and threshold multi-set occupancy statistics.

We prove that, on every connected bounded-degree graph, the infinite collision property holds in discrete time if and only if it holds in constant-speed continuous time, and the same equivalence holds for the finite collision property. The proof is based on a pointwise comparison between the Green kernels of the synchronous discrete-time pair chain and the asynchronous pair chain obtained by Poissonization. The main estimate exploits the binomial interlacing of the coordinate updates together with a local binomial estimate. We then use Martin capacities and collision zero–one laws to relate this Green-kernel comparison to infinite visits of the diagonal.

The classical Borell–Brascamp–Lieb inequality for multiple functions is an integral inequality relating finitely many input functions to a single output function. In this paper, we give an extension that allows a distinct output function for each input function. More precisely, we establish the following inequality. For a weight $ \lambda = ( \lambda_1 , \dots , \lambda_m ) $, we set $ z_\lambda ( x ) := \sum_{ i = 1 }^m \lambda_i x_i $ and denote by $ M_p^\lambda $ the weighted power mean of power $ p $. In addition, the power transform $ Q_d $ is defined as the continuous extension of $ p \mapsto p / ( 1 - d p ) $. We consider integrable functions $ f_1 , \dots , f_m , g_1 , \dots , g_m \colon \mathbb{ R }^d \to \mathbb{ R }_{ \geq 0 } $ satisfying $ 0 < \| f_i \|_1 < \infty $ for every $ i = 1 , 2 , \dots , m $. Then one has \[ \operatorname*{ess\,inf}_{ x = ( x_1 , \dots , x_m ) } M_p^\lambda \left( \frac{ g_1 ( z_\lambda ( x ) ) }{ f_1 ( x_1 ) }, \dots , \frac{ g_m ( z_\lambda ( x ) ) }{ f_m ( x_m ) } \right) \leq M_{ Q_d ( p ) }^\lambda \left( \frac{ \| g_1 \|_1 }{ \| f_1 \|_1 }, \dots , \frac{ \| g_m \|_1 }{ \| f_m \|_1 } \right) \] for any $ - \infty \leq p \leq 1 / d $, where the essential infimum is taken over the set of points satisfying $ f_i ( x_i ) > 0 $ for every $ i = 1 , 2 , \dots , m $. When all output functions $ g_1 , \dots , g_m $ are equal, this recovers the classical Borell–Brascamp–Lieb inequality for multiple functions. When $ p = 0 $ and the hypothesis is imposed pointwise, it coincides with a special case of the multi-output Prékopa–Leindler inequality of Cordero-Erausquin–Maurey.

We prove a free probabilistic Bakry–Émery criterion for logarithmic Sobolev inequalities and hypercontractivity. Our approach relies on a noncommutative version of the carré du champ operator and its iteration, which we are able to define in the free probability setting. This allows us to formulate a positive curvature condition, which is shown to be sufficient for logarithmic Sobolev inequalities and hypercontractivity in this setting. In the setting of free Gibbs measures, we also show that this curvature condition can be expressed in terms of the positivity of the Hessian of the associated potential.

Let the vertices of a graph be revealed one at a time in a uniformly random order, and let the last-isolation time be the last time at which the induced graph on the revealed vertices contains an isolated vertex. For the cycle C_n, write K_n for the number of vertices still unrevealed at this time. We obtain an exact finite-n tail formula for K_n in terms of Stirling numbers of the second kind, together with a compact bivariate generating function. The enumeration comes from reversing the process. An isolated revealed vertex then corresponds to the cyclic pattern 101, but the tail event requires this pattern to be absent from every earlier prefix, not merely from the final set. This prefix condition forces the final one-blocks to be separated by zero-gaps of length at least two and forces the reveal order inside each one-block to be peakless. Since a block of size m has 2^{m-1} peakless orders, summing over block sizes produces the Stirling numbers. We also prove K_n/sqrt(n) -> R, with P(R>x)=e^{-x^2}, and, for every fixed d>=1, K_{n,d}/n^{1-1/(2d)} -> W_d, with P(W_d>x)=e^{-x^{2d}}. Uniform stretched-exponential tail bounds imply convergence of all fixed positive moments. Thus the ordinary cycle has a Rayleigh limit, while its fixed powers give a Weibull family with shape parameter 2d.

This is the third paper in a series on threshold cascades of coupled Ornstein-Uhlenbeck diffusions on graphs converging Benjamini-Schramm to a unimodular Galton-Watson tree. In arXiv:2608.15987 the reduction of the cascade to a finite-type Galton-Watson process (front decoupling) was proved unconditionally in the dissipative regime kappa < 1, where kappa = gamma L_sigma Delta_max / alpha_min is the single-step dissipation ratio. At kappa >= 1 the single-step echo bound resonates, and the reduction was left conjectural. We replace it by a multiscale (renormalisation-group) analysis. Coarse-graining into space-time blocks of depth L and duration T, we record each block's boundary-to-boundary gains in a 2x2 matrix with a trajectory channel and a stored-mass channel, fan-out absorbed. Three theorems result. The gain matrices compose under depth and time concatenation, so a renormalised per-level dissipation ratio kappa_eff exists and equals its infimum (Fekete). If the spectral radius of the gain matrix is below 1 at a single finite scale, a finite-horizon, finite-volume condition in principle certifiable numerically, then front decoupling holds at all scales with a quantitative total-variation bound; echoes may resonate locally, yet independence is recovered macroscopically. Finally we split influence steps into front, quiescent and saturated phases, price each step by the adapted band occupation of its source rather than by a global worst case, and deduce kappa_eff <= rho(T) for an explicit channel matrix T. The region rho(T) < 1 strictly contains kappa < 1: near criticality the effective condition degrades from gamma L_sigma Delta_max / alpha < 1 to gamma L_sigma / alpha < 1, and the degree disappears. We do not prove kappa_eff < 1 throughout kappa >= 1, and give a heuristic for why the resonance regime should be a genuine phase boundary.

We prove a strong-universality theorem at the exact fourth-moment threshold for finite families of non-Hermitian random matrices assembled from independent unordered-pair vector atoms. Within one atom, the matrix colors and the two endpoint orientations may have arbitrary joint real covariance, subject to reversal consistency across ordered type pairs and endpoint exchangeability in same-type blocks; the law may also depend on finitely many endpoint types. The matrices may be adjoined to an arbitrary deterministic tuple that converges jointly strongly with the type projections. For every fixed matrix amplification and fixed noncommutative \(*\)-polynomial, the resulting tuple converges strongly to an explicit covariance-matched free Gaussian family. Cross-type blocks are described by masked circular variables, whereas same-type blocks split into independent endpoint-symmetric and endpoint-antisymmetric semicircular sectors. Only a finite radial fourth moment is assumed off the diagonal, and a finite second moment suffices on the diagonal. The mode of convergence depends on the coupling: corners of one infinite array converge almost surely on a common event, while fixed-law nonnested triangular arrays converge in probability for each fixed test. As an application, we obtain exact-fourth-moment strong limits for finite-separable continuous left/right profiles, including separately weighted literal-transpose terms. For the nested almost-sure formulation, the fourth-moment threshold is sharp already on the Wigner subfamily; at this threshold arbitrary fresh rows admit no coupling-invariant almost-sure upgrade of our in-probability conclusion.

Let $s(n)$ denote the number of ones in the binary expansion of an integer $n\in\mathbb{N}$, and let $\mu_t$ be the probability measure on $\mathbb{Z}$ defined by the asymptotic densities of the level sets of the function $\mathbb{N}\ni n\mapsto s(n+t)-s(n)\in\mathbb{Z}$. Let $P_t$ be the family of finitely supported measures defined by the convolution $\mu_t=\mu_1*P_t$. Recently, Tarlowski (2026) has shown that the family $P_t$ may be represented as a recursively grown binary tree $T_t$, and that the Cusick's conjecture - $\mu_t(\mathbb{N})>\frac12$, $t\in\mathbb{N}$, - follows from the asymmetry property of the family $T_t$, which was posed there as an open problem. Next, Cheng (2026) has provided the combinatorial description of the family $T_t$ in the language of principal subsequence ideals, and proved both conjectures. Both of these problems are directly related to the problem of determining the zeros of the function $\mathbb{N}\ni t \mapsto P_t(\mathbb{N})-\frac12\in[0,\tfrac12]$, a problem left open by Cheng (2026) as a saturation problem, and previously analyzed only numerically. In this paper we solve this problem completely. Writing an odd integer $t\ge3$ as $t=(1\,w\,1)_2$ with $w\in\{0,1\}^{\star}$, we show that $P_t(\mathbb{N})=\frac12$ if and only if $w$ is saturated in the following sense: in the block decomposition $w=1^{a_0}\,0\,1^{a_1}\,0\cdots0\,1^{a_k}$ with exactly $k$ zeros, every block of "1" satisfies $a_i\ge k$. Additionally, we show that the lower bound for $P_t(\mathbb{N})$ established by Cheng for $0$-initial words holds true for all non-saturated words.

We consider a many-server queue where tasks with general patience and service times arrive as a renewal process. Contrary to the standard assumptions, we allow for preemption and abandonment during service, relevant in applications such as electric vehicle charging and anytime algorithms in cloud computing. Moreover, we do not assume independence of patience and service times, which has been a critical assumption in the literature. We describe the state of the system using a discrete measure on a two-dimensional orthant, such that the coordinates of its atoms represent the attained service and time in the system of tasks. Under mild assumptions, we derive the fluid limit as the number of servers approaches infinity and the arrival rate of tasks grows proportionally. The limit is given by a measure-valued integral transport equation that we solve explicitly when the initial condition is the null measure. We also prove that all solutions, regardless of the initial condition, converge to the same fixed point, which admits a closed-form expression. Our results focus on the preemptive Last-Come-First-Served (LCFS) policy, which has been identified as practically appealing in recent work.

Recent work gives an all-$\beta$, all-order stochastic-zeta representation of the correlation functions of the $\Sine_\beta$ process and determines their leading Vandermonde asymptotics when several variables merge. We compute the first nontrivial correction throughout the regime $m\beta>1$. If $a_1,\ldots,a_m$ are distinct real numbers and \[ V(a)=\sum_{1\le i<j\le m}(a_i-a_j)^2, \] then, as $\varepsilon\to0$, \[ \rho^{(m)}_\beta(\varepsilon a_1,\ldots,\varepsilon a_m) =C^{(m)}_\beta |\varepsilon|^{\beta\binom m2}\prod_{i<j}|a_i-a_j|^\beta \left[1-\frac{\beta^2 V(a)}{8(m\beta-1)(2m+1)}\varepsilon^2+o(\varepsilon^2)\right]. \] For $m=2$ the normalized second-order coefficient is $-\beta^2/[40(2\beta-1)]$ for every $\beta>1/2$. The proof combines a finite-$N$ rotational Ward identity, exact Hua–Pickrell trace moments, compact moment bounds for the stochastic-zeta entire function and its derivatives, and a quantitative multivariate expectation–Taylor lemma. As a by-product we evaluate \[ \E_{\HP_{\beta,m\beta/2}}\sum_x x^{-2}=\frac{m\beta}{4(m\beta-1)(2m+1)}. \] The pole at $m\beta=1$ marks the boundary of the present second-moment argument and suggests a transition in the form of the next fusion correction.

We discuss the famous conjecture {$\theta(p_c,{\mathbb Z}^d)=0$} for the Bernoulli site percolation model on $\Z^d$ with $d\geq 3$. We prove that, for $d\geq 3$, in the Bernoulli percolation model on ${\mathbb Z}^d$, at the critical point $p_c$, either there is no infinite cluster or the tail of the finite cluster distribution is not a stretched exponential.

In the Eulerian walker model (also known as rotor walk), each site of the square lattice begins with an arrow pointing to one of its four neighbors. A walker that starts at the origin repeatedly turns the arrow at its current site clockwise by $90^\circ$ and steps in the new direction. Priezzhev, Dhar, Dhar, and Krishnamurthy (1996) introduced this as a model of self-organized criticality and conjectured that, for independent uniform initial directions, the region explored in the first $t$ steps has radius of order $t^{1/3}$. We establish this conjecture and further show that the walker visits every lattice site infinitely often, and that the region it has visited by time $t$, rescaled by $t^{1/3}$, converges to a convex body.

We establish the existence of a stochastic flow on $L^{\infty} (\mathbb{T})$ for the stochastic Allen-Cahn equation with multiplicative noise \[ (\partial_t - \partial_x^2) u = u - u^3 + \sigma (u) \xi \quad \text{on} \quad \mathbb{R}_+ \times \mathbb{T}, \] where $\xi$ is space-time white noise and $\sigma : \mathbb{R} \rightarrow \mathbb{R}$ is sufficiently smooth, bounded, and has bounded derivatives. Our strategy is to obtain pathwise a priori estimates via regularity structures. In fact, we consider a general singular multiplicative equation with superlinear damping, driven by noises of parabolic regularity $\alpha - 2$, for ${\alpha \in (0, 1)}$, which can be lifted to a weakly admissible model. We show that the required estimates hold whenever \[ m > \frac{2 - \alpha}{\alpha} \varepsilon_{\alpha}, \quad \text{where} \quad \varepsilon_{\alpha} = 1 - \alpha \left( 1 - \frac{2}{3 - \alpha} \right) \in (0, 1) . \] Thus the strength of the damping needs to be chosen only as a function of the regularity of the driving noise. Under an additional smoothness assumption on $\sigma$, we show that the stochastic flow is differentiable with respect to its initial condition.