Papers

We define and investigate two families of dual polynomials associated with the Gauss–Legendre polynomials, which have recently found interesting applications in computer graphics. Using the presented results, one can derive representations of the Gauss–Legendre polynomials, construct the dual bases for Lagrange bases and solve certain approximation problems arising, for example, in CAGD.

We study the Threshold Group Testing (TGT) problem in the noiseless and non-adaptive setting, where the objective is to exactly recover a sparse binary vector from pooled tests, using as few tests as possible. In TGT, each test applied to a subset of items returns a positive outcome if the number of 1's (defective items) in that subset meets or exceeds a specified threshold, and has a negative outcome otherwise. We investigate how the complexity of TGT compares to that of Classical Group Testing (CGT), corresponding to the special case of the threshold equal to one, and analyse the impact of increasing the threshold on the required number of tests. Our main contribution is the derivation of a sharp information-theoretic phase transition at $c_{\mathrm{inf}}^{\mathrm{TGT}}k\log(n/k)$ (non-adaptive) tests for TGT within the constant-column test design. The threshold constant $c_{\mathrm{inf}}^{\mathrm{TGT}}$ is expressed as a function of the prevalence of defectives and the threshold value. Our upper bound is derived under an analytic assumption, and we verify that this assumption is satisfied for a threshold value of 2. The value of $c_{\mathrm{inf}}^{\mathrm{TGT}}$ reveals that TGT on the constant-column design has the same information-theoretic behaviour as CGT in the low-prevalence regime. Yet, strikingly, at higher prevalences, the threshold leads to a significant reduction in the number of tests. On the other hand, we provide evidence that when the asymptotic proportion of defective items is positive, TGT actually becomes strictly harder than CGT (excluding trivial reductions).

In the upcoming 5G Advanced and 6G technologies, joint sensing and communication (JSAC) will play a pivotal role in enabling the simultaneous utilization of hardware and spectrum resources for communication and sensing tasks. While current algorithms primarily focus on designing beampattern invariant covariance matrices for transmitting various symbols for communication, they often overlook the distances among these symbols. While these covariance matrices effectively facilitate ranging operations, they have adverse effects on communication performance. Designing beampattern invariance covariance matrices with maximal distances among themselves poses a challenging non-convex problem. In this paper, we introduce a novel waveform design method based on McCormick relaxation called McCormick-based JSAC (MJSAC). MJSAC sequentially solves an optimization problem to generate a set of covariance matrices by maximizing the distances (Frobenius norm) among themselves while ensuring a consistent beam pattern. Also, MJSAC eliminates the requirement for channel information to generate the covariance matrices. Through simulations, we demonstrate that MJSAC outperforms conventional algorithms, even those utilizing channel information at the transmitter.

We propose annealed entropic allocation, an adaptive sampling policy based on an annealed, weighted soft-min formulation of static budget allocation. We replace the maximin large-deviation rate objective with a weighted log-sum-exp surrogate that blends challenger-specific pairwise scores through soft-min weights, avoiding hard switching when several challengers are nearly active. To capture tail behavior beyond the leading exponent, the surrogate incorporates saddlepoint prefactors from refined pairwise tail asymptotics. Because these corrections are subexponential, decreasing the annealing temperature with the budget preserves the same first-order target allocation. For the static problem, we prove uniform convergence to the hard minimum, concentration of soft-min weights on active challengers, and continuity of the induced target-allocation map under fixed weights. Experiments show that the proposed methods are consistently competitive: the no-saddlepoint ablation performs best in symmetric Gaussian and exponential slippage settings, while saddlepoint weighting can help in heterogeneous or asymmetric cases.

We study distributed optimization with stochastic gradients and finite-bit communication modeled by random (unbiased) quantization. We propose q-PDGD, a quantized stochastic primal-dual method, and analyze it under relaxed global geometry. Under restricted secant inequality (RSI), a constant step-size yields linear contraction to an explicit neighborhood determined by gradient noise, quantization distortion, and network connectivity, while a diminishing step-size achieves O(1/k) convergence without shared-minimizer assumptions. Under Polyak-Lojasiewicz (PL) inequality, we obtain linear-to-neighborhood convergence in the same stochastic quantized setting. Our results match the best-known centralized stochastic rates in oracle complexity, and are supported by experiments demonstrating the predicted tradeoffs between quantization level, step-size choice, and graph structure.

This paper studies a lattice structure for almost perfect matchings on certain planar, bipartite (plabic) graphs embedded in a disk. Postnikov's boundary measurement map, and subsequent related work, yielded that plabic graphs parameterize positroid cells within the totally nonnegative Grassmannian with the map itself given in terms of almost perfect matchings with fixed boundary condition. For finite planar bipartite graphs, Propp introduced a distributive lattice structure on their set of perfect matchings. Subsequently Muller–Speyer, provided this distributive lattice structure on the aforementioned almost perfect matchings with fixed boundary condition. Their work also identified the extremal matchings of this lattice for boundary conditions that coincide with face labels of the plabic graph given by the positroid structure. We extend this by giving an explicit construction of extremal matchings in terms of height functions and show that all possible boundary conditions of an almost perfect matching can be obtained within this construction.

Finite-dimensional operator algebras can be viewed as $\mathrm{C}^*$, $\mathrm{W}^*$, or $\mathrm{H}^*$-algebras, leading to different notions for their categories of modules and correspondence 2-categories. In this article, we show how these differences can be understood systematically using the notion of $G$-dagger category from arXiv:2403.01651 for different subgroups $G\leq O(2)$. To do so, we first introduce $G$-Hermitian $2$-vector spaces using fixed points of a certain $O(2)$-action on $2\mathsf{Vect}$. We then propose criteria for when such pairings are `positive', generalizing the passage from Hermitian vector spaces to Hilbert spaces. Finally, we outline an inductive approach to defining higher Hilbert spaces in arbitrary dimension, suggesting an extension of these ideas beyond the 2-categorical setting.

In this paper, we study the sharp bounds of the first three logarithmic inverse coefficients for Janowski convex class $\mathcal{C}(A, B)$. We also derive sharp upper and lower bounds of $\bigl|\,\gamma_2 \,\bigr|-\bigl|\,\gamma_1\,\bigr|$ and $\bigl|\,\Gamma_2 \,\bigr|-\bigl|\,\Gamma_1\,\bigr|$ for functions in the class $\mathcal{C}(A, B)$. Furthermore, a sharp estimate for the second Hankel determinant associated with the logarithmic inverse coefficients for functions in $\mathcal{C}(A, B)$ is obtained.

The Terwilliger algebras of association schemes over an arbitrary field $\mathbb{F}$ were called the Terwilliger $\mathbb{F}$-algebras of association schemes in [10]. In [7], He and Jiang studied the Terwilliger $\mathbb{F}$-algebras of factorial association schemes. In this paper, we continue studying the Terwilliger $\mathbb{F}$-algebras of factorial association schemes. We get all block idempotents of the Terwilliger $\mathbb{F}$-algebras of factorial association schemes. We get the $\mathbb{F}$-dimensions, the centers, the Jacobson radicals of the block algebras of the Terwilliger $\mathbb{F}$-algebras of factorial association schemes.

We study the Alexander polynomials of the 4-strand Turk's head knots $Th(4,2n+1)$, defined as the closures of the braid $(\sigma_1\sigma_2^{-1}\sigma_3)^{2n+1}$. Using the reduced Burau representation, we derive an annihilating recurrence of order at most 8 and a rational generating function for the resulting polynomial sequence. By executing a multivariable resultant elimination over the reciprocal constraint, we obtain an exact factorization of the normalized Alexander polynomial in terms of Chebyshev polynomials. This factorization produces a binomial convolution formula for an associated coefficient sequence and a representation by a terminating ${}_4F_3$ hypergeometric series. We evaluate the continuous approximation of this representation using the saddle-point method, demonstrating negative curvature in the asymptotic main term. Finally, we describe analytic obstructions to extracting global discrete error bounds via this method, leaving the formal proof of Fox's Trapezoidal Conjecture for this family open.

Classical Game Theory underpins much of AI and multiagent research, but hybrid Human AI systems require a framework in which execution authority can alternate within a digital environment. We introduce NeoGame Theory, an extension of classical Game Theory for hybrid Human AI coalitions operating under Virtual Nature, the algorithmic analogue of classical (physical) Nature. The framework combines a lexicographic coalition utility with a delegation rule based on the Jensen-Shannon divergence between Human and AI policies. Two thresholds define agreement, contextual, and disagreement regions. In the contextual region, execution follows a scenario specific rule. Apart from the theory, in this paper we develop the first regime, Human arbitration, in which the AI learns by observation and frequency matching while the Human retains final execution authority. We establish the axiomatic basis of the framework and characterize a frequency convergence equilibrium, providing the foundation for later extensions and computational validation.

This monograph develops probability and stochastic-process ideas as a translation language for statistics: from designed observations and data objects to targets, stability statements, inference, and use. The chapters move from motivating examples and randomization through probability measures, kernels, likelihoods, data objects, weak convergence, empirical fields, functional data, M- and Z-estimation, testing, local approximations, event-time processes, and prediction. Historical and biomedical examples are used to keep abstract objects tied to records, mechanisms, and decisions. The aim is to give readers a common grammar for classical probability, modern data structures, and statistical practice.

In 1916, Hausdorff proved that the Baire hierarchy on $\mathbb{R}$, starting with the continuous functions, generates all Borel functions on $\mathbb{R}$. It remained open whether, starting with the a.e. continuous functions, the corresponding hierarchy generates all Lebesgue measurable functions on $\mathbb{R}$. We prove that, assuming the Axiom of Choice, the answer is negative.

The joint rate-distortion framework of Stavrou and Kountouris (IEEE Transactions on Communications 2023) characterises dual-fidelity tradeoffs for semantic communication on stochastic semantic sources. Many task-oriented communication systems instead use designed sources, where the semantic object is a deterministic oracle allocation $\phi^(t)$ rather than a stochastic quantity given by nature. We isolate the subclass of designed sources under smooth concave utility with assumptions A1, A2 and Euclidean allocation codomain, and restrict the encoder class to deterministic common-category mappings. Within this subclass the SK exponential-tilting decoder and generalised Blahut–Arimoto iteration specialise to conditional-mean decoding and Lloyd–Max stationarity on $\phi^(t)$. When the second fidelity is a monotone single-letter distortion, the joint problem stays inside the SK admissible class; the common-category SK rate is lower-bounded by the max of the corresponding Shannon rate-distortion functions, with equality only when the common-category reconstruction is compatible and RDF-optimal. When the second fidelity is aggregate verification, the joint problem leaves the SK single-letter class and admits a constrained-design feasibility band $R_{\min}(\varepsilon^) \leq R \leq R_{\max}(\beta^)$ of width $\log_2(K_{\max}/K_{\min})$ bits in partition cardinality. The reduction and the band are scope statements on the SK apparatus, not modifications to it. A smart-grid economic-dispatch example with a non-technical-loss-detection contrast illustrates the band.

We tackle two key difficulties in the simulation of the viscous hydrodynamics of a large dense collection of rigid particles: (i) the poor convergence rate of an iterative solution of the discretized linear system as particle gaps shrink, and (ii) the large number of unknowns needed to accurately discretize the resulting lubrication-driven flows. Our focus is the 2D Stokes resistance and mobility boundary value problems for nearly-touching disks. To address both challenges, we introduce a general two-body preconditioning strategy, and implement it with the method of fundamental solutions. For each close particle pair, the hard-to-resolve interaction is represented in a basis precomputed by solving a local boundary value problem on a fine grid. In an iterative solve, the resulting flow field corrects that obtained from a coarse representation of all particles. The local fine-grid correction can even be compressed so that all particles except the pair itself are affected by an equivalent set of coarse sources. Numerical experiments demonstrate rapid GMRES convergence in challenging multi-particle settings, with iteration counts remaining low even in densely packed suspensions. For example, the mobility problem is solved for a random close packing with area fraction $\varphi = 0.65$, $P = 10000$ monodisperse disks, and minimum separation $10^{-3}$, in just 47 GMRES iterations, achieving five digits of accuracy with 72 vector unknowns per body.

The $K$-th nearest neighbor random walk $(X_n)_{n \geq 0}$ on a homogeneous Poisson point process $\chi$ on $\R^d$ ($d\geq 1$), starts at the origin and at each step picks its next Poisson point among its closest neighbors according to i.i.d. labels having the same distribution as $K$. Our main result (Theorem 1) states that the number of Poisson points visited by $(X_n)_{n \geq 0}$ admits an exponential decay whenever the random variable $K$ has a bounded support (BS). In particular, the $K$-th nearest neighbor random walk visits finitely many Poisson points if and only if $K$ satisfies Assumption (BS). To prove it, we introduce the key notion of pioneer point which allows us to deal with the region of $\R^d$ already explored by $(X_n)_{n \geq 0}$. Still under Assumption (BS), we also prove an exponential decay for the Euclidean length of the trajectory performed by $(X_n)_{n \geq 0}$ (Theorem 2). Finally, and quite surprisingly, we exhibit an example of label distribution with bounded support for which the $K$-th nearest neighbor random walk discovers new Poisson points after a number of steps whose tail distribution is at least polynomial (Theorem 3).

Spectral methods rely on the stability of principal eigenspaces under random perturbations. Classically, this is quantified by the Davis-Kahan and Wedin theorems, which bound the eigenspace error via the operator norm of the noise and the relevant spectral gaps. While sharp for arbitrary deterministic perturbations, these worst-case bounds can be wasteful in the low-rank signal-plus-noise setting, as they fail to capture the interaction between the signal geometry and the noise distribution. We study the spectral perturbation of signal-plus-noise matrices corrupted by sparse random noise with an arbitrary, inhomogeneous variance profile. Under heterogeneous variances, the empirical eigenvectors suffer a systematic, deterministic geometric bias invisible to classical bounds. Leveraging the Quadratic Vector Equation (QVE) and fine-grained isotropic local laws, we derive near-optimal, non-asymptotic bounds for the leading eigenspaces in the operator and 2-to-infinity norms. These separate the usual signal-to-noise contribution, stochastic fluctuations, and structured geometric bias terms determined by the alignment between the signal eigenspaces and the row-wise variance profile. We further develop refined rowwise bounds that adapt to the variance-weighted leverage of the signal space, yielding sharper guarantees in delocalized regimes. As applications, we establish strong consistency of adjacency spectral clustering for degree-corrected stochastic block models with heterogeneous degrees and unbalanced communities, recovering the logarithmic expected-degree scale in the regular balanced case. We also study spectral embedding for generalized random dot product graphs, showing that the full signal embedding admits sharp rowwise control, whereas spectral truncation can retain a systematic geometric bias determined by the omitted signal directions and the variance profile.

For groups without strict hierarchy, collective decisions often emerge through compromise. We develop a second-order network model of collective decision-making using a dissipative Hamiltonian formulation, in which informed agents introduce preferred directions while uninformed participants contribute only direction-free dissipation. We show that under low conflict, the model admits a locally unique, exponentially stable compromise state. Using a structured modular network we further show that as conflict increases the local compromise branch terminates through a saddle-node fold rather than through a smooth mean-field symmetry-breaking transition. Modular polarized states persist on branches that are locally separated from the compromise branch. Direction-free dissipation does not shift the static structural threshold, but it delays escape from the saddle-node ghost and pushes the observable onset of polarization to larger conflicts. Our work identifies a dissipation-mediated mechanism, complementary to connectivity-based accounts, through which uninformed participants stabilize collective behavior in biological and engineered swarms.

We present a new proof that an integer rectifiable current with finite mass, and whose boundary has also finite mass, is integral. We deduce the result from De Giorgi's structure theorem for integer-valued $BV$ functions and a cylindrical projection argument. As a consequence, we also give a new proof of the compactness of integral currents that is ultimately based on the $BV$ theory.

We initiate the study of nonparametric empirical Bayes denoising methods in the setting where both the latent variables and their measurements lie on a compact Riemannian manifold, and where the likelihood is a Riemannian Gaussian distribution. Our starting point is a novel Tweedie-Eddington formula for Riemannian Gaussian mixture models which identifies a certain surrogate oracle denoiser in terms of the marginal distribution of the measurements; it avoids the explicit computation of the posterior Fréchet mean (as required by the Bayes denoiser) via a first-order approximation, hence we refer to it as the "tangential" Bayes denoiser. We show that this surrogate oracle achieves nearly the Bayes risk in a low-noise regime, we construct a fully data-driven approximation of it using the spectral theory of the Laplace-Beltrami operator, and we establish finite-sample rates of convergence for the distance between the the surrogate oracle and its approximation. Contrasting the nearly-parametric rates from the Euclidean setting, the rates in the Riemannian setting are slower due to the singularities of the Riemannian Gaussian density at the cut locus of its Fréchet mean; in the special case of the circle we establish matching lower bounds which show that our proposed denoiser is minimax-optimal, and that the denoising problem exhibits a genuinely nonparametric rate of convergence. Lastly, we implement our methodology in two scientific applications: in astronomy, the sphere-valued problem of denoising the locations of gamma ray bursts; in structural biology, the torus-valued problem of denoising pairs of torsion angles of adjacent amino acids in a protein (i.e., the Ramachandran plot).

We prove an equivariant Darboux theorem for $-1$-shifted contact derived Artin stacks. Such a stack admits a smooth atlas by contact Darboux charts: the derived discriminant locus $\Delta\mathrm{loc}(s)$ of a relative section $s$, extended in degree $-2$ along the relative dimension of the chart. Near a point with linearly reductive stabilizer $G$ acting on the contact line through a character $\chi$, the stack is étale-locally the quotient $[\Delta\mathrm{loc}_G(s)/G]$, whose degree $-2$ generators are indexed by the Lie algebra of $G$ with differential a contact moment map. The cotangent complex sees these generators as $\mathrm{Lie}\,\ker(\chi)$ in degree $-2$; they vanish exactly when $\ker(\chi)$ is finite, and the quotient of the discriminant locus alone is a local model at no point where they are nonzero. For a quiver with potential the contact moment map is the moment map of the doubled quiver, the classical truncation is the representation space of the Jacobi algebra, and the degree $-2$ generators are carried exactly by the locus of positive-dimensional stabilizer.

A central open question in extremal design theory is Nash-Williams' Conjecture from 1970 that every triangle-divisible graph on $n$ vertices (for $n$ large enough) with minimum degree at least $0.75 n$ has a triangle decomposition. In this paper, we prove this conjecture in full. In 2016, Barber, Kühn, Lo, and Osthus proved that if the fractional relaxation of Nash-Williams' Conjecture holds for minimum degree $cn$ for some constant $c\ge 0.75$, then Nash-Williams' Conjecture holds for any constant $c' > c$. The previously best-known bound on the fractional relaxation was due to Delcourt and Postle from 2021 with $c= \frac{7+\sqrt{21}}{14} \approx 0.82733$. This bound on the fractional relaxation has grown in importance over the years as it has been directly tied to bounds for a number of other problems in extremal design theory. This paper consists of three parts. In Part I, our first main result is a proof of the Fractional Nash-Williams' Conjecture: if $G$ is a graph on $n$ vertices with minimum degree at least $\frac{3n}{4}$, then $G$ has a fractional triangle decomposition. In Part II, our second main result is a Fractional Stability Theorem for Nash-Williams' Conjecture: if a graph $G$ on $n$ vertices has minimum degree close to $\frac{3n}{4}$ but no fractional $K_3$-decomposition, then $G$ is close (in edit distance) to the join of two $\frac{n}{4}$-regular graphs each on $\frac{n}{2}$ vertices. We use this to prove that if a triangle-divisible graph $G$ on $n$ vertices has minimum degree close to $\frac{3n}{4}$ but no $K_3$-decomposition, then $G$ is close (in edit distance) to the join of two $\frac{n}{4}$-regular graphs each on $\frac{n}{2}$ vertices. In Part III, our final main result is a proof of Nash-Williams' Conjecture in full.

Equilibrium quantum many-body methods differ not only in approximation, but in which information they represent explicitly. We formulate this distinction by fixing an equilibrium specification and viewing every representation as an encoder from admissible states to represented variables. The identity encoder gives a full-state representation, whereas a non-injective encoder gives a reduced representation whose value labels a fiber of compatible states. For a specified task, an exact decoder exists on a state class if and only if the task is constant on the encoder fibers within that class. Variational principles, reconstruction correspondences, functionals, kernels, and closures are different realizations of additional structure used to select, restrict, or approximate the task-relevant content of a fiber when the retained variable alone is insufficient. Static moments and imaginary-time correlation functions are unified as restrictions of a complete equilibrium readout functional to different probe families. Within the same principle, quantum embedding can be viewed as consistency or replacement between global and local descriptions through reduced interface encoders and their conjugate fields.

In this paper, we prove that for every $k$ and every graph $H$ with $m$ edges and no isolated vertices, the Ramsey number $R(C_k,H)$ is at most $(k-1)m+1\le km$. This settles a problem of Erdős, Faudree, Rousseau and Schelp, which is listed as problem 34 in the graph theory collection.

We introduce Bellman-sufficient information complexity for minimax analysis of sequential decision problems. A Bellman-sufficient state retains enough of the history to close the controlled recursion, while an index $Y=\chi(\Omega)$ specifies the decision-relevant information being charged. The upper bound is a log-penalized Bellman program; the lower bound is a Bellman–Fano comparison along an algorithm-dependent reference trajectory. If the two values match at a common localization scale and the stated admissibility, calibration, and growth conditions hold, they form an information-risk sandwich. UCB, E2D, and AMS/EBO control or relax the upper Bellman bracket in different ways. For the main application, we give a negative answer to a widely studied form of the GP–UCB minimax-optimality question. For every $0<\alpha<1/4$, we construct one bounded continuous kernel whose minimax regret is $\Theta(T^{1-\alpha})$ along an infinite sequence of horizons, while two globally calibrated GP–UCB rules incur linear regret under one fixed truth. An epochwise finite-marginal action-index AIR Bellman policy, implemented through robust AIR/AMS/EBO control, attains the minimax order. The construction separates realized information from the cost of uniform optimism: many low-value directions inflate the exploration multiplier and change the trajectory. Through the canonical RKHS feature map, it also yields a finite-horizon polynomial minimax separation for the specified maximal-information-calibrated LinUCB rule. A reproducible experiment illustrates the mechanism.

Using the theory of complex spinorial forms, we prove that an odd-dimensional Riemannian manifold admits a pure spin-c Killing spinor with a real Killing constant $\alpha\in\mathbb{R}^{\ast}$ if and only if it is $\alpha$-Sasakian, thereby obtaining an extension of a well-known result by A. Moroianu that, under the purity assumption, does not require simple connectivity or completeness.

We develop a global wellposedness theory for weak solutions of the pressureless Euler Alignment system with measure-valued density $\rho$, bounded and unidirectional velocity $\mathbf{u} = (u,0, \ldots, 0)$, and a communication protocol $\phi$ that may be bounded or weakly singular. This appears to be the first such theory that admits shocks outside one space dimension. We recast the system as a family of nonlocally coupled scalar balance laws—one for each horizontal slice of $\mathbb{R}^d$—and establish existence, uniqueness, and stability of entropy solutions of the reformulated system, before translating back to the level of $\rho$ and $u$. This grants us access to certain key one-dimensional tools in the direction of the flow. The analysis in the directions transverse to the flow, however, still presents substantial challenges: The horizontal slices are coupled and therefore cannot be treated independently, and two solutions need not distribute their mass over horizontal slices in the same way, so that there is no canonical way to compare them slicewise. Accordingly, our analysis is centered around the nonlocal alignment force, and our framework leverages optimal couplings between the projections onto $\mathbb{R}^{d-1}$ of the density profiles under consideration. We construct our solutions as limits of atomic density and momentum profiles whose atoms follow sticky particle Cucker–Smale dynamics, and under additional regularity assumptions, we obtain quantitative rates of convergence. Finally, our investigation of the long-time behavior of unidirectional solutions highlights the role of lateral communication in driving the system toward a limiting density profile: We prove that flocking occurs, at a rate independent of the number of agents, under assumptions on $\phi$ that bound its size from below only outside a cylindrical neighborhood of the axis parallel to the flow.

Given a family of semi-stable curves together with two degree 0 line bundles, the double double ramification cycle measures the locus where both line bundles are trivial on the fibers. When the two line bundles come equipped with natural roots, we provide a refinement of the DDR-class using the Weil pairing of the roots. We prove that the refined classes satisfy a multiple cover formula analogous to the one for correlated invariants of projective bundles on elliptic curves proved in [BC25b]. As a consequence, we prove that log-GW invariants of toric surfaces can be refined taking into account the position of the points mapped to the toric boundary, and that these refined invariants also satisfy a multiple cover formula; the latter is as a variation of the N. Takahashi conjecture for genus zero maximal contact curves for P2 relative a smooth elliptic curve E.

A new type of Yang-Baxter equation (YBE) for $R$-operators depending on three spectral parameters is constructed from the star-triangle relation for the chiral Potts model. As the $Z_N$ symmetric generalization to the Ising model, its Boltzmann weights are known to depend on two variables describing a curve with genus larger than one for $N>2$, except for the self-dual point corresponding to the Fateev-Zamolodchikov chain. Combined with the fact that quantum Hamiltonians of edge-type models such as the Ising model contain both nearest-neighbor interaction and onsite potential terms, this leads naturally to an additional spectral parameter in the associated $R$-operator. The construction extends the edge-vertex correspondence of solvable lattice models, and provides a bridge between the Bazhanov-Stroganov four-parameter $R$-matrix—realized as an intertwiner of cyclic representations of $U_q(\mathfrak{sl}_2)$ at a root of unity—and Shastry's two-parameter $R$-operator obtained from the decorated YBE.

In this paper, we construct strong approximations for discrete-time Markov chains weakly converging to continuous diffusion processes, as well as for their perturbed counterparts. Under the assumption of bounded coefficients, we construct closely coupled versions of these processes on a shared probability space. In particular, for both non-degenerate and degenerate cases, we maximize the probability of their exact pathwise coincidence on discrete time grids. Moreover, we construct such probability space that the probability of small deviation of the interpolated Markov chain from the continuous diffusion trajectory is small on the entire time interval if the perturbation is small enough.