Papers

Nonlinear least-squares objectives form the foundation of scientific machine-learning tasks, yet even curvature-aware optimizers remain geometrically inconsistent at finite step sizes: Levenberg-Marquardt (LM) derives its direction from local Riemannian metrics but realizes it as a straight parameter update. Here we introduce RNC-LM, which carries the LM direction along a locally constructed curved trajectory in Riemann normal coordinates. A recursive reformulation of the geodesic equation generates arbitrary finite-order corrections while reusing the same damped Gauss-Newton matrix factorization, and curve length is controlled separately from damping. On a reaction-diffusion physics-informed neural network benchmark, RNC-LM reduces relative $L^2$ errors below $8\times10^{-3}$, whereas L-BFGS, LM and LM with geodesic acceleration remain near one. On a large-scale machine-learning potential fitting task with 985,160 configurations, fourth-order RNC-LM reaches a fixed training-error target with a $34\times$ wall-clock speedup over LM. These results establish finite-step geometric realization as a distinct optimizer design principle.

A classical notion of a factor complexity of an infinite word is defined as a function $p(n)$ counting, for each $n$, the number of distinct factors (or blocks of consecutive letters) of the word of length $n$. The notion has various generalizations and variants. For example, the abelian complexity $p_{ab}(n)$ counts the number of distinct factors of each length $n$ up to abelian equivalence, i.e., only the numbers of occurrences of letters are taken into account, and not their order. The notion of a group complexity generalizes both notions of a factor and an abelian complexities. Namely, given a sequence $\omega=(G_n)_{n=1}^{\infty}$ of subgroups of the symmetric group $S_n$, the group complexity $p_{\omega}(n)$ of a word counts the number of classes of factors of each length $n$ of the word, where words obtained from one another by permutations from $G_n$ are put in the same class. Taking $G_n=S_n$, we obtain the abelian complexity, and taking $G_n=Id$, we recover the factor complexity. Clearly, the group complexity value is between the abelian and the factor complexities. In this paper, we are interested in the following property of words. We say that an infinite word has universal group complexity if for each length $n$ and for each $k$ satisfying $p_s^{ab}(n) \leqslant k \leqslant p_s(n)$, there exists a group $G \in S_n$ such that $p_s^G(n) = k$. In other words, all “intermediate” values of complexity can be obtained. We show that Sturmian words satisfy the universal group complexity property, while they are not the only ones. We also study the universal group complexity property for aperiodic ternary words of minimal complexity and for eventually periodic words.

We investigate the pathwise well-posedness issue of the stochastic nonlinear wave equation (SNLW) with a multiplicative noise. While the Ito solution theory (= random field solution theory) was established in the '80s, its pathwise well-posedness has remained a challenging open problem for over forty years. By building a unified framework for the Fourier restriction norm method adapted to the $U^p$- and $V^p$-spaces, due to Koch and Tataru (2007), and the Young/rough integration theory via the sewing lemma and controlled paths due to Gubinelli (2004) along with the random tensor estimate for multiple stochastic integrals with respect to (fractional) Brownian motions, we establish pathwise local well-posedness of SNLW in optimal regularity ranges. In particular, in the one-dimensional case with a white-in-time noise, our result covers the case of an almost space-time white noise, which is optimal within the framework of one-parameter rough paths.

We prove that the universal local point processes of random matrix theory are characterized by their loop equation hierarchies. More precisely, for every rational $\beta>0$, the $\mathrm{Sine}_{\beta}$ point process is the unique solution of the bulk loop equation hierarchy, and the $\mathrm{Airy}_{\beta}$ point process is the unique solution of the edge loop equation hierarchy. These uniqueness results provide a direct route to universality: it suffices to verify the corresponding approximate loop equations for the ensemble. In many models, these equations follow from local laws and integration by parts.

The Ising correlation function can be expressed in terms of connectivity probabilities of a random graph known as the double random current. In 2016, Duminil-Copin asked whether this random graph only has one topological end. One-endedness implies that free and wired random-cluster measures are equal and that any translation-invariant Gibbs measure of the Ising model is a convex combination of the $+$ and $-$ Gibbs measures. In the general setting of transitive amenable graphs, these properties have been established by Raoufi, effectively circumventing the question of one-endedness. In this paper, we prove one-endedness of the double random current on any transitive, amenable one-ended graph and, relying on Raoufi's work, we answer the question of Duminil-Copin in the affirmative in this general setup. The results of this paper do not give a new proof of uniqueness of random-cluster measure or the characterisations of Gibbs states for the Ising model.

We prove the existence of time-periodic solutions of the two-dimensional incompressible Euler equations bifurcating from a translating vortex pair. The reference configuration consists of two symmetric vortex patches of equal strength and opposite sign traveling at constant speed. In a regime of large separation between the vortices, the dynamics may be viewed as a small perturbation of an integrable system. Working in a co-moving frame and using the contour dynamics formulation, we reduce the problem to a nonlinear transport equation for the vortex boundaries. The linearized operator exhibits degeneracies associated with symmetries and transport effects. By combining a Lyapunov-Schmidt reduction, Nash-Moser scheme, and spectral analysis with sharp asymptotic expansions of the eigenvalues in order to overcome the degeneracy, we construct families of non-rigid, time-periodic vortex patch solutions for a large Cantor set of parameters. The analysis reveals that translating dipoles possess a surprisingly rich nearby dynamics, far beyond the classical rigid paradigm usually associated with vortex patch motion. More generally, the approach developed in this work is flexible and robust, and is expected to extend to a broader class of nonlocal PDEs from Fluid Mechanics with degenerate behaviours.

In this note, we study degenerate complex Monge-Ampère equations on singular Stein spaces with right-hand sides depending on the unknown. First, we prove continuity up to the boundary for solutions. Next, under a holomorphic peak-set condition, we prove local Hölder estimates on the regular locus, allowing the boundary datum to fail to be Hölder continuous at the singular boundary. This includes the case of finite singular boundary sets.

We study non-exchangeable interacting diffusions with pairwise interaction strengths encoded by a sequence of matrices. Under suitable structural and denseness conditions on these matrices, we prove a universal Central Limit Theorem for the global fluctuation field. As the number of particles $n$ becomes large, it converges in distribution to the unique solution of a stochastic partial differential equation (SPDE), the same Gaussian limit as in the exchangeable mean field case. The result applies, for instance, to scaled adjacency matrices of $m_n$-regular graphs when $m_n/\sqrt{n}\to\infty$. A spatial interaction model shows that the $n^{-1/2}$ denseness threshold is sharp. The proof proceeds with an analysis in negative Sobolev spaces, building on sharp quantitative propagation of chaos results together with functional inequalities.

In this paper, we introduce a framework that aims to unify classes defined by forbidden induced subgraphs or induced minors with classes defined by the existence of certain structured tree decompositions. Let $\mathcal{H}$ be a fixed family of graphs. We define \emph{induced-$\mathcal{H}$-packing treewidth}, a tree-decomposition-based graph parameter that, for each bag, measures the maximum number of pairwise anticomplete induced copies of graphs from $\mathcal{H}$ intersecting that bag. This notion generalizes some previously studied parameters: when $\mathcal{H}=\{P_1\}$, it is equivalent to tree-independence number, and when $\mathcal{H}=\{P_2\}$, it is equivalent to induced matching treewidth. We show that bounded induced-$\mathcal{H}$-packing treewidth yields new algorithmic consequences for a range of choices of $\mathcal{H}$. In particular, we prove the following results for graphs of bounded induced-$\mathcal{H}$-packing treewidth. Our results partially answer and substantially extend a question of Bodlaender, Fomin, and Korhonen [SODA~2026] on the tractability of MWIS for graphs of bounded induced-$\mathcal{H}$-packing treewidth for $\mathcal{H}=\{P_3\}$ and for $\mathcal{H}$ equal to the family of all cycles.

In two-type first passage percolation on $\mathbb{Z}^2$, two entities compete to capture the sites of the lattice. The entities spread between nearest neighbor sites at times specified by random passage times associated with the edges. We consider the case when both types have the same passage time distribution, with one type starting at the origin and the other from an infinite cone with tip at the origin and pointing in direction $\theta$. Itai Benjamini has suggested that the type starting at the origin can grow unboundedly if and only if the slope of the cone is strictly smaller than $\pi/2$, so that the cone does not fill a whole half-plane. The main result is that this is correct for any $\theta$ such that the asymptotic shape of the one-type process has a tangent line with direction $\theta$. The proofs are based on a description of infinite time-minimizing paths in terms of Busemann functions together with local modification arguments.

For genus $\geq 4$, it is an open question whether the mapping class group of a handlebody contains a finite-index subgroup with nontrivial rational abelianization. In this paper, we provide evidence that no such subgroup exists. First, we prove that, for all such finite-index subgroups $\Gamma$, meridian multitwists vanish in $H_1(\Gamma; \mathbb{Q})$. Next, we show that $H_1(\Gamma; \mathbb{Q}) = 0$ for finite-index subgroups $\Gamma$ containing the handlebody Torelli group, or large enough subgroups of the twist group or the handlebody Johnson kernel.

In this paper, we introduce the moving-anchored extra-gradient (MAEG) method for solving monotone inclusion problems involving the sum of a continuous monotone operator and a maximal monotone operator. Notably, the distance from the anchor point to the solution set is designed to be monotonically non-increasing. Under Lipschitz continuity of the forward operator, MAEG attains an $\mathcal{O}(1/k)$ non-asymptotic iteration complexity, and when a positive anchor-update parameter is used, it further achieves an $o(1/k)$ asymptotic rate. Furthermore, leveraging the specific behavior of the anchor point, we propose a tailored restart strategy. We demonstrate that this strategy ensures convergence even in the absence of local Lipschitz continuity, while preserving the original iteration complexity guarantees whenever the Lipschitz condition holds.

We study multiplicative graph spanners in the $f$-degree fault tolerant ($f$-DFT) model, in which the spanner must approximately preserve distances even after any subset of edges of maximum degree $f$ temporarily "fails" and is removed from the graph. We prove that there are $n$-node lower bound graphs for which any $f$-DFT $(2k-1)$-stretch spanner $H$ must have size $$|E(H)| \ge \Omega\left( f^{1-1/k} n^{1+1/k}\right).$$ This matches a lower bound that was previously only known to hold conditionally, under the 1963 girth conjecture of Erdős. It also matches the current upper bounds, up to a factor of $\texttt{exp}(k)$. Our proof is an analysis of the so-called Wenger graphs (J. Comb. Theory 1991), via their recent reinterpretation by Szabó and by Conlon (Am. Math. Monthly 2021).

In nonlinear optimal control, uncertainties in system dynamics may affect not only closed-loop stability but also the achieved optimality properties of the resulting solutions. This paper develops a systematic robustness analysis for nonlinear optimal control beyond the conventional focus on stability in robust control theory. First, we demonstrate that the optimal value function retains its Lyapunov property under a quantifiable criterion, thereby guaranteeing the preservation of closed-loop stability. Building upon this foundation, we establish explicit characterizations for optimality deviations induced by model mismatch in both closed-loop performance and optimal controllers, and further reveal their consistency with classical linear-quadratic regulator (LQR) results. In addition, the robustness analysis admits a unified computational formulation that gives rise to an iterative scheme with guaranteed convergence, enabling quantitative assessment of optimality robustness in nonlinear control systems. Numerical examples validate the theoretical analysis.

For a finite irreducible real reflection group, $W$, we triangulate its permutahedron and use this to give an explicit homotopy equivalence between two known classifying spaces for the associated Artin group, $B(W)$. In the process, we characterise the Bruhat intervals from $w_1$ to $w_2$ in $W$, where $w_1^{-1}w_2$ is a Coxeter element.

The Mertens function is defined as $M(x)=\sum_{n\leq x}\mu(n)$, where $\mu(n)$ is the Möbius function. This paper describes a practical implementation of the classical $O(x^{2/3+\varepsilon})$ algorithm for computing $M(x)$ at isolated values, together with the segmented Möbius and Mertens sieve on which it relies. The implementation was used to compute $M(10^{24}) = 7189337839$ and $M(10^{25}) = -258560632948$, taking $7.0$ days and $34.6$ days, respectively. These computations extend the previous record of $M(10^{23})$ by two orders of magnitude. Run standalone, the segmented sieve computed all Mertens values through $10^{16}$ in approximately $7.4$ days, compared with the $7.5$-month runtime of the author's 2018 computation. In pursuit of the most practical isolated-value method, an optimized implementation of the asymptotically faster Helfgott-Thompson algorithm is also presented, running roughly four times faster than the original implementation. With both implementations optimized, the choice between methods depends on input size and hardware. The contribution is a reproducible computation and implementation study, rather than a new asymptotic algorithm.

In a 2021 survey on Katona's circle method, Frankl conjectured that every family $\mathcal{F}\subseteq 2^{[n]}$ in which any two members intersect and no two members cover $[n]$ satisfies the sharp binomial norm bound $ \lVert\mathcal F\rVert_n :=\sum_{F\in\mathcal F}\binom{n}{|F|}^{-1} \leq \frac{n+1}{6}. $ This improves the earlier estimate $\frac{n}{4}$ obtained by the circle method. In this paper, we prove Frankl's conjecture and determine all extremal families. Our proof develops a continuous $p$-biased measure approach in place of the circle method. The intersection and union conditions lead to a sharp estimate for $ \mu_p(\mathcal F)+\mu_{1-p}(\mathcal F). $ Integrating this estimate over $p$ converts it directly into the desired binomial norm bound and recovers the optimal coefficient $\frac{1}{6}$. This continuous averaging is the key new ingredient of the proof and also yields the characterization of all extremal families.

Tate produced a table for elliptic curves over local fields that beautifully summarises their arithmetic invariants in terms of the Kodaira type. We present an analogous set of tables for curves of genus 2. The invariants addressed include: reduction type of the minimal regular model, reduction type of the minimal regular model with normal crossings, Néron component group of the Jacobian, conductor exponent, valuation of the discriminant of a minimal Weierstrass model, and cluster pictures.

We define an invariant of the module category of a representation-finite algebra by the magnitude of its Auslander algebra. This invariant will be called the magnitude of the module category. For bound path algebras, it can be computed as the Euler characteristic of the Auslander–Reiten quiver, in a suitable sense. To aid the computation of our invariant, we define the Auslander–Reiten–Euler characteristic of a translation quiver. We build on classical results in Auslander–Reiten theory to determine the magnitude of module categories of biserial algebras, hereditary path algebras, radical square zero bound path algebras, and self-injective bound path algebras. In these cases, we express our invariant in terms of other known quantities, notably the rank of the Grothendieck group and Coxeter numbers of Dynkin quivers. Based on our calculations and results, we obtain a conjectural characterisation of representation-finite biserial algebras in terms of the magnitude of the module category and the rank of the Grothendieck group.

We study the weak convergence rate of a drift-implicit discretisation scheme for the Cox-Ingersoll-Ross (CIR) process in the regime where the process remains strictly positive. Specifically, we consider scheme~(4) of Alfonsi~\cite{A}, which arises naturally from applying a drift-implicit Euler step to the SDE satisfied by the square root of the CIR process and admits a unique positive closed-form solution at each time step. Using a PDE approach combined with a continuous-time SDE representation of the discretised process, we prove that the weak convergence rate is $\mathcal{O}(1/N)$ under the Feller condition $2\alpha\geq\theta^2$ and mild polynomial growth conditions on the payoff function. The proof requires only elementary techniques and, in particular, avoids the semi-exact simulation machinery used in earlier work. The methodology is expected to extend to a broader class of diffusion processes.

In this paper, we prove the existence of $\mathbf{m}$-fold doubly-connected stationary vortex patches for the quasi-geostrophic shallow-water equations. The solutions are obtained through a bifurcation analysis based on the Crandall-Rabinowitz theorem, with either the inner radius of an annulus or the Rossby deformation length serving as the bifurcation parameter. A central feature of the work is the highly nontrivial analysis of modified Bessel functions arising in the spectral study of the linearized operator. The proof requires delicate and extensive manipulations of these special functions, including precise asymptotic expansions, differentiation formulas, recurrence identities, monotonicity properties and the analysis of singular quantities governing the bifurcation mechanism. These ingredients are essential for characterizing the bifurcation points and establishing the transversality conditions. Finally, we investigate the radial symmetry of stationary and uniformly rotating simply-connected vortex patch solutions, therefore motivating the previous bifurcation results.

We propose a novel approach to the minimax estimation of high-order functionals from the perspective of quantum computing. Specifically, for any real number $\alpha \gg 1$, we present two estimators, one for the classical functional $\mathrm{F}_\alpha(P) = \sum_{i=1}^S p_i^\alpha$ of a discrete distribution $P$ and the other for the quantum functional $\mathrm{F}_\alpha(\rho) = \operatorname{tr}(\rho^\alpha)$ of a mixed state $\rho$. These functionals have close connections with the Rényi entropy and the Tsallis entropy. We show that both estimators achieve the minimax optimal $L_2$ rate $\alpha \mathsf{n}^{-1}$ in the range $\alpha \lesssim \mathsf{n} \lesssim \alpha^{3-o(1)}$, where the support size $S$ of $P$ or the dimension of $\rho$ can be much larger than the number of samples $\mathsf{n}$. As a result, both estimators achieve the optimal sample complexity $\mathsf{n} \asymp \alpha$, improving upon the prior best upper bounds $O(\alpha^2)$ established by Jiao, Venkat, Han, and Weissman (IEEE Trans. Inf. Theory 2017) for classical functionals and Chen and Wang (COLT 2025) for quantum functionals. Our estimators are constructed under a unified framework using quantum primitives and run in linear time on a quantum computer. This work reveals an unexpected path from quantum computing to statistics, suggesting a conceptually new methodology for functional estimation. It adds to the growing list of quantum proofs for classical theorems.

Training a model with noisy gradient descent can be idealized as overdamped Langevin dynamics, and a natural safety question is to bound the probability $\nu_t(\mathcal{A}_H) = \mathbb{P}(Q_t \in \mathcal{A}_H)$ that the trajectory lies in a designated failure region $\mathcal{A}_H$. We study this for a smooth, strongly convex loss in $d$ dimensions, with $\mathcal{A}_H$ separated from the minimizer by an energy gap. At the end of training, the equilibrium mass $\pi(\mathcal{A}_H)$ is exponentially small in $d$, with a complementary energy-barrier rate when the noise is small. Along the trajectory, a shape-free bound $\nu_t(\mathcal{A}_H) \le \pi(\mathcal{A}_H)(1 + \sqrt{\chi_0^2/\pi(\mathcal{A}_H)}\,e^{-mt})$ shows the in-set probability relaxes to (twice) the static value after a burn-in of order $d$, using only the global spectral gap $m$. A worked Ornstein-Uhlenbeck example shows this burn-in is necessary: an angular slice of the equilibrium shell can transiently swell by a factor exponential in $d$, though its equilibrium mass is tiny. To rule this out we introduce a local relaxation rate, defined through the spectral measure of the region's centered indicator rather than a Dirichlet-form Rayleigh quotient. For geometrically isolated regions this rate exceeds the global one, shrinking the burn-in, and with a maximum-principle ceiling it caps the trajectory probability uniformly in time. Strong convexity sets how fast training relaxes, but the shape of the unsafe set decides whether the trajectory bulges through it on the way to equilibrium.

A strong symmetric monoidal functor $F\colon (Ab^{free,fg},\oplus)\to (Mod(k)^{flat},\otimes)$ is determined by the Hopf algebra $F(\mathbb{Z})$ over the ring $k$. We will show that the algebra structure on the left derived functor $L^* F(P)$ can be recovered from the augmented coalgebra structure on $F(\mathbb{Z})$ for $P\in D^{>2}_{perf}(Ab)$. Using a similar technique we will prove that the multiplicative Dold-Puppe-Thom isomorphism $H_*(K(A,n);\mathbb{Z})\simeq L_*Sym A[n]$ is functorial in $A\in Ab^{fg}$ whenever $n\ge 2$. By contrast, if $n<1$, this is known to be false in general.

We generalize the function field analogue analogue of the Gross-Zagier and Waldspurger formulae due to Yun and Zhang relating the (higher) derivative of an automorphic (base change) L-function to intersection numbers of Heegner-Drinfeld cycles on moduli spaces of shtukas to not necessarily square-free level. The main new input is the study of the geometry and cohomology of the appropriate “integral models” of moduli spaces of shtukas at deeper level.

The hyper-Kummer construction, discovered by the first-named author, associates a hyper-Kähler manifold of $\mathrm{K}3^{[3]}$-type with any hyper-Kähler sixfold of generalized Kummer type. We regard this construction as a higher-dimensional analog of the classical Kummer construction of K3 surfaces from abelian surfaces. In this spirit, we prove several results which parallel those known for the classical Kummer construction: we characterize the hyper-Kummer $\mathrm{K}3^{[3]}$-manifolds up to birational equivalence in terms of their Hodge lattices, and establish a McKay correspondence for their derived categories and Chow motives. We propose a recipe to construct locally complete families of projective varieties of $\mathrm{Kum}^3$-type starting from a family of varieties of $\mathrm{K}3^{[3]}$-type equipped with 16 prime divisors in a certain Kummer lattice configuration. We also compare hyper-Kummer $\mathrm{K}3^{[3]}$-manifolds with the Mongardi-Rapagnetta-Saccà double covers of O'Grady's six-dimensional hyper-Kähler manifolds. The hyper-Kummer construction produces a rich configuration of hyper-Kähler manifolds of $\mathrm{K}3^{[2]}$-type and K3 surfaces canonically associated with a manifold of $\mathrm{Kum}^3$-type, in particular the hyper-Kummer K3 surfaces, which form countably many $4$-dimensional families of generic Picard rank 16. We prove abelianity of Chow motives for infinitely many 4-dimensional families of hyper-Kummer K3 surfaces, thereby proving Kimura-O'Sullivan finite-dimensionality conjecture for many new K3 surfaces of Picard rank 16. As other applications, we prove Beauville's weak splitting conjecture for all varieties of $\mathrm{Kum}^3$-type, and, building on previous results, we prove the Hodge and Tate conjectures for all powers of any of the varieties involved in the hyper-Kummer construction.

We study quantitative convergence rates of the vanishing viscosity approximation of first-order time-dependent Mean Field Games with regularizing coupling acting in the Hamilton-Jacobi equation. Under standard structural assumptions on the Hamiltonian ensuring convergence of the vanishing viscosity approximation, previous results provide either qualitative convergence for the two unknown of the system or quantitative estimates merely for solutions of the Hamilton-Jacobi equation. In this work, we improve these convergence rates under the same assumptions and establish, in addition, quantitative estimates for the convergence of the associated forward Fokker-Planck equation. As a consequence, we obtain quantitative convergence rates for the full Mean Field Game system, thus extending and strengthening the existing theory for the first-order limit.

Let $\mathfrak{G}$ be a connected split $p$-adic group of type $B_n$, $C_n$ or $D_n$. Amongst the tempered representations of $\mathfrak{G}$ a key role is played by the Iwahori-spherical block. We provide a compact Hausdorff model for this space which allows us to compute the $K$-theory ranks for the corresponding $C^*$-algebra. The model, an extended quotient, is stratified by sectors. Underlying the approach of this paper is the interplay between the geometric extended quotient and the spectral extended quotient in the context of the ABPS conjecture. We give geometric descriptions of every sector thus equipping the spectrum with a cellular structure. We classify the geometric structures arising in our model, and specifically discover real projective spaces along with cones and suspensions of these. These examples require a minor modification to our homotopy sector conjecture: we show that Langlands dual sectors are rationally (indeed dyadically) homotopy equivalent in all cases ($A_n, B_n, C_n, D_n, E_6, E_7$ and $E_8$).

We address the problem of efficiently sampling multimodal probability distributions, where standard Markov Chain Monte Carlo methods often suffer from poor mixing and mode trapping. To mitigate these issues, we propose Gradient-free Riemannian Langevin Sampler (GRiLS), a novel proposal that improves exploration without requiring gradient evaluations of the target density. Our approach introduces a Riemannian metric which reshapes the local geometry in order to facilitate transitions across modes. The resulting gradient-free MCMC algorithm is particularly suitable for complex, computationally expensive targets where derivatives are unavailable or impractical. The GRiLS proposal requires knowing the mean and covariance of the target density, which we estimate using an ensemble of interacting particles. Empirical results on multimodal benchmarks demonstrate that GRiLS achieves improved mixing compared to existing gradient-based and gradient-free MCMC approaches.

We study bimatrix two-player games and investigate the last-iterate convergence and stability of equilibria for the iterates generated by the optimistic exponential weights method. In contrast to prior work, we allow the step sizes $\eta_x$ and $\eta_y$ to differ. Our first main result establishes, under the assumption that the set of fixed points is finite, a sufficient condition for global last-iterate convergence in the special case of zero-sum games, which constrains only the product $\eta_x\eta_y$ of the step sizes. This condition is practically relevant and partially explains empirically observed behavior. Our second main result provides an almost-tight threshold for asymptotic stability and instability, again in terms of products of the step sizes, for general bimatrix games. This result is primarily of theoretical interest. We derive several known results and practically relevant step size bounds for special cases and illustrate our results by experiments.