Papers

This study investigates a virtual multiuser multiple-input multiple-output (MU-MIMO) system with PSK modulation, realized with a reconfigurable intelligent surface (RIS)-based transmitter. The study focuses on minimizing transmit power under quality-of-service (QoS) constraints while addressing the associated computational complexity. A discrete phase-shift RIS model is considered, and the power minimization problem is formulated in two scenarios. First, for QPSK user data, the symbol-error probability (SEP) is adopted as the QoS criterion. Second, for general $M$-PSK modulation, the union-bound SEP (UBSEP) is used to define the QoS constraints. Based on the considered formulations, a partial branch-and-bound (PBB) approach is developed, which improves on full branch-and-bound (FBB) methods in the sense of allowing for favorable complexity performance trade-offs. For the special case of high-resolution RIS, the discrete phase-shift set is approximated by its continuous counterpart, enabling the reformulation of the original problems as constrained optimizations on an oblique manifold, which are solved with reduced computational complexity with the proposed bisection method. Numerical results demonstrate the effectiveness of the proposed approaches in minimizing the transmit power for different SEP requirements and showcase the balance between power efficiency and computational complexity

We establish a mesoscopic local law and quantitative eigenvector-delocalization estimates for the Sachdev–Ye–Kitaev model (SYK) Hamiltonian of $N$ interacting Majorana fermions subject to a $q$-body interaction. For even $q\ll N^{\frac{1}{2}}$, we prove that the normalized Stieltjes transform is approximated, uniformly on bounded energy intervals, by an explicit deterministic profile which we construct as an asymptotic expansion in the proportion $\alpha_{N,q}$ of anticommuting $q$-body interaction terms. The result holds both on the full Hilbert space and in each fermion-parity sector. We derive mesoscopic eigenvalue counting and spectral form factor estimates. For fixed $q$, we further obtain high-probability $\ell^\infty$-delocalization bounds in any deterministic orthonormal basis for individual bulk eigenvectors. These are the first mesoscopic laws and eigenvector delocalization estimates for the SYK Hamiltonian, which we obtain using the resolvent method.

Let $\{0, 1\}^n$ be the Boolean cube, endowed with the probability product measure, where ${\Bbb P}(1)=p$ and ${\Bbb P}(0)=q$ with $0 < p \leq q$ and $p+q=1$. For $i=1, \ldots, m$, let $\phi_i: \{0, 1\}^n \longrightarrow {\Bbb C}$ be $L_i$-Lipschitz functions in the Hamming metric, such that each $\phi_i$ depends on at most $r$ coordinates of $x \in \{0, 1\}^n$, where $rp \geq 12$. For $j=1, \ldots, n$, let $I_j $ be the set of indices $i$ such that $\phi_i$ depends on the $j$-th coordinate. We prove that $E \exp\left\{ \sum_{i=1}^m \phi_i \right\} \ne 0$ provided $\sum_{i \in I_j} L_i \leq {1 \over 10 \sqrt{rp}}$ for all $j$. This translates into a regime for $\pm 1$ spin systems, where a linear increase in the energy of multi-spin interactions requires only a logarithmic increase of the external field to keep the partition function zero-free and the system away from the phase transition. As a corollary, we obtain efficient deterministic algorithms to approximate the partition function in the zero-free region.

We develop a unified geometric and probabilistic theory of conformal type for unimodular random infinite trivalent hyperbolic polyhedra in $\mathbb{H}^3$. By corresponding these with dual angled disk triangulations and regular circle patterns, we associate to each face an intrinsic geometric characteristic number $L_f(P)$, determined entirely by local dihedral geometry. For the root face $f$, we establish the unimodular Gauss-Bonnet formula $\mathbb{E}[L_f(P)] = 2\pi - (\pi/3)\mathbb{E}[deg(f)]$. Under natural tameness and admissibility assumptions, this yields a sharp dichotomy: a unimodular random trivalent hyperbolic polyhedron is parabolic precisely when $\mathbb{E}[L_f(P)] = 0$, and hyperbolic when $\mathbb{E}[L_f(P)] < 0$. Thus, global conformal type is governed by the expectation of a local geometric quantity. We also investigate the approximation of infinite polyhedra by finite ones. We prove that every admissible Benjamini-Schramm limit of uniformly face-rooted finite trivalent hyperbolic polyhedra is necessarily parabolic, revealing a geometric and topological obstruction to the existence of hyperbolic unimodular polyhedral limits. To study stochastic behavior in the hyperbolic regime, we overcome the failure of classical circle packing tools for unbounded degrees by establishing a refined ring lemma for regular circle patterns. This yields effective exponential control of adjacent circle radii via local flower degrees. Combined with boundary methods, we identify the Poisson boundary with the circle at infinity and prove positive hyperbolic speed for the face random walk. These results provide the first quantitative framework connecting local three-dimensional dihedral geometry, global conformal type, and asymptotic stochastic behavior of unimodular random infinite hyperbolic polyhedra.

We study self-avoiding walks on the canonical one-sided graphs of Lindstrom nested fractals. We prove that the connective constant $\mu$ exists and identify $\log\mu$ with the critical inverse temperature of a finite-dimensional boundary-state renormalization. If the boundary-state partition vectors are bounded at criticality, then the fixed-length counts $c_n$ satisfy two-sided polynomial bounds around $\mu^n$. We also prove that $h$-flexibility implies $c_{n+h}/c_n\to\mu^h$. For regular polygonal $N$-gaskets, we derive exact crossing recursions, determine the smallest flexibility step $h$, and obtain explicit algebraic connective constants for the $6$- and $9$-gaskets. The Vicsek graph has no flexibility step, and its successive ratios do not converge.

We revisit the “collapse to the mean” phenomenon, which refers to mild structural conditions, such as local linearity, that force a law-invariant functional $\ph$ defined on finite-mean random variables to depend solely on the expectation of its argument $X$, and not on any other distributional feature. Starting from a concise characterisation of the convex order, our simplified approach unifies and extends existing results without assuming the functional to be convex or monotone in the almost-sure order, and clarifies the conceptual foundations of the “collapse to the mean" phenomenon. In addition, we establish a new “dual collapse” result for quasi-star-shaped functionals.

In recent work, Marcussen, Rubinfeld, and Sudan introduced the notion of quality control problems, which aim to capture the task of determining if a given input is truly random. Formally, their goal is to accept typical inputs from the specified distribution while rejecting every input whose value of a specified statistic is far from the distributional baseline. This captures the empirical practice of using specified statistics as a proxy for the quality of randomness. Empirical algorithms, however, have not exploited the asymmetry in the definition of quality control problems, which require soundness guarantees in the worst-case while only seeking average-case completeness. Their work abstracted a problem definition emphasizing this asymmetry and used it to give efficient quality control algorithms for assessing the randomness of graphs. In this work, we introduce and study quality control problems over sequences, where the goal is to distinguish a sequence of i.i.d. characters from sequences where some specified pattern appears too often (or too infrequently) as a subsequence. We consider this problem in both the finite-alphabet setting and for real-valued sequences. We refer to the former setting as the pattern counting problem. In the latter case, the natural notion of a pattern is to consider the relative ordering of the characters in the subsequence, and we refer to this as the permutation pattern counting problem. Algorithms to approximately count (permutation) patterns of length $k$ in a worst-case sequence of length $n$ can provably require exponential in $k$ queries into the sequence. In contrast, we show that by taking advantage of the asymmetry in the definition of quality control, we give algorithms that run in poly$(k)$ time to solve these problems. We also prove that any quality control algorithm (over some natural distributions) requires superlinear queries in $k$.

We introduce a McKean–Vlasov semigroup in the presence of common noise and study its first-order differential structure. Given a terminal functional $G$, the semigroup is defined by propagating the initial law through a reference conditional McKean–Vlasov flow. Using a finite-particle approximation and a diagonal/off-diagonal decomposition of the tangent processes, we identify the Lions derivative of the propagated functional and derive a backward representation along arbitrary admissible conditional law flows. This formula involves only first-order derivatives in the measure variable and yields both a Feynman–Kac representation for linear equations on the Wasserstein space and, in the driftless case, a martingale representation problem whose unique solution is given by the McKean–Vlasov heat semigroup and its Lions derivative. We also discuss mixed finite-dimensional/Wasserstein representations and discrete path-dependent extensions.

We establish a simple characterization of amenability of second countable locally compact quantum groups $\mathbb{G}$ in terms of the long-time behavior of convolution semigroups of states on $\mathbb{G}$. We then investigate the deeper problem of combining amenability of $\mathbb{G}$ with other approximation properties of the dual quantum group $\hat{\mathbb{G}}$. Specifically, we characterize the combination of ("strong") amenability of $\mathbb{G}$ with either the failure of property (T) or the presence of the Haagerup property for $\hat{\mathbb{G}}$. These results, which appear to be new even in the classical setting of locally compact groups, produce convolution semigroups on $\mathbb{G}$ whose behavior is controlled both as time goes to zero and as time goes to infinity.

Environmental DNA (eDNA) has emerged as a novel tool for quantifying the seasonal abundance of aquatic species in water bodies; however, its mathematical modeling is still at a germinating stage because of its mechanistic uncertainties. We propose a first-step mathematical and computational framework for the eDNA dynamics of migratory fish based on a novel stochastic partial differential equation model with a delayed source input. The model governs spatiotemporal eDNA concentration in rivers where the source comes from a stochastic differential equation for the migration dynamics of the fish. The affine nature of the model facilitates its theoretical analysis, including the guarantee of well-posedness and the closed-form derivation of the Laplace functional despite the proportionality coefficient of the multiplicative noise term being non-Lipschitz. We also propose a discretization scheme for the model that theoretically generates nonnegative numerical solutions. We finally apply the proposed model to eDNA concentration data sampled from midstream reaches of a river system and perform sensitivity analysis.

For a Markov kernel $T$ with an invariant probability measure $\pi$, we give a self-contained proof of the Markov chain convergence theorem via a criterion called asymptotic equivalence with the target. It assumes two parts about the Lebesgue decompositions of $T^n_x$ and $\pi$ for every starting point $x$: 1.) asymptotic absolute continuity: the singular mass $\mathrm{sing}(T^n_x \mid \pi)$ tends to $0$; and, 2.) asymptotic domination of the target: the singular mass $\mathrm{sing}(\pi \mid T^n_x)$ tends to $0$, as $n \to \infty$. Assuming a jointly measurable density for the absolutely continuous part of each iterate $T^n$ w.r.t. $\pi$, this criterion is sufficient and necessary for convergence. A positive minorant density version of it is verified in three cases: i.) $T$ has a positive transition density w.r.t. $\pi$; ii.) $T$ consists of an absolutely continuous part with positive transition density together with an atom at the starting point, which covers the Metropolis-Hastings algorithm; iii.) the transition density is positive only after a finite number of steps that may depend on the starting point $x$. To demonstrate our general criterion, we investigate the Gibbs sampler with random scan and the parallel tempering algorithm. Furthermore, we show that in all mentioned settings Birkhoff's ergodic theorem applies, so as to obtain the strong law of large numbers. Throughout this paper, neither irreducibility, nor aperiodicity, nor recurrence, nor couplings, nor splitting constructions, nor small sets are used. In all results, the state space is a general measurable space with no structure beyond a $\sigma$-algebra. That joint measurability is assumed of the Markov kernel, not of the space; countable generation supplies it. None of the theorems proved here is new; what is offered is a short route to a single, widely applicable Markov chain convergence criterion.

The classic embedding problem for finite-dimensional Markov matrices has a natural counterpart for sub-stochastic matrices, which is analysed and discussed here. One necessary and sufficient characterisation of embeddability can be given via the unique extension of a sub-stochastic matrix to a stochastic one with one added state in conjunction with the embedding of this extension. This is then explicitly treated for $d\leqslant 3$.

For a rectangular box with edges $a_1,a_2,a_3$, the width in a uniform random direction $u \in S^2$ is $w=\sum_i a_i|u_i|$. Under a bijective change of parameters this is also the projected area of a rectangular parallelepiped, whose distribution was derived by Walters. We give a direct co-area derivation that recasts the law as one global positive-part formula and integrates it to an elementary closed cumulative distribution function. The density is real-analytic except at the edge lengths, face diagonals and space diagonal. We classify every singularity as a corner, square-root fold, superposition or terminal jump and compute its coefficient. A Gaussian representation gives all moments from one generating function. The degree of the $n$th cumulant in $\pi^{-1}$ is exactly $\lfloor n/2 \rfloor$; at every order the mean-normalised cumulant is a polynomial in two scale-invariant combinations of the intrinsic volumes, with explicit formulas given through fifth order. The first three cumulants determine the box, and we characterise the admissible triples. Beyond boxes, intrinsic volumes determine neither width variance nor brightness variance. The sign-pattern representation extends to bodies whose central symmetral is a zonotope, while the cellwise co-area method applies to every full-dimensional polytope.

We study some geometric consequences of the discrepancy between weak and total variation convergence of probability measures. We consider a sequence of probability measures on $\mathbb R^d$, admitting densities with respect to the Lebesgue measure, that converge weakly to a limiting measure but stay bounded away from it in total variation distance. We show that the sets on which the sequence passes from below to above the limiting density must grow unboundedly in perimeter, as measured by the $(d-1)$-dimensional Hausdorff measure. Moreover, this growth persists within a fixed compact set, so that it must reflect an increase in the geometric complexity of these sets rather than only an unbounded expansion in ambient space. We further provide a sufficient condition under which the number of connected components of the sets diverges, recovering a behavior that is closely reminiscent of the one-dimensional case, in which the number of oscillations of the sequence of densities around the limit grows without bound. We also show that this condition cannot be dispensed with in general, by means of an explicit sequence of measures in the plane whose passing sets remain connected in a single component at every stage while growing in length and complexity. Another sequence, built from cosine oscillations, illustrates the complementary behavior, in which the number of components diverges.

In this paper, we study the asymptotic behavior of deterministic identification (DID) over binary symmetric channels (BSCs) under vanishing error constraints. By introducing a minimum error parameter, we characterize how different error-decay regimes affect the achievable DID rate. General achievability and converse bounds are derived, with explicit asymptotic characterizations in the large-deviation, moderate-deviation, and central-limit regimes. The achievability analysis combines coding-theoretic constructions with probabilistic concentration techniques, while the converse links statistical distinguishability to the minimum-distance structure of DID codes via total variation and Hamming-type bounds. Our results show that the asymptotic behavior of DID over BSCs is governed by a Hamming-shell concentration geometry of channel outputs, offering insights into the finite-blocklength behavior of deterministic identification over discrete-output channels.

In this paper the asymptotic exponent of coupled Harish-Chandra-Itzykson-Zuber (HCIZ) integrals is studied. We consider $L$ rank-one HCIZ integrals which are coupled through additive exponent terms, with $L>1$ finite. We show that the normalized asymptotic concentration exponent in the rank-one case is, under some constraints, characterized by the operator-valued R-transform. In this way the coupled HCIZ integral provides the operator-valued free analog of the classical Laplace transform. The result can be used to recover the asymptotics of rank-one non-self-adjoint spherical integrals, if the constant matrix is R-diagonal in the sense of free probability. Furthermore, we show that $L$ coupled HCIZ integrals of rank $M(N)=O\left(N^{1/2-\epsilon}\right)$ can be factorized into $M(N)$ rank-one $L$-fold coupled spherical integrals, and characterize the operator-valued R-transform for R-cyclic matrices on the algebra of real, diagonal $L\times L$ matrices. The treatment of coupled spin glasses with correlated disorder is enabled by our asymptotic exponent.

We construct a compatible sequence of resistance forms on the graphs approximating the infinite Sierpinski gasket and show that the resistance form given by the limit of the compatible sequence induces the Dirichlet form associated with the Brownian motion on the infinite Sierpinski gasket. Moreover, using this characterization of the Dirichlet form, we obtain an explicit expression of the jump kernel associated with the trace process of the Brownian motion onto the half-line $[0, \infty)$.

We study stopping rules in which each family is assigned a positive integer and stops the first time its number of girls equals that integer plus a fixed nonnegative integer multiple of its number of boys. After all families have stopped, we ask for the expected proportion of boys and the expected boy-to-girl ratio in the combined population, and show that both expectations depend on the assigned integers only through their sum. We obtain exact series for both expectations, allowing the probabilities of a boy and a girl to be unequal. Our formulas recover the known formulas for stopping at the first girl and for stopping when girls first outnumber boys. These expectations can also be recovered from earlier general results on Galton–Watson degree profiles and shifted inverse moments, but the resulting series do not appear to have been recorded in these forms. To our knowledge, our two elementary proofs, one bijective and one probabilistic, are also new.

We give an alternative proof of Yu's Thinning Lemma and of the Law of Thin Numbers using a stochastic variational formula for the relative entropy. We show that this approach yields new convergence rates for the Law of Thin Numbers, complementing and extending previously known results.

We prove a non-lattice local central limit theorem and Edgeworth expansions for the logarithm of the norms of products of invertible independent random matrices. Our conditions include a contraction assumption, an assumption that supports of the matrices are “large enough" and their distributions are sufficiently regular. As a byproduct of the proof we are also able to provide a different proof to the optimal rates in the CLT proved in \cite{MatBE}. Like in \cite{MatBE} we provide several sufficient conditions for contraction.

We introduce a non-commutative Laplace transform between functionals on path space and formal series in a tensor algebra, under which a natural convolution of path functionals becomes an algebraic product of series. Applying it to a random unitary matrix-valued path development - the quantum path signature - we show that the governing planar loop equations take a non-commutative spectral form. We then extend the loop equations to a $1/N$ genus expansion, organised by topological recursion, and obtain a hierarchy of integral equations on path space for the corrections.

Observation-driven filters typically use likelihood-score updates, corresponding to the logarithmic scoring rule. We generalise these updates to negative parameter derivatives of differentiable proper scoring rules, under a declared working family and predictable scaling. The rule determines the conditional risk projection and tail response, scaling converts its derivative into the update, and the autoregressive component determines the composite dynamic centre. Locally, risk curvature and scaling govern mean reversion, while the variance of the scaled innovation governs update noise. Under correct specification, unscaled curvature and score variance coincide for the log score by the information identity, but generally differ for other proper rules. For static parameters, we establish consistency and asymptotic normality under explicit stability and fixed-tuning conditions. In high-frequency scale models, centred updates converge to diffusions, whereas non-centred updates follow deterministic mean flows. For time-varying rule-specific projections, the local tracking error admits an Ornstein-Uhlenbeck approximation up to a stopping time, allowing correlated update and target shocks. Simulations and an international-equity application illustrate how criterion choice affects robustness, adaptation, variance-forecast loss, value-at-risk calibration and probability-integral-transform diagnostics. Predictive density and updating criterion are distinct design choices whose relative performance depends on the target and disturbance.

Score-based diffusion models are typically formulated using continuous-time stochastic differential equations and measure-theoretic stochastic calculus. In this paper, we develop a hyperfinite formulation of score-based generative modeling within the framework of Nonstandard Analysis. Starting from an internal diffusion process on a hyperfinite grid, we derive the associated infinitesimal generator and establish its correspondence with the classical Fokker–Planck equation. We then obtain a hyperfinite backward-mean identity that yields the reverse-time drift and provides a constructive derivation of the reverse-time SDE. Building on these results, we show that minimization of an internal score-matching objective recovers the score function required by the reverse-time dynamics, thereby connecting score estimation with generative sampling directly at the hyperfinite level. Under suitable assumptions, we further derive a hyperfinite Girsanov formula and establish a relationship between likelihood optimization and Fisher-divergence objectives. Finally, we analyze the second-order consistency of the hyperfinite dynamics and show that the leading correction term depends explicitly on the fourth moment of the increment distribution, with the Gaussian value $\kappa=3$ eliminating the leading dispersion contribution. Taken together, these results provide a unified hyperfinite framework for diffusion-based generative modeling–while laying foundations for further extensions–that links discrete grid dynamics, reverse-time diffusion, score matching, and likelihood-based formulations within a common nonstandard setting.

This work introduces an algebraic framework yielding explicit, closed matrix differential-difference equations for eighteen models in the exactly solvable sector of the KPZ universality class across four scaling regimes. By organizing Fredholm determinant data into an overdetermined linear problem on a directed lattice graph, we derive a compatibility system termed the diamond equations. Elementary seed data extracted from the shift structure of the Fredholm kernel provides simple solutions to this system. We then construct a Darboux transformation to compress the infinite-dimensional Fredholm data into a finite-dimensional matrix observable. We show this dressing procedure preserves the diamond equations; consequently the resulting matrix observable obeys the same nonlinear structure as the initial seed data. Verifying a closed nonlinear equation for any specific model thus reduces to checking a handful of linear conditions on its kernel data. Under a scalar reduction, the framework produces variable-coefficient Hirota-Miwa equations for Fredholm determinants, recovering the one-point bilinear equations of the author's earlier work as specializations. To supply the necessary seed data, a product graph construction with admissible propagators builds multipoint data in the fully discrete regime, while Euclidean division in a polynomial quotient algebra handles vertex and polymer models. Finally, we demonstrate the diamond equations are a gauge-equivalent reparametrization of the non-abelian Hirota-Miwa system, a central system in classical integrability theory.

Imaginary Liouville theory was recently proposed as a path integral construction of a non-rational conformal field theory (CFT) with central charge less than one (Usciati et al. 2026). In the present work, we establish Ward identities and Belavin-Polyakov-Zamolodchikov differential equations in this framework, assuming a precise conjecture on the decay of the Laplace transform of Imaginary Gaussian multiplicative chaos. These results provide a first step towards a rigorous implementation of the conformal bootstrap program for conformal field theories with central charge lower than one. This work is the first to investigate the intrinsic properties of Imaginary Liouville theory beyond exact computations.

Recently, rectified flow has emerged as a fundamental framework for large-scale image generation, powering state-of-the-art systems such as FLUX.1 and Stable Diffusion 3. Despite its remarkable empirical success, the computational and statistical guarantees of iterative rectified flow have remained largely unexplored. We address this problem by studying c-rectified flow, a cost-aware class of rectified flow that projects velocity fields onto a gradient class while preserving endpoint marginals. The ordinary rectified flow can fail to recover the optimal transport coupling: in a Gaussian case study, the iteration converges to the optimal coupling if and only if the source and target covariance matrices commute. In contrast, under suitable compactness and uniform-integrability assumptions, iterative c-rectified flow always converges to the optimal transport coupling. We further establish quantitative one-step contraction and exponential convergence guarantees under projection-stability assumptions for both quadratic and strongly convex displacement costs. Finally, under a Hölder ball assumption, we develop new minimax-optimal score estimation rates and show that, when combined with iterative c-rectified flow, they yield a rate-optimal estimator of the optimal transport for the dimension \(d \ge 3\) and a nearly parametric rate for \(d=1,2\).

We develop a perturbative response calculus for spectral problems obtained by changing the speed measure of a fixed symmetric energy form in a Gaussian environment. If Cameron–Martin translation acts by $\mu_{x+f}=e^{\kappa f}\mu_x$, unitary transport identifies the varying $L^2$ spaces and produces a common-domain analytic family. For a positive eigenvalue $\Lambda$, the first-order operator is $-\kappa\Lambda$ times the compression of multiplication by $f$ to the $\Lambda$-eigenspace; its eigenvalues are the derivatives of the analytic branches issuing from $\Lambda$. A countable separation condition and finite-dimensional Gaussian disintegration then give almost-sure simplicity; a response-transversality condition and the inverse function theorem give joint densities for all finite vectors of positive ordered eigenvalues. We verify these hypotheses for every $0<\gamma<2$ in two models: Dirichlet Liouville Brownian motion on an arbitrary bounded connected planar domain, and the Liouville–Cauchy operator on the circle. In the first model the whole spectrum is almost surely simple; in the second the constants form the deterministic zero mode and the positive spectrum is almost surely simple. Transversality follows from a local eigenfunction-square identity in the Brownian case and its nonlocal jump-form analogue in the Cauchy case.

Recovering a complex-valued signal from coded diffraction patterns, namely the Fourier intensities obtained after modulating the signal with a collection of masks, is a fundamental structured phase retrieval problem arising in diffraction imaging and related applications. Despite its practical importance, the theoretical analysis of this structured framework remains scarce. In the standard random mask model, the optimal sampling rate achievable by computationally tractable recovery methods has remained open. In this paper, we establish the optimal sampling rate for the PhaseLift feasibility program. More precisely, PhaseLift achieves exact recovery of an unknown signal $\pmb{x}_0\in\mathbb{C}^n$, up to a global phase, from $\mathcal{O}(\log n)$ random masks, with polynomially decaying failure probability. Since $\Omega(\log n)$ masks are necessary to identify certain signals under the erasure mask ensemble, our result thereby achieves the optimal mask complexity. Equivalently, PhaseLift attains the optimal total sampling rate of $m=\mathcal{O}( n\log n)$ scalar intensity measurements. The proof is based on an approximate dual certificate construction via a refined golfing scheme that combines adaptive mask allocation with a dimension-independent truncation threshold.

We provide new estimates in Wasserstein distance for the asymptotic bias of the unadjusted Langevin algorithm, in the classical setting of log-smooth strongly log-concave measures. Our bound implies a Wasserstein mixing time of order $\kappa \sqrt{d}/\varepsilon$, where $\kappa$ is the condition number, $d$ is the dimension, and $\varepsilon$ is the target precision: this improves by a factor of $\sqrt{d}/\varepsilon$ over the previous state-of-the-art results.

We prove a Bismut-Elworthy-Li formula for linear stochastic evolution equations in Banach spaces. The admissible directions are those for which the deterministic orbit is square integrable with values in the Hilbertian range of the noise coefficient. Under this square function condition, the derivative of the semigroup is represented by a stochastic integral against an explicit deterministic control. The result is a concrete control realisation of the Cameron–Martin formula for the fixed-time Gaussian transition measures.