Papers

Motivated by the concept of numerical hypercyclicity, in this paper, we introduce three new notions in linear dynamics: topological numerical transitivity, generalized numerical transitivity, C*-transitivity. The first notion is defined for operators on general Banach spaces, whereas the latter two are formulated in the setting of operators on Banach algebras. The concept of C*-transitivity is also applicable to operators on the space of Hilbert-Schmidt operators. We prove that these new notions are mutually different, and we also show that they differ from the standard notions in dynamics. In particular, while topological numerical transitivity implies numerical hypercyclicity, as we argue in the paper, we provide examples of numerically hypercyclic and strongly numerically hypercyclic operators that are not topologically numerically transitive. Also, we construct nonsupercyclic C* transitive and generalized numerically transitive operators on the C*-algebra of compact operators on a separable Hilbert space, as well as C*-transitive nonsupercyclic operators on the standard Hilbert C*-module. In addition, we study diagonal operators on finite-dimensional and separable Hilbert spaces, and we obtain complete characterizations of both numerical hypercyclicity and topological numerical transitivity in terms of coefficient-simplex criteria. Further, we provide some sufficient conditions for diagonal operators on the standard Hilbert C*-module to be C*-transitive. All these results are illustrated with concrete examples. At the end of the paper, we compare topological numerical transitivity and C*-transitivity with numerous standard notions in linear dynamics, and we prove that topological numerical transitivity and C*-transitivity genuinely differ from all these standard concepts in dynamics.

Let $(M,g)$ be an asymptotically flat manifold such that its tangent bundle $TM$ is diffeomorphically asymptotically Euclidean. We prove a positive mass theorem and a positive mass rigidity theorem for a natural class of asymptotically flat metrics $\widetilde g$ on $TM$ exhibiting slow asymptotic decay. These metrics are constructed by interpolating, along the horizontal distribution of $TM$, between the Euclidean metric and the base metric $g$ near infinity. The main difficulty is that the induced metrics on $TM$ decay below the standard threshold for the ADM mass in dimension $2n$; nevertheless, we show that their asymptotic mass is well-defined in a generalized sense and is determined by geometric data on the underlying manifold $M$.

The Cardy-Smirnov function of a four-marked planar domain encodes the conjectural scaling limit of the crossing probability for critical percolation. Cardy predicted a closed formula for this limit on the rectangle in the cases of both site and bond percolation, and Smirnov proved the formula together with conformal invariance for critical site percolation on the triangular lattice. Kleban and Zagier later showed that on the rectangle the function admits a modular interpretation: it is determined by a modular functional equation together with a mild analytic ansatz. We carry out the analogue on the $\pi/3$ parallelogram. We derive a closed conformal-map formula for the Cardy-Smirnov function as an incomplete beta integral, prove a closed modular formula realizing it as an integral of the modular form $\eta(\tau)^2\eta(3\tau)^2$, and establish a uniqueness theorem showing that a single functional equation, together with a $q$-expansion ansatz and a nondegeneracy condition, determines the function uniquely.

This article studies transmissive reconfigurable intelligent surface (RIS)-assisted architectures. It compares them with electronically steered phased arrays for the deployment of vehicular direct-to-satellite (D2S) communications in future satellite networks. Rather than treating RIS as a direct replacement for phased arrays, we clarify the operating regimes in which RIS can serve as a low-power wavefront-shaping aperture and those in which phased arrays remain preferable because of their high gain and mature beam-tracking capability. Moreover, phased arrays can support multi-beam operation, which is particularly beneficial for dual connectivity and seamless handover. We distinguish analog, digital, and hybrid phased arrays, discuss the relationship between transmissive RIS and reconfigurable transmitarrays, and highlight practical profile, tracking, and link-budget constraints for mobile terminals. The comparison shows that passive RIS offers attractive power efficiency and aperture scalability, active RIS can partially improve the link budget, and phased arrays remain preferable for high-throughput.

We investigate the three dimensional Boussinesq equations in the planetary geostrophic scale, where the buoyancy/gravity driven flow is near geostrophic balance in the large horizontal spatial scale. By establishing the uniform (with respect to the characteristic number) estimates, we are able to obtain the global weak solutions and the unique global strong solutions to the Boussinesq equations without restriction on the initial data. Furthermore, sending the characteristic number to zero with the established compactness rigorously justifies the planetary geostrophic approximation of the Boussinesq equations. We employ the novel buoyancy flow-free flow decomposition to overcome the singularity caused by the buoyancy/gravity.

We study a null model of high-dimensional projection pursuit: we are given $M$ points sampled i.i.d. from a standard gaussian in $N$ dimensions, where $M,N\to\infty$ with $M/N\to\alpha\in(0,\infty)$. Our goal is to characterize the possible empirical distributions of these points' projections along a data-dependent direction $x$, which ranges over either the sphere $S_N=\sqrt{N}\mathbb{S}^{N-1}$ or cube $\Sigma_N=\{-1,+1\}^N$. We consider this problem in an algorithmic setting, where $x$ must be the output of an algorithm with dimension-free Lipschitz dependence on the input; this class of algorithms includes general gradient-based methods such as Langevin dynamics and approximate message passing (AMP). Our main result exactly characterizes the set of empirical distributions attainable by this class in terms of a one-dimensional stochastic control problem. As a consequence of our main result, we obtain exact algorithmic thresholds for optimizing the Hamiltonian of a spherical or Ising perceptron model with general bounded continuous activation. For the spherical problem, independent work of Montanari and Zhou (2024) characterized the empirical distributions attainable by a related two-stage AMP algorithm, also in terms of stochastic control. Our proof of hardness builds on the branching overlap gap property introduced in earlier work by the first two authors. Our main innovation is to develop stochastic control theory within the branching OGP framework, significantly expanding the settings in which it locates an exact algorithmic threshold. Notably, our methods apply even though the non-algorithmic problem of characterizing all feasible projections remains a major outstanding challenge. For the matching algorithmic result, we construct a new incremental AMP algorithm that acts on a Brownian-bridge revelation of the gaussian disorder and simulates the same family of controlled SDEs.

This paper investigates physical-layer security (PLS) over correlated composite generalized-Gamma (GG)/GG fading channels, where both shadowing and small-scale fading follow GG distributions. Using Mellin transforms and Fox-H functions, closed-form expressions are derived for the single-link probability density function (PDF), joint distribution, survival function, and zero-rate secrecy outage probability (SOP)/probability of non-zero secrecy capacity (PNZSC). The general-rate SOP is expressed as an exact double series with one residual onedimensional integral per term. The model includes the Nakagamim/GG and Nakagami-m/Gamma channels as special cases. Numerical results validate the analysis and demonstrate the impact of the fading parameters on secrecy performance.

Partially totally real (PTR) submanifolds were introduced in Kähler geometry by distinguishing a totally real distribution and leaving its orthogonal complement unrestricted. In this paper we develop the corresponding framework for submanifolds of Sasakian manifolds tangent to the Reeb vector field. After separating the Reeb direction, we define the totally real and ambiguous distributions and show that anti-invariant, contact CR, hemi-slant and pointwise hemi-slant submanifolds occur as special cases of the Sasakian PTR framework. We establish the basic tangential and normal decompositions, study maximality and integrability, and derive the additional restrictions produced by the Reeb field. We then investigate the canonical morphisms \(P\) and \(F\), the geometry of the associated distributions, and PTR-submanifolds in Sasakian space forms. Explicit models are included to illustrate the principal structures and the differences from the Kähler case.

Let $N=\binom n2=\dim\mathrm{SO}(n)$. We prove that the coordinate-plane Kac walk on $\mathrm{SO}(n)$ has total-variation mixing time of order $N$: for every fixed $0<\varepsilon<1$, \[ t_{\mathrm{mix}}^{(n)}(\varepsilon)=\Theta_\varepsilon(n^2). \] The lower bound is the dimensional singularity obstruction before $N$ steps. The upper bound removes the final logarithm from the previously known $O(n^2\log n)$ estimate. The proof combines the discrete Malliavin coupling and low-degree pseudo-mixing inputs with a new log-free analysis of the derivative shells. Its static core is a circuit-anchored, arbitrary-spectrum root/pass identity for the physical five-box prime. Keeping one normalization base per original circuit permits simultaneous scalar regluing without paying for artificial cuts. Its temporal core is an exact chronological calculus: passive singleton runs acquire a coboundary/Riesz gain, while root-interrupted components are allocated by vertex-labelled packets before absolute values are taken. The curvature split into pure-Weyl and Ricci parts is kept at its physical tensor type. All-Weyl packets retain a full $N^{-1}$ resource; mixed packets contain a typed $O(n^{-1/2})$ Ricci debit; and the final packetless Ricci cell is closed by a joint invariant-column estimate on its two root-hit circuits and an exact causal restoration of the marked root time. These estimates yield an $O(n)$ squared first-derivative shell and a summable all-order marked-shell expansion through logarithmic degree. The resulting score energy is $O(n/c^2)$ after $cN$ steps. A weighted submersion integration-by-parts argument and the Haar log-Sobolev inequality then give the uniform total-variation upper bound. No cutoff profile or cutoff window is asserted.

As several mathematical conjectures have recently been settled using large language models (LLMs), the mathematical community has formulated norms and recommendations regarding the publishing of such results. These norms do not cover the disclosure of the prompts and precise software setup used to obtain those results, nor do they require that results be formalized in a manner that allows for machine verification. I argue that both of these are essential. In addition, since LLM-obtained results may be hard to understand, human authors have the responsibility to invent intuitive explanations.

This article presents a systematic study of the critical cohomology of $\QM^{\xi}(\bbP^1, X)$, the moduli space of based quasimaps from $\bbP^1$ to a Nakajima quiver variety $X$. We develop two derived equivalent presentations of this moduli space as a global critical locus. Accordingly, we obtain two equivalent explicit realizations of the canonical DT sheaf of the moduli space. Using each critical presentation, we describe canonical actions of shifted Yangians and quantized Coulomb branches on the quasimap critical cohomologies. We produce, by geometric means, a canonical morphism from an appropriate Hall algebra to the Coulomb branch, and show that the two actions we describe are compatible via this morphism under an additional hypothesis. We conjecture the compatibility holds generally, and reduce the claim to a sheaf-theoretic statement that may be approached via Joyce conjecture.

In this note, we prove a Liouville type rigidity result for smooth entire solutions of the $k$-Hessian equation. The proof makes use of a concavity inequality for the $k$-Hessian operator.

We introduce the Test and Find (TF) problem, where a decision maker (DM) faces the following situation: $k$ identical objects are randomly allocated to $n$ distinct boxes (sites) according to some distribution $\pi$, with no more than one object to a box. The DM tests all of the boxes. However, the tests are imperfect: they can give false positive or false negative results. DM has $m$ tags, $1\leq m\leq n$, and, after testing all boxes, she can place a tag on any box that she thinks has a hidden object. She is rewarded $c_i$ for a correct guess and penalized $d_i$ for a wrong guess in box $i$. DM knows all of the parameters of the model and her goal is to maximize the expected reward. We give an explicit solution to this problem. We then turn to the symmetric case, for which we derive more computationally efficient results. We also consider several extensions of the TF model and give detailed solutions. One of these extensions is to the realistic case where $k$, the number of objects, is unknown and random.

The standard formulation of function-correcting codes uses systematic encodings. We study the redundancy cost of this constraint when both the prescribed function and the encoding are linear. We introduce the function-separation distance and show that several classical unequal error protection parameters are special cases. For linear functions and encodings, this distance is the first relative generalized Hamming weight of the code relative to the encoded kernel. We formulate free and systematic linear separation problems. We prove that the optimal free redundancy depends only on the rank of the function, and determine the optimal systematic redundancy for prescribed separations $d\leq3$.

We establish a Grothendieck six-functor formalism for Kummer étale cohomology including Poincaré duality for every separated vertical exact log smooth morphism of noetherian fs log schemes $f\colon X\rightarrow S$ when the coefficient ring $\Lambda$ is killed by an integer invertible on $S$. This is done via log étale rigidity \[\mathrm{D}_{\mathrm{l\acute{e}t}}(S,\Lambda)\simeq \mathrm{DA}_{\mathrm{l\acute{e}t}}(S,\Lambda).\] To achieve this, we also prove that Kummer étale cohomology satisfies $\mathbb{A}^1$-invariance, invariance under virtual isomorphisms, log cdh-descent, and invariance under verticalization.

Let $M_d(X)$ denote the average number of rational preperiodic points among degree-$d$ polynomials over $\mathbb{Q}$ with vanishing $z^{d-1}$ coefficient, constant term $1$, and height at most $X$. We prove that, for every integer $d\ge2$, $M_d(X) \sim \gamma_d X^{-1}$ for an explicit constant $\gamma_d>0$. For $d\ge4$, the error term is $O_{d,\varepsilon}(X^{-2+\varepsilon})$ for every $\varepsilon>0$; for $d=3$ the error term is $O(X^{-3/2})$; and for $d=2$ the error term is $O(X^{-3/2}\log X)$.

Mohsen Hachtroudi (1908–1976) was one of the founders of modern mathematics in Iran and the first Iranian mathematician to gain international recognition for original research in modern times. A student of Élie Cartan, he introduced the canonical projective connection now known as the Hachtroudi connection, whose significance has become increasingly apparent through later developments in Cartan geometry, CR geometry, and parabolic geometry. This article combines a historical and mathematical study of\\ Hachtroudi's life and work. It examines his education in Iran and France, his role in establishing modern mathematical research and higher education in Iran, and his influence as a teacher, essayist, and public intellectual. It also presents a modern account of his mathematical contributions and their continuing impact on the geometry of differential equations and the theory of geometric structures.

This is the first publication, in which an inverse problem of the simultaneous reconstruction of an unknown (x,t) - dependent coefficient and an unknown initial condition in a general linear parabolic equation of the second order is considered. The input data depend on (n+1) variables and are, therefore, formally determined ones. Stability estimates and uniqueness theorem are obtained for this inverse problem. As a by-product, logarithmic stability estimates for initial conditions of parabolic equations and inequalities with time reversed data are obtained for the first time. All known publications about inverse problems for parabolic equations with formally determined input data and unknown coefficients assume that those coefficients depend either only on x or only on t. In addition, the initial condition is assumed to be known, except of two publications cited in the text. The inverse problem of this paper has two potential applications. The first one is in forecasting of public opinions in the framework of the Mean Field Games theory. The second one is in tracking spatiotemporal outbreaks of epidemics.

For a graph $G$, a proper coloring of $G$ is called proper conflict-free if for every non-isolated vertex $u$, there is at least one color appearing exactly once in $N_G(u)$. A graph $G$ is proper conflict-free $f$-choosable if for every list assignment $L$ with $|L(v)|\ge f(v)$ for each vertex $v$, $G$ admits a proper conflict-free $L$-coloring. Recently, Kashima, Škrekovski, and Xu proposed a conjecture on proper conflict-free list coloring. For a graph $G$, let $\kappa_G:V(G)\to \mathbb{N}$ be defined by \[ \kappa_G(v)= \begin{cases} 4, & \text{if } d_G(v)=2,\\[4pt] d_G(v)+1, & \text{if } d_G(v)\neq 2. \end{cases} \] They conjectured that every connected graph other than $C_5$ is proper conflict-free $\kappa_G$-choosable. In this paper, we confirm this conjecture in two classes of graphs with bounded average degree, thereby generalizing results of Kashima, Škrekovski, and Xu and of Wang and Zhang. We prove that every connected graph $G\neq C_5$ with either $\operatorname{mad}(G)<\frac{12}{5}$ or $\Delta(G)\le3$ is proper conflict-free $\kappa_G$-choosable. To prove these results, we introduce a method based on systems of proper conflict-free representatives and develop a construction of auxiliary graphs that preserves the maximum average degree bound.

On the space of density matrices $D(H)$, Kadison's theorem establishes that all invertible convex transformations $K:D(H)\to D(H)$ are unitarity or antiunitary. For a finite Hilbert space, we propose a similar theorem which replaces the invertibility requirement with that of entropy conservation. Thus, we argue, the supposition of reversibility in quantum mechanics may be replaced with a principle of entropy conservation.

We prove a sharp higher-order Rademacher-chaos inequality on the Hamming cube using random-reveal martingales. As a consequence, we obtain the sharp metric $X_p$ inequality for $L^p$, resolving an open question posed by Naor.

ISO New England's (ISO-NE) Day-Ahead Ancillary Services Initiative settles reserve products as financial call options on real-time energy prices. The system-wide strike price creates an efficiency-reliability tradeoff: increasing it lowers competitive reserve offers, but weakens resources' incentives to incur preparation costs and remain available for real-time performance. The existing strike price rule does not explicitly account for heterogeneous resource incentives and reserve requirements. We develop an optimization framework that selects the highest strike price while ensuring that enough resources retain an incentive to prepare and collectively satisfy the reserve requirements. Because a resource's preparation decision may affect the resulting real-time price distribution, its incentive depends on an unobservable counterfactual. To address this, we derive a tight lower-bound certificate using only the available conditional price distribution and a bound on the resource's price impact. We show that each resource enters the optimization through a single incentive threshold and that any finite optimal strike price occurs at one of these thresholds. This yields a tractable exact solution method based on threshold calculations and a small number of linear feasibility checks. Using reconstructed ISO-NE conditional price distributions and representative gas-fired resources, we find that higher heat-rate combustion turbines are more likely than combined-cycle resources to constrain the strike price choice.

We show that the $\gamma$-LQG metric (appropriately re-normalized) for $\gamma \in (0,2)$ converges in law to the critical 2-LQG metric (appropriately re-normalized) with respect to the local uniform topology of continuous functions on $\mathbb{C} \times \mathbb{C}$. Combining with the results in arXiv:2509.15544, we obtain that the law of the $\gamma$-LQG metric (appropriately re-normalized) is continuous in $\gamma \in [0,2]$ with respect to the local uniform topology of continuous functions on $\mathbb{C} \times \mathbb{C}$, where the 0-LQG metric corresponds to the Euclidean metric on $\mathbb{C}$.

We develop a method for evaluating finite and infinite series involving Catalan numbers by specializing alternating binomial identities at half-integer parameters. The key ingredient is an explicit representation of the generalized binomial coefficients $\binom{m+\frac12}{k+1}$ in terms of Catalan numbers and products of odd linear factors. Combining this representation with Pascal-type identities and two identities of Bat{\i}r and Sofo, we derive several families of closed-form summation formulas involving Catalan numbers, harmonic numbers, and odd harmonic numbers. As special cases, we obtain evaluations of finite sums together with their corresponding infinite-series analogues. The approach provides a systematic mechanism for transforming identities for generalized binomial coefficients into identities for Catalan-number series.

We first give a complete classification of bi-affine representations of mapping class groups of surfaces with finitely many boundary components or punctures. We also show that every linear representation of the mapping class group of a genus-$g$ surface with two boundary components of dimension at most $3g-3$ is bi-affine. We then classify low-dimensional symplectic representations of the mapping class group associated to triple covers. Let $[\beta]\in H_1(S_g;\mathbb{Z}/3\mathbb{Z})^*$, and let $\widetilde{S}\to S_g$ be the corresponding triple cover with deck transformation $\sigma$. For $h\le g$, every non-abelian homomorphism from either $\mathrm{Mod}(S_g,[\beta])$, the stabilizer of $[\beta]$ in $\mathrm{Mod}(S_g)$, or $\mathrm{Mod}(\widetilde{S},\sigma)$, the centralizer of $\sigma$ in $\mathrm{Mod}(\widetilde{S})$, to $\mathrm{Sp}_{2h}(\mathbb{Z})$ is, up to conjugation, the standard symplectic representation on $H_1(S_g;\mathbb{Z})$. As an application, we obtain a rigidity theorem for holomorphic maps from the moduli space $R_g^{(3)}$ of genus-$g$ curves equipped with a $3$-sheeted (unbranched) normal covering to the moduli space $\mathcal{A}_h$ of $h$-dimensional principally polarized abelian varieties. We prove that, for $g\ge 6$ and $h\le g$, the unique nonconstant holomorphic map from $R_g^{(3)}$, equipped with either of its two natural complex-orbifold structures, to $\mathcal{A}_h$ is the period map sending a cover $Y\to X$ to the Jacobian of the base curve $X$.

A parametrized quantum circuit reports its state geometry through a quantum Fisher information matrix (QFIM), often singular. A small Fisher value can reflect exact state-preserving redundancy, compression by the circuit chart, or weak intrinsic distinguishability, and these mechanisms call for different numerical treatments. We show that the circuit metric factors as $F=B^{*}MB$, where $B$ is the state-level circuit-to-orbit differential and $M$ is the intrinsic Fisher operator on the reachable orbit. The factorization identifies the exact kernel as $\ker B$, separates coordinate transfer from intrinsic geometry, and yields the condition for a circuit to realize an orbit-level quantum natural-gradient (QNG) direction. When the prescribed redundancy exhausts the Fisher kernel, the Moore–Penrose update is the minimum-norm horizontal lift of the quotient Riemannian gradient. At critical points with a locally diffeomorphic quotient-to-orbit map, chart singular values cancel from the linearized QNG operator while intrinsic anisotropy remains; in the trace-orthonormal full-control generator frame, excitation-gap anisotropy gives $\kappa_{\mathrm{QNG}}=\kappa_{\mathrm{Eucl}}$. Representation theory makes $M$ explicit on highest-weight, Slater, and fermionic-Gaussian orbits, and cominuscule fidelity flow becomes integrable, with conserved principal-defect ratios, cubic Lie-retracted convergence at $\eta=2$, and stability boundary $\eta=4$. Finite data impose a second boundary: an estimated QFIM and its confidence radius alone cannot distinguish an exact zero from a small physical mode, so the estimated spectrum alone cannot license hard projection. Under depolarization, inverse-Fisher scaling amplifies mean updates and fluctuations together and cannot restore update signal-to-noise. A redundant Slater/Givens circuit confirms exact transfer identities and illustrates finite-shot tradeoffs.

Dynamical localization requires all position moments of a quantum wavepacket to remain bounded in time, but for quasiperiodic Schrödinger operators such bounds are generally not uniform in phase. In the positive Lyapunov exponent regime, the best known phase-uniform estimates instead grow on a logarithmic scale. We prove that both the logarithmic scale and the dependence on the moment order are sharp for a class of one-frequency quasiperiodic Schrödinger operators with even potentials. Our main ingredient is a reflective version of semi-uniformly localized eigenfunctions, adapted to the two localization centers forced by a completely resonant phase, from which we obtain matching logarithmic lower bounds along sequences of times.

In this article we generalise a classical Kolyvagin's result on the vanishing of the $p$-part of the Shafarevich–Tate group of an elliptic curve (defined over $\mathbb{Q}$) over an imaginary quadratic field $K$ to modular abelian varieties of $\mathrm{GL}_2$-type (defined over a totally real number field $F$) and a CM field $K/F$. Combining this result with the abstract Iwasawa-theoretical argument of [MN19], we show that a similar vanishing holds over the layers of a suitably defined anticyclotomic multi-$\mathbb{Z}_p$-extension of $K$.

We prove that the full Dirichlet-to-Neumann map on a finite time interval globally determines the Lorentzian metric \(\mathbf G_{g,c}=-c(x,t)^2\mathrm{d}t^2+g(x)\) encoded by the principal symbol of the wave operator. The result holds in spatial dimensions \(n\geq3\) for multiplicatively separable wave speeds \(c(x,t)=a(x)b(t)\), provided the accumulated effective time during the experiment exceeds the maximal travel time from the boundary to the interior and back. The data determine both the unknown Riemannian metric \(g(x)\) and the wave speed \(c(x,t)\) throughout the observation cylinder, up to a spatial change of coordinates that fixes the boundary and leaves the measured time unchanged. In particular, the temporal factor \(b\) is not prescribed but is recovered from the same measurements. We also construct several examples demonstrating that the dimensional and visibility conditions for uniqueness are sharp.

We present a systematic Eulerian study of passive scalar advection in rough two-dimensional Kraichnan flows, covering the full range of velocity roughness exponent $h\in(0,1)$. The advection–diffusion equation is integrated directly using a pseudo-spectral method at resolutions up to $2048^2$ grid points, with a frozen-noise Runge–Kutta scheme consistent with the white-in-time construction of the carrier flow. The simulations recover the duality between advected scalar and advecting flow—the smoother the carrier, the rougher the scalar—together with the predicted scaling laws for the second-order statistics. Once the scaling range is properly identified, the fourth-order flatness anomaly is measured across the whole range of $h$, in agreement with the Lagrangian estimates of Frisch et al. (1999) and with the perturbative predictions of Bernard et al. (1998) and Pumir et al. (1997). Benefiting from the fact that our Eulerian simulations give direct access to the full scalar field, we also examine the probability density functions of scalar increments and the corresponding higher-order statistics, which show systematic departures from Gaussianity and from log-normality, with the strongest deviations manifesting for intermediate values of $h$. A central outcome of this work is a systematic account of how the simulation parameters, in particular the molecular diffusivity, must be adjusted with $h$, providing practical guidelines for reliable simulations; we further show that the residual deviations from the theoretical scaling laws are quantitatively accounted for by the finite spectral representation of the carrier flow. Our analysis also serve as a numerical baseline for simulating more realistic extensions of the Kraichnan model, where the carrier flow is coupled with a Gaussian multiplicative chaos and theoretical developments are limited.