Papers

We study deterministic relative approximation of the total variation distance between high-dimensional distributions given by succinct descriptions. We develop an abstract deterministic approximation framework based on representing the total variation distance as a support function of a low-dimensional zonotope. As applications, we obtain FPTASs for several models. Given two mixtures of product distributions over $[q]^n$ with a total of $K$ component distributions, our algorithm approximates their TV-distance within a factor of $1+\varepsilon$ in time $\widetilde O_K(nq(n/\varepsilon)^{2K})$. We also give an FPTAS for mixtures of $n$-step Markov chains over $[q]^n$ with a total of $K$ component distributions, with running time $\widetilde O_K(nq^2(n/\varepsilon)^{2K})$. Finally, for two latent-tree Ising models with the same underlying tree topology, we give an FPTAS for the TV-distance between their leaf marginals in time $O(|V|^{13}\varepsilon^{-12})$.

We prove the Courtade-Kumar conjecture: among all Boolean functions $f\colon \{-1,1\}^n\to\{-1,1\}$, a dictator retains the most information about a uniformly random input observed through independent binary noise.

We study the fixed-$q$ right-edge asymptotics of the ordinary $q$-Krawtchouk ensemble. For $0<q<1$, finite polynomial duality gives a direct proof of convergence of the correlation kernels to an explicit limiting kernel. The local Jacobi limit is a shifted continuous $q^{-1}$-Hermite expression with an indeterminate moment problem, so the limiting Jacobi expression does not determine a unique self-adjoint operator. A single spectral atom, asymptotically saturating the mass bound, selects the realization and yields strong-resolvent convergence. The limiting kernel is its positive spectral projection and admits spectral-series, Christoffel–Darboux, and partial-theta representations, and the associated dual operator agrees with a two-step spatial Markov operator of Borodin–Corwin. A companion $q>1$ limit is instead essentially self-adjoint, of Al-Salam–Carlitz~I type.

Let $K \subset \mathbb{R}^N$ be a convex body containing the origin in its interior. In this work, we study the diameters of random sections of $K$ and derive upper and lower bounds for them in terms of geometric parameters of $K$. Our bounds hold with large probability, and they offer new insights into this widely studied subject. Our upper bound complements the so-called low $M^*$-estimate and in many cases it is much sharper. The two lower bounds that we give are each of a different nature: depending on the body in question each time, either could be better, and in many interesting cases it matches the upper bound too. Subsequently, we apply our results to determine random diameters of $p$-ellipsoids (images of $\ell_p$ balls under diagonal operators), improving upon previously known results and achieving sharp estimates in many cases. One notable application is to Information-Based Complexity Theory, where we manage to establish a simple (and essentially optimal) dichotomy in response to a very natural conjecture posed by Hinrichs, Prochno and Sonnleitner in 2023. Our solution settles precisely when it is useful to replace the optimal information used for the recovery of vectors from a $p$-ellipsoid with random (Gaussian) information, which can be more practical to obtain.

We show the height at time $t$ of the standard continuous-time ballistic deposition process has variance at most $C_d t, t \geq 1$ in every dimension $d \geq 1$. The crucial idea is to adapt a martingale argument due to Aldous, which gives tail bounds on first hitting times for monotone surface growth processes. Inverting these tail bounds and applying a duality theorem due to Penrose yields a linear variance bound for the height of ballistic deposition.

A random group in the square model $\mathcal{Q}(n,d)$ is given by a presentation with $n$ generators and a uniformly random set of $(2n-1)^{4d}$ cyclically reduced relators of length 4 over them. We prove that if $d>\frac{5}{12}$, then a random group in $\mathcal{Q}(n,d)$ has Kazhdan's property (T) asymptotically almost surely (a.a.s.). This provides a new construction of infinite word-hyperbolic Kazhdan groups when $d\in\left(\frac{5}{12}, \frac{1}{2}\right)$. Moreover, also for $d>\frac{5}{12}$, we show that a random group in the square model a.a.s. has property $(\textrm{F}L^p)$ for an increasing range of values of $p$, and property $(\textrm{F}_X)$ for any fixed uniformly curved Banach space $X$. These results lead to bounds on the conformal dimension of the boundary of a random group in the square model. In the process we devise a method of applying the spectral link criteria for property (T), and other fixed point properties, to the groups presented by relations of length 4. In order to use it for the square model, we verify the required spectral gap hypothesis by the trace method, in the spirit of the work of Broder-Shamir.

Ultra-log-concave distributions are ubiquitous in probability, combinatorics, and statistical mechanics and have been studied extensively. In this paper, we introduce a quantitative high-dimensional extension of this notion, called $\delta$-ultra-log-concavity, for probability measures on $\mathbb{N}^d$ with downward closed support. When $\delta = 1$, this notion coincides with the class studied by Gurvits (2009) via strongly log-concave generating functions, and with the class defined by Anari, Oveis Gharan, and Vinzant (2021) via completely log-concave generating functions; in one dimension, it reduces to classical ultra-log-concavity. We establish several functional inequalities, including a weighted Poincaré inequality, a discrete Brascamp–Lieb inequality, and a weighted Wu-type modified log-Sobolev inequality. Our approach combines integrated Bakry–Émery calculus for a canonical birth-death chain with Poisson stochastic localization, which arises as the time reversal of coordinatewise binomial thinning. We further establish concentration of measure, maximum-entropy principles, and several closure properties for ultra-log-concave measures, and develop applications to queueing models, polymatroids, antiferromagnetic Potts models, and hardcore models. Finally, a lattice scaling limit of the discrete theory yields Poincaré and Brascamp–Lieb inequalities for Laguerre diffusions.

We study minimal sets of torus homeomorphisms admitting an irrational circle factor whose fibres are thin essential annular continua. For totally irrational pseudo-rotations and homeomorphisms in a nontrivial Dehn-twist class, we prove uniqueness of the minimal set when the fibres are Jordan curves on a residual set of base parameters. The same conclusion holds if the fibre cores are Jordan curves, or if the fibres are locally connected, on a nonmeagre set of parameters. The residual hypothesis cannot be replaced by a full-measure hypothesis, even under area preservation and topological transitivity. For every totally irrational rotation vector, we construct such a map with Jordan-curve fibres almost everywhere and uncountably many pairwise disjoint uniquely ergodic minimal Cantor sets. We also construct examples in every nontrivial Dehn-twist class, with prescribed irrational vertical rotation number and bounded deviations. In both families, the Jordan-curve parameters form a meagre set of full Lebesgue measure.

We study the one-parameter family of Hessian metrics $g_\alpha=\alpha h_++(1-\alpha)h_-$, $\alpha\in[0,1]$, formed by convexly combining the log-partition functions (not the densities) of the multinomial and negative multinomial Fisher information Hessians. Four results are proved in full. (1) $g_\alpha\succ0$ for all $\alpha\in[0,1]$; its Gaussian curvature ($n=2$) is $-\tfrac14$ at $\alpha=0$ and $+\tfrac14$ at $\alpha=1$, but non-constant for intermediate $\alpha$, changing sign at a closed-form critical value $\alpha^\ast(w)$, with an analytic uniqueness proof. (2) Viewing $g_\alpha$ as the spatial metric of a Kasner-type volume equation, positive-definiteness forces one conserved quantity $\beta\le0$, while a second, independent quantity $C=4\det K$ governs the equation's branch (bounded oscillatory, parabolic, or unbounded growth) and is not fixed by $\beta$ or positive-definiteness alone; we show this discriminant dictionary is a theorem about the abstract matrix equation, to which $g_\alpha$ supplies initial data of either sign of $C$ without the trajectories remaining on $\{g_\alpha(\theta)\}$ itself. (3) $\Psi_\alpha$ is identified as the log-partition function of a compound distribution: a negative-binomial latent count $M$ (parameter $r=1-\alpha$), augmented by a Bernoulli parity bit and split multinomially, with non-negative base measure for every $\alpha$. (4) The associated Bregman divergence obeys a generalised Pythagorean theorem whose projection onto the symmetric locus is the arithmetic mean, independent of $\alpha$; the endpoints' curvature sign matches the Gauss–Bonnet excess of genuine geodesic triangles. Numerical checks are kept separate from analytic proofs throughout.

This paper presents a probabilistic perspective on large language models (LLMs), developed with the aim of bringing together, in a single self-contained account, tools that are usually treated separately across the literature. LLMs are described through probability measures on the set of sequences of tokens, specified via their autoregressive conditional distributions. Training is formulated as a maximum-likelihood estimation problem, addressed by stochastic gradient methods, while text generation is viewed as the sequential simulation of the resulting stochastic process. The role of the asymmetry of the Kullback–Leibler divergence in text generation is examined in relation with characteristic phenomena such as hallucination and the distinction between statistical plausibility and truth. As a complementary illustration of the same viewpoint, we also discuss diffusion models, built around the score function, which cast generation not as sequential token prediction but as the simulation of a reverse-time stochastic process transforming noise into data both in discrete and continuous time.

We construct a measurable aggregator for the value processes associated with stochastic control problems, where the criterion to optimise is given by solutions to semi-martingale backward stochastic differential equations (BSDEs) with jumps. The results can be applied to control problems where the triplet of semi-martingale characteristics is controlled in a possibly non-dominated case or where uncertainty about the characteristics is present in the optimisation. The construction also provides a time-consistent system of fully nonlinear conditional expectations on the Skorokhod space. We also construct an appropriate path-regularisation of the value function and prove a corresponding dynamic programming principle. The generality we seek allows for the treatment of controlled diffusions, pure-jump processes, and discrete-time processes in a unified setting.

This note provide a proof of Conjecture 2.5 in Fang et al. (J. Appl. Probab., 58(4):851–867, 2021), with the help of GPT-6 Astra. We show that the number of edges lying above zero of a random walk generated from the uniform permutation has the discrete arcsine law. The proof is short but hinges on an unexpected representation. We also make several comments on solving the problem using large language models.

We study row-softmax self-attention with independent Gaussian query and key weights in the proportional regime, at fixed inverse temperature. Hayase, Collins, and Karakida proved Gaussian equivalence for the empirical squared singular-value distribution after removal of the Perron direction. A global law alone does not exclude finitely many nonleading outliers. We prove that no such outliers persist: the rescaled squared singular value $\ell s_k(A)^2$ converges in probability to the upper edge of their bulk law for every fixed $k\ge 2$. In fact, this convergence is uniform over any deterministic sublinear number of leading non-Perron indices. The proof uses an exact decomposition of the softmax normalization, conditions on the key matrix, identifies the conditional covariance exactly with a diagonally conjugated inner-product kernel, linearizes that kernel in operator norm, and applies the outside-support local law of Fan, Ma, Paquette, and Wang. A separate stability argument identifies the finite conditional deformed Marchenko-Pastur edge with the limiting bulk edge. We also derive a scalar formula for that edge throughout the proportional regime and recover the explicit square-model formula of Hayase, Collins, and Karakida, including its physical branch.

We propose a new framework for noncommutative stochastic integration in $L^p$, inspired by Bichteler's notion of $L^p$-integrators, that massively generalizes our previous theory of $L^2$-valued stochastic integration against $L^2$-decomposable processes. Central to our framework is a noncommutative analog of a predictable integrand we call a predictable linear process. We develop basic properties of stochastic integrals against "noncommutative $L^p$-integrators" and establish a useful criterion for a noncommutative stochastic process to be a noncommutative $L^p$-integrator. This criterion applies, in particular, to a rich class of processes we call measured decomposable processes, which includes free Brownian motion and, more generally, the $q$-Brownian motions. We also show that, when specialized appropriately, our framework recovers the classical notion of an $L^p$-integrator up to the fact that the noncommutative theory sees only the modification class of a classical stochastic process. As an application, we use our theory to study noncommutative stochastic differential equations (SDEs) in $L^p$. Under natural local-Lipschitz and boundedness assumptions, we establish the existence and uniqueness of maximal solutions and show that a maximal solution with finite lifetime must blow up in $L^p$ norm. As corollaries, we obtain new existence and uniqueness results for free SDEs and SDEs driven by $q$-Brownian motion. Notably, in the free case, our results apply to coefficients arising through the continuous functional calculus from merely locally Lipschitz scalar functions, as opposed to locally operator-Lipschitz functions.

Let $\Pi$ denote the independence copula and $M,W$ the upper and lower Fréchet–Hoeffding copulas. Catalano and Lavenant (2025) conjectured that $M$ and $W$ maximize the quadratic Wasserstein distance from $\Pi$ among all bivariate copulas. We prove this conjecture and characterize all equality cases: $\mathcal W_2^2(C,\Pi)\leq 1/10$, with equality if and only if $C\in\{M,W\}$. We also determine explicitly the optimal Monge map from $\Pi$ to $M$. The proof is constructive and combines the optimal transport from independence to the diagonal, a sharp convex-order inequality for 1-Lipschitz functions, the conditional convex order, and a coupling construction based on conditional comonotonicity and the supermodular order. As a consequence, we obtain a normalized Wasserstein-based dependence measure that characterizes independence and attains its maximal value exactly for comonotone and countermonotone dependence.

Let $p_n$ be a random polynomial of degree $n$ whose roots are independent and identically distributed according to a rotationally invariant probability measure $\mu_0$ on the complex plane with finite logarithmic moment. If $k_n/n\to t\in(0,1)$, we prove that, as $n \to \infty$, the empirical zero measure of the $k_n$-th derivative of $p_n$ converges weakly in probability to a deterministic rotationally invariant probability measure. We describe the limiting measure explicitly in terms of the radial quantile function of $\mu_0$. This proves a conjecture of Hoskins and Kabluchko [Exp. Math. 32 (2023), no. 4].

We classify, up to quasi-isometry, the large-scale geometry of Galton–Watson trees for every finitely supported offspring distribution. Apart from the trivial finite diameter regimes, we condition on infinite diameter. We find that the remaining classes are the ray class, the full tree class (which includes the binary tree), one chain class $(\mathrm{C}_\Lambda)$ for every possible branching semigroup $\Lambda$, and the bushy class. Two independent trees almost surely admit a root-preserving quasi-isometry when their offspring distributions belong to the same class and are almost surely not quasi-isometric when they belong to different classes. Any two survival-conditioned supercritical realisations with finitely supported offspring laws nevertheless a.s.~admit quasi-isometric embeddings in both directions. We prove that the class of a realisation depends only on the support of the offspring distribution, not its specific distribution. For offspring distributions supported on $\{1,2\}$, we additionally obtain an explicit exponential tail bound for the probability of non-existence of a root-preserving $D$-quasi-isometry. The classification also implies that the corresponding random Cantor boundaries are almost surely quasisymmetrically equivalent. Conditioned on nonextinction, this applies to strongly separated fractal percolation, even when the underlying self-similar iterated function systems and retention parameters differ. We also classify two families of trees with continuous random branching times. Our proofs use new automorphism-matching theorems for random graph labellings of trees, based on contraction estimates for mismatch potentials. These matching results for Markov labellings on the binary tree may be of independent interest. The quasi-isometry classification, embeddability, and matching theorems are formally verified in Lean.

Let $d \geq 1$ be fixed and let \[ D(n,d) := \sum_{j=0}^{d} \binom{n-1}{j}. \] We show that if $x^{(1)}, \dots, x^{(m)}$ are independent uniform points of $\{\pm 1\}^n$ then uniformly for $m \leq D(n,d)$, there exists a constant $C_d > 0$ such that \[ \mathbb{P}((x^{(1)})^{\otimes d}, \dots, (x^{(m)})^{\otimes d} \text{ are linearly independent}) = 1 - O_d\left(\frac{\log^{C_d} n}{n^{1/2}} \right). \] This achieves the exact dimensional threshold and answers a question asked by Baldi and Vershynin. We discuss applications of the result to the semidefinite relaxation of the cut-polytope and to matrix factorization.

We introduce and study multiradial regular variation of random fields, allowing componentwise thresholds to diverge at unrelated rates. The limit measures are homogeneous in each component and finite on events where every component exceeds a positive level in absolute value somewhere on a compact window. We characterise convergence by anchor exceedance masses and conditional whole-path laws, together with a compact-window mass bound in the continuous-parameter setting. For fields indexed by a countable discrete abelian group or by R^m every shift-invariant tail measure in this class admits a strictly stationary realisation. Finite moving averages with shared volatility show that scalar row-tail measures and simultaneous-exceedance tail masses on every consecutive finite window can coincide while relative-lag tail masses differ.

Blackwell and Cover both proposed the same method for guessing the larger of a pair of two numbers when one can only observe one and the other of the two numbers remains unobserved. As originally stated, the procedure ended with a computation showing that the probability of correctly guessing the larger number was greater than one-half. In this paper, we investigate what we can say when about the number in the pair that was not observed we employ the Blackwell/Cover procedure. Let {\alpha}1 and {\alpha}2 be two different numbers in (0,1]. Let G be the uniform distribution on [0, {\alpha}1] and H be the uniform distribution on [0, {\alpha}2]. We think of G and H as being represented by the heads probabilities of coins. We use a variant of the Blackwell-Cover strategy ro take pairs of coins chosen from distributions created from G and H such that one of the coins is flipped, and as a result of that flip the unobserved (unflipped) coin is placed in one of two sets A and B. We show that there are situations in which this can be done so that the mean of the heads probabilities of the unobserved coins in A is less than the mean of the heads probabilities of the unobserved coins in B. We show that there are situations in which recursive application of this process results in a set consisting almost completely of coins from one of the two distributions G and H, and another set consisting almost completely of coins from the other distribution.

Dictionary learning seeks to recover an unknown dictionary $A$ from observations ${\bf y}_i = A{\bf x}_i$ with sparse coefficient vectors ${\bf x}_i$. We introduce the Iterative Atom Refinement (IAR) algorithm, a simple procedure for recovering individual dictionary atoms. Starting from a random direction, IAR repeatedly selects the observations most strongly correlated with the current iterate and updates the direction by averaging the selected data. Our main contribution is a rigorous convergence theory of IAR. Using high-dimensional probabilistic estimates and a novel monotonicity principle for atom-selection probabilities, we show that a small initial advantage of one atom is amplified until that atom is isolated. Under our model assumptions, IAR identifies a generating atom after only three refinement steps. Numerical experiments support the theory and show that the resulting dynamics accurately capture the behavior observed in dictionary refinement.

We study the graph distance $D_n(p)$ and effective resistance $R_n(p)$ between the boundary vertices of a depth-$n$ random series–parallel graph, obtained by recursively joining two independent copies in series with probability $p$ and in parallel with probability $1-p$. We prove the existence of deterministic logarithmic speeds: for every $p\in[0,1]$, $n^{-1}\log D_n(p)$ and $n^{-1}\log R_n(p)$ converge to deterministic limits $v_D(p)$ and $v_R(p)$, respectively, almost surely and in $L^1$. The limiting speeds agree with the corresponding first-moment logarithmic rates for every $p\in[0,1]$ in the distance case and for $p\in[1/2,1]$ in the resistance case. We further determine the near-critical behavior of the resistance speed: $v_R(\frac12+\delta)\sim 2\zeta(3)^{1/3}\lambda_*\delta^{2/3}$ as $\delta\downarrow0$, where $\lambda_*>0$ is characterized by an explicit nonlinear boundary-value problem. This exponent $2/3$ contrasts with the exponent $1/2$ for distance obtained by Chen, Derrida, Duquesne, and Shi (2026). The main idea of this work was proposed by ChatGPT 5.6 Sol, and the authors take full responsibility for the mathematical content. The three main theorems and their supporting proof dependencies have been formalized in Lean 4, relative to two explicitly documented external mathematical inputs.

We study optimal stopping under dynamic risk measures with simultaneous ambiguity in the probability model and the discount rate. We introduce a paired ambiguity framework combining Girsanov model uncertainty with cash subadditive risk evaluation and characterize the stopping value by an upper reflected backward stochastic differential equation (BSDE). We establish structural properties of the resulting stopping operator and study quadratic drivers associated with entropic risk measures, obtaining explicit stopping rules in several benchmark cases. We then develop a deep learning scheme for the reflected quadratic BSDE. The convergence analysis uses discrete reflection and truncation to reduce the quadratic problem to a globally Lipschitz system and combines reflected BSDE discretization estimates with neural network approximation errors. Numerical experiments for American options illustrate the effects of discount rate and entropic ambiguity on stopping values and exercise decisions.

We establish global well-posedness and a small-noise large deviation principle for the two-dimensional Dean-Kawasaki equation with Coulomb interaction $\partial_t\rho^\varepsilon=\Delta\rho^\varepsilon-\nabla\cdot(\rho^\varepsilon(V*\rho^\varepsilon))-\sqrt{\varepsilon}\nabla\cdot(\sqrt{\rho^\varepsilon}\circ\xi^{K(\varepsilon)})$, where $V=\lambda_V(\cos(\alpha_V)\nabla G+\sin(\alpha_V)J\nabla G)$. Here $\lambda_V\geq0$ is the interaction strength, $\alpha_V$ is the interaction angle, $G$ is the mean-zero Green function of $-\Delta$ on $\mathbb{T}^2$, $J$ is rotation by $\pi/2$, and $\xi^K$ is a finite-mode approximation of space-time white noise. For finite-entropy initial data of mass $M$ satisfying $\lambda_V\max\{\cos(\alpha_V),0\}M<8\pi$, the equation with the exact square-root coefficient is globally pathwise well posed at every finite Fourier cutoff in the class of stochastic kinetic solutions. Following Fehrman and Gess (arXiv:1910.11860), we prove a large deviation principle on $L^1((0,T)\times\mathbb{T}^2)$ under the scaling $K(\varepsilon)\to\infty$ and $\varepsilon K(\varepsilon)^4\to0$, with good rate function given by the quadratic control problem for the skeleton equation.

We establish quantitative propagation of chaos in total variation for the unregularized three-dimensional repulsive Vlasov–Poisson–Fokker–Planck (VPFP) particle system. For tensorized initial data satisfying weighted Sobolev regularity and a Gaussian moment, we prove that for every $0<b<1/3$ there exist $\tau_b>0$ and $C_b\ge1$, independent of $N$ and $k$, such that \[ \sup_{0\le t\le\tau_b}\|F_{N,k}(t)-f_t^{\otimes k}\|_{\mathrm{TV}} \le C_b^kN^{-b}, \qquad N\ge1,\quad 1\le k\le N. \] More generally, for repulsive Riesz potentials with singularity $|x|^{-s}$ on $\mathbb{T}^d$ and $\mathbb{R}^d$, $0<s<d/2$, we obtain $C^k\bigl(N^{-1/q}+N^{-(d/s-2)}\bigr)$ for $q\ge2$ and $q>d/(d-s-1)$. The proof combines conditioning of the product initial law, auxiliary $k$-particle Fokker–Planck equations, and weighted $L^q$ estimates for the BBGKY hierarchy. The level-$(k+1)$ interaction term is controlled by a weighted Hölder estimate under the local condition $K\in L^{q'}_{\mathrm{loc}}$; for $d=3$ and $s=1$, this condition is $q>3$. On $\mathbb{R}^d$, polynomial spatial weights control the far field. For each fixed particle number, the singular particle dynamics are globally well posed and have no collisions. This permits the force regularization to be removed.

Let \(\{X_n:n\ge0\}\) be a controlled branching process with controls \(\phi_n\), and let \(N_n=\phi_n(X_n)\) be its progenitor count. We study large deviations of \(X_{n+1}/N_n\), \(N_n/X_n\), and \(X_{n+1}/X_n\), on their natural positive-denominator events, in supercritical and critical regimes. We treat generation-wide random slopes and asymptotically deterministic first-order slopes. For asymptotically affine random-slope controls, an affine recursion and a random environmental product yield a sharp Bahadur–Rao–Petrov upper-tail asymptotic. A positive Cramér tilt identifies the prefactor via a nontrivial normalized limit, and conditional on the upper-deviation event the three ratios concentrate around their environmental centers. With individual-sum controls, the process is an exact branching process in a random environment with immigration (BPREI). Motivated by this connection, we establish sharp upper large-deviation asymptotics for BPREI and transfer its positive-transform and harmonic-moment profiles to the controlled process. In the nondecreasing supercritical case, harmonic moments exhibit an exact environmental–boundary–persistence trichotomy. In a centered finite-variance critical class, finite-horizon-minimum methods yield exact \(n^{-1/2}\)-scale limits for all positive harmonic orders and for the three ratio-deviation probabilities. For deterministic first-order slopes, the environmental-product mechanism disappears. Matched lower and upper probability-generating-function envelopes transfer auxiliary exact-recursion profiles to controls without an exact branching recursion, producing distinct supercritical and critical harmonic-moment trichotomies. Thus the location of first-order randomness determines the lower-tail mechanism, while immigration may change a sharp constant, a phase boundary, or the decay exponent.

We describe what a typical covering of a closed 2-orbifold looks like on a small scale. Specifically, we prove that a uniformly random covering of degree n contains in expectation $\Theta(n^0)$ short closed geodesics, potentially $\Theta(n^0)$ pairs of nearby order-2 cone points, in general $\Theta(n^{\frac{1}{m}})$ cone points of order m, and with high probability no small regions with more complicated topology. This generalises many results of Magee and Puder (2023) from orientable surfaces to possibly non-orientable orbifolds, and answers some of the questions raised by Puder and Zimhoni (2024).

We reconsider the theory of probabilistic formal languages generated by n-gram models and by probabilistic context-free grammars (PCFGs). The expected hierarchy of probabilistic grammars is established by proving that every probabilistic language generated by an n-gram model is also generated by some PCFG, while some probabilistic languages generated by PCFGs cannot be generated by any $n$-gram model. We introduce the notion of fully connected PCFGs, namely PCFGs in Chomsky normal form where every production rule only involving non-terminals has non-zero probability. Our main result shows that any probabilistic language generated by an $n$-gram model differs from any probabilistic language generated by a fully connected PCFG. Therefore, the class of probabilistic languages generated by $n$-gram models is not a subset of the class generated by fully connected PCFGs.

We prove that an operator-norm separated family of matrices satisfies $\mathbb{E}\sup_{A\in T} G^{T}AG' \geq ca\log |T|$, where G,G' are independent standard Gaussian vectors and a is the separation. The main information estimate concerns arbitrary separated coisometries: conditional entropy is bounded by a source-dependent operator energy times $\log|T|$, up to an additive quadratic term in the common row dimension. An adaptive Gaussian experiment proves this estimate by charging actual information increments to one weighted posterior-entropy potential. Convex separation and a Gaussian covering estimate then yield a bounded-radius result. To reach the general case, we first choose an operator scale preserving the Sudakov ratio, apply the known Hilbert-Schmidt minoration, and recompute a common Gaussian block compression at the retained entropy. This ordering preserves the normalization needed by the coisometry argument.

We prove weak existence and uniqueness in law of $[0,1]$-valued solutions for a class of scalar stochastic heat equations with measurable drifts $b : [0,1] \to \mathbb{R}$, driven by Wright–Fisher noise. The only assumptions we require on the drift are Borel-measurability, and the one-way inequalities $-Cu\leq b(u)\leq C(1-u)$ for some $C<\infty$, compatible with the necessary condition $b(1)\leq 0\leq b(0)$. These are vastly more general conditions than were previously available. Our proof relies on a stochastic duality between the solution of the stochastic heat equation, and a voting scheme running along the graph of a branching-coalescing particle system, generalising the class of drifts for which a dual process is available. Our results and their proof put the regularisation-by-noise result of Barnes, Mytnik and Sun, and its explanation, on a much more general footing.