Probability Papers

ALT We study rough differential equations (RDEs) driven by Brownian (Itô) rough paths with uniformly locally integrable drifts. The diffusion coefficient is assumed to be uniformly elliptic, of class C²⁺ in space and of bounded variation in time, while the drift is q-integrable in time and p-integrable in space, uniformly on balls of a fixed radius, for some p,q satisfying the Krylov–Röckner condition (d/p)+(2/q)<1. We show that, for almost every realization of the Itô rough path lift of the Brownian motion, such an RDE has a unique solution for every initial value. In other words, for almost every realization of the Itô lift, the corresponding rough differential equation has exactly one solution in the class of Davie-type solutions, for every initial value; this is the appropriate notion of path-by-path uniqueness for the corresponding stochastic differential equation (SDE) with multiplicative noise. The proof follows the strategy of Shaposhnikov (2016). The principal new ingredients are

ALT We study the large size dynamics of a fully asymmetric spherical Sherrington- Kirkpatrick model, allowing for a distinguished random signal that interacts with the evolving spin configuration. We first derive the large-N dynamical mean-field limit and characterize the law of a typical coordinate by an effective Gaussian process whose covariance must satisfy a self-consistency relation. The resulting formulation is intrinsically nonlocal in time and is not, in its usual form, a closed finite-dimensional dynamical system. Our main contribution is an autonomous spectral representation of this limiting dynamics. We introduce an auxiliary stationary chain of Gaussian replicas whose one-step marginal reproduces the effective field appearing in the dynamical mean- field equation. Applying a matrix-valued Herglotz representation, or equivalently Fourier analysis in the replica index, converts the covariance structure of the chain into a family of finite-dimensional matrix-valued evolution equa

ALT We present a unified treatment of various large deviation principles in the context of random walks induced by actions of countable groups. The fundamental notion undergirding the global perspective is that of weakly almost additive random walks, inspired by earlier works on the metric geometry of group actions as well as by recent developments in the theory of nonadditive thermodynamic formalism. We treat large deviations of the renormalized vector-valued distance function from the origin for groups acting simultaneously, by isometries and in a weakly almost additive fashion, on an arbitrary number of metric spaces. The foremost class of examples is provided by group actions with contracting elements, which encompasses, among others, actions on Gromov-hyperbolic spaces and mapping class group actions on Teichmüller spaces. We further establish large deviation principles for quasimorphisms on countable groups, and for random walks generated by compositions of Euclidean isometries; also

ALT We introduce a general form of Ising–Kac interactions through heat-kernel averaging, with the heat-time parameter replacing the role of the interaction range. From the viewpoint of Dirichlet forms theory, this construction provides an intrinsic analogue of the mesoscopic averaging in classical Kac models when translation invariance is unavailable. We then investigate limits of this model on graph approximations of the Sierpiński gasket. For the associated Glauber dynamics, we establish a hierarchy of scaling limits for the magnetization field. First, at fixed interaction heat time and inverse temperature, we derive a nonlocal hydrodynamic equation on the continuum gasket. Second, as the heat time vanishes and the inverse temperature approaches its mean-field critical value, suitably rescaled hydrodynamic solutions converge to the solution of the deterministic Allen–Cahn equation. Finally, under a joint scaling of graph refinement, interaction heat time, and inverse temperature near cri

ALT A binary-output dice enterprise uses independent rolls of a die with unknown face probabilities 𝐩 (along with fair coin flips) to simulate a coin with bias f(𝐩). We determine which power-law tails, uniform over the closed probability simplex, are attainable for the number N of rolls. For continuous f with values in (0,1) and q>0, some enterprise for f satisfies sup\ₘₐₜₕbfₚℙ\ₘₐₜₕbfₚ(N>n)=O(n⁻q) if and only if f can be approximated uniformly by polynomials of degree k with error O(k⁻²q). When this holds, the same tail bound is attained by an oblivious enterprise, which chooses its number of rolls before rolling. The characterization extends to multiple dice, with total degree on a product of simplices; it then suffices to roll the dice in a fixed cyclic order. For q>1, these enterprises have a uniformly bounded expected number of rolls.

ALT We show that, for any infinite transitive graph G, the infinite cluster after supercritical percolation almost surely inherits the anchored isoperimetry of G. Moreover, the probability of violating the corresponding isoperimetric inequality decays exponentially in the isoperimetric profile of the host graph. This was previously known only for nonamenable transitive graphs, ℤᵈ, and very recently for transitive graphs of polynomial growth. Our proof combines the thinning-and-sprinkling technique with a novel counting algorithm.

ALT We study heterogeneous bootstrap percolation on uniform simple graphs with prescribed degrees of mean d→∞ and deviations at most C√ d. Thresholds are independently one with probability (1-c/d)/d and two otherwise; seeds form an independent uniform set of prescribed size a. Let Tₐ be the terminal number of active vertices and b be one minus the mean number of threshold-one descendants of a reached vertex. Let K∼ d² be the quadratic response coefficient. If b>0 and nb³/K→∞, we prove, uniformly over seed counts a, ℙ(Tₐ≥ n/2)=Φ(ᵃ⁻ⁿᵅᵈᵉᵗ⁄w)+o(1) with w=√(nb/K), where α_det is the exact empirical response-barrier center and Φ is the standard normal distribution function. Put a₀=(n/K)¹/³. If d⁴/n→0, a/a₀→ x>0 and λ=ba₀→ℓ∈ℝ, then ℙ(Tₐ≥ n/2)→ p(x,ℓ), the one-sided Brownian parabolic survival probability. A sequential absolute-error law joins these regimes. A graphical counterexample shows that replacing the exact center by its leading term can cause limiting probability error 1/2. The proof bala

ALT We give an analytic proof of the comparison conjectured by Das, Łochowski, Matsuda and Perkowski for the fractional Brownian level-crossing constant: for standard fractional Brownian motion, c_H>E|Z|¹/ᴴ when 0<H<1/2 and c_H<E|Z|¹/ᴴ when 1/2<H<1, where Z∼ N(0,1). At H=1/2, both constants equal one. The main tool is Gaussian comparison under ordering of increment variances. It applies because truncated variation is translation invariant and lattice-submodular, as follows from its minimum-total-variation characterization. A tangent variogram produces Brownian motion with an independent Gaussian drift on the upper side, and a periodic Brownian bridge on the lower side. The former has an explicit stationary reflection rate. For the latter we calculate an integrated cyclic crossing count by first-passage Laplace transforms. Elementary hyperbolic, theta-series, and Gamma inequalities then give strict comparisons on both full parameter ranges. No numerical integration, simulation, interval cer

ALT Randomly connected networks are ubiquitous in computer science, physics or biology. A central question is whether, and how, their collective dynamics depend on the choice of the distribution of the random couplings. Forty years ago, Geman and Hwang formulated a precise version of this universality question for randomly connected neural networks. We come back to these conjectures, review what was proven since, provide a few new extensions and propose new conjectures in this paper. For n neuron networks with couplings with a mean scaling as 1/n and light enough tails, two classes of limits are expected. For weakly heterogeneous couplings, with variance o(1/n), the network is predicted to converge to a McKean–Vlasov equation, and a broad literature has established this under general assumptions. For strongly heterogeneous couplings, with variance of order 1/n, the limit is a non-Markovian equation driven by a self-consistent Gaussian process. After landmark results for Gaussian couplings,

ALT We determine the leading small-time Wasserstein action of the full spatial laws of centered KPZ and dynamical Φ⁴₃, both started from zero. The leading term is Gaussian, with an explicit heat-trace constant. A common comparison also determines the ratio of length to displacement. For KPZ, a first-chaos cancellation yields a quadratic coupling error. For Φ⁴₃, two weighted entropies and an adjoint estimate give comparison bounds uniform in the spectral cutoff. Integrating a Gaussian rotation gives a differentiable realization of the full law; the comparison holds at every positive time.

ALT The present work discusses the birthday problem and the birthday person problem, two classical problems in probability theory widely used to illustrate situations in which human intuition diverges from mathematical results. Taking the work of [1] as its main reference, the study analyzes the probability of birthday coincidences within groups of people, considering a year with 365 days and assuming that all birth dates are equally likely. The birthday problem seeks to determine the probability that at least two people in a group share the same birthday, showing that, even in relatively small groups, the probability of a coincidence increases rapidly due to the large number of possible pairwise comparisons among individuals. In contrast, the birthday person problem investigates the probability that a previously selected person shares the same birthday as another member of the group. This probability increases more slowly, since the number of relevant comparisons is restricted to the pair

ALT We study directed last passage percolation with geometric weights in a spatially inhomogeneous environment. We derive an explicit limit shape and establish fluctuation limits for the passage time, showing that, depending on the strength of the inhomogeneity, the limiting fluctuations are governed by Fredholm determinants associated with either the Airy or Bessel kernel. Our approach is based on an exact relation between the passage time and the smallest particle of the q-deformed Jacobi unitary ensemble (q-JUE). As part of our analysis, we derive the macroscopic limiting density of the q-JUE, which provides a natural q-deformation of the classical Wachter distribution, as well as of the limiting distribution of the Meixner ensemble. We further establish Airy and Bessel scaling limits for the smallest particle, which are of independent interest.

ALT [Kosygina et al. 2023] showed that the rescaled polynomially self-repelling walk does not converge to any Brownian motion perturbed at extrema (BMPE). Motivated by this result, we introduce an explicit candidate for the scaling limit, the polynomially self-repelling motion (PSRM), and study its properties. The construction relies on a new family of continuous stochastic flows, called polynomially self-repelling flows, introduced in [Yu 2026]. A key ingredient in relating the limiting process of the rescaled walk to the PSRM is a joint Ray-Knight theorem for rescaled local time processes, extending the classical Ray-Knight theorems of [Tóth 1996].

ALT We study conditioning on long survival for time-inhomogeneous Markov chains on finite state spaces. We assume that a sequence of substochastic transition kernels converges to an irreducible substochastic kernel and that the transition probabilities admit asymptotic expansions up to order n⁻². Under these assumptions, we construct a positive space-time harmonic function and prove that the limiting path distributions under conditioning on long survival are given by the corresponding Doob h-transform. We also investigate the long-time behavior of the resulting conditioned process. Finally, we apply our general theory to an elephant random walk, a generalized Pólya urn model, time-inhomogeneous Wright–Fisher and Moran models, and noncolliding random walks.

ALT Let p∈(0,1/2] be fixed, and let d,n be positive integers satisfying d=pn. Let Qₙ be an n× n random combinatorial matrix whose rows are independent and uniformly distributed over the vectors in {0,1}ⁿ having exactly d ones. For an arbitrary deterministic invertible n× n matrix M, we establish two-sided bounds for the smallest singular value of QₙM. More precisely, for every ε∈(0,1), ℙ( ε≤ sₙ(QₙM)⋅ ‖M⁻¹‖ₕₛ ≤ ε⁻² ) ≥ 1-C( ε+d⁻¹/²+e⁻ᶜ ˢʳ⁽ᴹ⁻¹) )-2e⁻ᶜⁿ, where positive constants C, c depend only on p and sr denotes the stable rank. Consequently, for any sequence of deterministic invertible matrices M satisfying sr(M⁻¹)→∞, sₙ(QₙM) is typically of order ‖M⁻¹‖ₕₛ⁻¹. In particular, taking M=𝐈ₙ recovers the known n⁻¹/² scale for sₙ(Qₙ).

ALT We study clusters of the Brownian loop soup on mathbbR³ and of random walk and metric-graph loop soups on mathbbZ^d and mathbbZ̃^d within a unified framework. For the continuum model, when the one-arm exponent ξ(α) is positive, we prove that almost surely every cluster has Hausdorff dimension 3-ξ(α). A key step is to show that, conditioned on the one-arm event, a crossing cluster has capacity comparable to the crossing scale with uniformly positive probability. This gives one-arm estimates of order αr^-ξ(α), uniformly down to zero intensity, and shows that ξ(α) is strictly decreasing and left-continuous. Moreover, 1-ξ(α)asympα as αdownarrow0. As another application of the one-arm estimates, we prove that several critical values coincide, resolving a conjecture of Jego and Lupu in arXiv:2601.04840. The proof combines a decoupling technique with a multiscale recursive argument to control the combined effects of large and small loops. These methods also extend to loop soups on mathbbZ^d a

ALT We introduce a new framework for stochastic variational inequalities with oblique reflections generated by non-symmetric matrix coefficients. The loss of symmetry fundamentally changes the structure of the reflected dynamics: besides the dissipative component, it produces a genuinely gyroscopic component describing rotational effects that are invisible to the associated energy forms. To capture this hidden component, we consider planar obstacle problems and we lift the dynamics to an extended phase-space system, where the dissipative and gyroscopic mechanisms are naturally separated. This leads to a new notion of solution, for which we establish existence and uniqueness. The resulting framework allows for genuinely rotated reflections, which can lead to angle-controlled rotations.

ALT This thesis studies finite projections of transient regenerative processes, constructed by conditioning regeneration cycles to be finite. It develops comparisons between the original and projected laws, including finite-time bounds and best-approximation properties for specified metrics and comparison classes. Applications concern birth-and-death processes, Markov chains and semi-Markov processes, with explicit queueing examples. Structural results describe projections associated with different regeneration states, invariant measures and connections with Martin boundary kernels. The thesis also studies reconstruction of birth-and-death rates from passage-time laws and continued-fraction representations of Laplace-Stieltjes transforms for skip-free semi-Markov processes, together with truncation-error bounds. This is a corrected English edition of a Romanian doctoral thesis completed in 1998 and defended in 1999. The preface distinguishes the historical work from corrections and later a

ALT We analyze the limiting dynamics of two copies of a disordered mean-field spherical p-spin system coupled non-reciprocally to one another and initialized independently of the disorder. We prove that, on every finite time interval, as the system size tends to infinity, the empirical correlation and integrated response functions converge almost surely and in Lq for q ≥ 1, uniformly in time, to the unique non-random solution of a system of coupled integro-differential equations. For even p-spin models, we establish stationarity and exponential decay of the correlation and response functions in the high-temperature regime. In the special case of the spherical Sherrington–Kirkpatrick model, we derive explicit representations of the limiting correlation and integrated response functions. These representations allow us to characterize the long-time dynamics throughout the full range of temperatures. In particular, we prove the occurrence of aging in the low-temperature regime and fluctuation-

ALT We establish strong Gaussian approximations for centered stationary random fields on ℤᵈ, d≥2, represented as measurable functions of i.i.d. innovations. Under finite pth moments, p>2, and sufficiently fast decay of spatial dependence, we construct a single coupling of the field with an i.i.d. Gaussian field. Its partial sums, scaled by the long-run standard deviation, approximate those of the original field almost surely, uniformly over all rectangular prefixes and all observation scales. Over a cube of side length n, the approximation error is Oₐ.ₛ.(nᵐᵃˣ{ᵈ/ᵖ⁽ᵈ⁻¹/2})) up to logarithmic factors. While d/p is the familiar moment term, the surface term (d-1)/2, absent for independent planar arrays, emerges as a cost of dependence, and cannot be removed even for finite-range Gaussian fields. The assumptions accommodate nonlinear and noncausal fields without imposing mixing. The proof combines a permanent dyadic blocking, a local Gaussian coupling in tensor Haar coordinates driven by a Doob