arXiv subjects
Publications and source records attributed to Keheng Zhu.
In this paper, we establish two Turán-type results for signed graphs. We first generalize the classical Turán theorem to signed graphs and then extend Nikiforov's spectral Turán theorem to signed graphs. Moreover, we determine the second maximum spectral radius among all unbalanced signed graphs that contain no balanced complete signed subgraph on \(r+1\) vertices.
We study the quenched pressure of the nearest-neighbor Edwards--Anderson Ising model with free and periodic boundary conditions. First, we prove that the infinite-volume pressure is $βd$-Lipschitz in the coupling law for the $1$-Wasserstein distance, yielding quantitative thermodynamic limits for spatially inhomogeneous disorder. Second, for periodic volumes, we prove that almost-sure convergence under every joint coupling of the finite-volume disorder arrays is equivalent to complete convergence of the one-volume pressure laws. Third, we prove that $\mathbb{E}\vert{}J\vert{}^{1+1/d}<\infty$ guarantees this universal-coupling conclusion. We further show that this exponent is optimal among uniform power-moment assumptions: for every $1\leq q<1+1/d$, there is a centered symmetric law with finite $q$-th moment for which canonical nested volumes converge almost surely, whereas independently resampled volumes with the same fixed-volume marginals converge in probability but not almost surely. Finally, in dimension one, we prove that the first-moment condition is also necessary for a finite limiting pressure.
Fix $k\ge 3$, and let $f_k(N)$ be the largest harmonic sum of a subset of $[N]$ containing no $k$ distinct integers with a common pairwise least common multiple. We prove that $f_k(N)=(\log N)^{γ_k+o(1)}$ for a well-defined exponent $γ_k\in(0,1]$. Following the weighted-pressure idea of Chojecki, we give a self-contained proof of variational formulas for $γ_k$ in terms of weighted sunflower-free families. We then eliminate the continuous weight: if $M_k(n,r)$ is the largest size of an $r$-uniform $k$-cosunflower-free family on $[n]$, then \[ γ_k=\sup_{n\ge1,\ 1\le r\le n}\frac{r}{en}M_k(n,r)^{1/r}. \] This finite-block formula yields a direct transfer principle from uniform set-system constructions, recovers the Tang--Zhang bounds, and gives $0.438899\ldots<γ_3\le0.889881\ldots$.
Accurate electron-ion collision models are central to predicting transport in fusion and space plasmas, yet most practical formulations rely on binary-collision assumptions and impact-parameter cutoffs whose quantitative accuracy is difficult to assess directly. We develop a first-principles simulation framework for collisional transport by solving the Newton-Lorentz equations for test electrons in the many-body electric field of a Debye-screened ion background, without imposing binary-collision closures or artificial lower cutoffs. The method combines explicit force summation within a Debye sphere, a volume-preserving particle pusher, and adaptive time stepping, enabling stable and scalable simulations in both unmagnetized and magnetized plasmas. Using simulation-based measures of momentum relaxation and cross-field diffusion, we recover the classical scalings for the electron-ion collision frequency and perpendicular diffusion coefficient, namely $ν_{ei} \propto v_{th}^{-3}$ and $D_\perp \propto B^{-2}$. Within the parameter range studied, both simulated coefficients are lower than their corresponding classical estimates by approximately 15-25%. These regime-specific benchmark results indicate that classical transport theory captures the leading scaling behavior, but that the corresponding quantitative prefactors can remain sensitive to many-body and near-field effects in the simulated regime. The framework therefore provides a computational benchmark for testing and improving reduced collision operators and transport models.
We study boundary free energies and half-space responses in nearest-neighbor disordered Ising models. First, we prove the existence of fixed-depth tangential pressure and identify its derivative almost everywhere with the limiting boundary response. Second, under quenched exponential boundary-response mixing, we prove Gibbs-state selection independence and exponential convergence of finite-depth pressures. We further show that uniform one-spin mixing yields DLR uniqueness and Gaussian normal localization, and verify the required mixing conditions whenever $(2d-1)\mathbb E\tanh(β\vert{}J\vert{})<1$. Finally, we establish an all-face surface limit in the bounded Dobrushin regime and provide counterexamples that delimit general surface and stiffness claims.
We present a novel efficient technique for hybrid (kinetic ions, quasi-neutral fluid electrons) simulations of non-relativistic magnetized collisionless plasma shocks, frequently observed near the Sun, in the solar system, and beyond. This Adaptive Frame-Of-Reference Algorithm (AFORA) enables multi-dimensional simulations of plasma shocks along with concomitant ion acceleration in the shock frame, where shock evolution remains quasi-steady. Compared to moving shocks, this technique allows us to reduce the simulation time and domain size to a minimum while achieving converged shock dynamics and spectra of energetic ions. Using an event-driven (asynchronous) hybrid code, HYPERS, we demonstrate this approach in two spatial dimensions for different orientations of the background magnetic field with respect to the shock normal. Our results show excellent agreement of simulation shocks with observations of interplanetary (IP) shocks. We verify that different shock configurations (quasi-parallel, oblique, and quasi-perpendicular) convert bulk plasma flow energy into ion acceleration with varying degrees of efficiency. These findings underscore the importance of efficient and robust numerical algorithms for future high-resolution modeling of plasma shocks and ion acceleration in three dimensions. In addition to enabling efficient computational studies of collisionless shocks in general, this work paves the way for accurate prediction of seed populations of Solar Energetic Particles (SEPs), generated by coronal mass ejection (CME) shocks. The characteristics of seed ions can be used as inputs to Fokker-Planck models that simulate long-term transport and acceleration of ions along magnetic field lines through their interactions with background solar wind turbulence.
We study boundary free energies in disordered Edwards--Anderson Ising models. For rectangular strips of fixed width, we prove that the expected free-to-fixed boundary correction, divided by the tangential length, converges for every finite inverse temperature and at zero temperature. The proof combines a uniform almost-additivity estimate with product-measure concentration, yielding self-averaging along tangential intervals. We derive an exact finite-volume Gaussian interpolation identity and prove that a uniform exponential boundary-mixing condition implies boundary localization with an explicit exponential rate. The tangential pressure theorem extends to all dimensions and symmetric coupling laws with finite first moment; under a product-concentration hypothesis, self-averaging is obtained along tangential cubes. Finally, a subcritical open-bond criterion, verified explicitly for sparse signed couplings, gives low-temperature localization without assuming an infinite-volume Gibbs-state property.
We prove that for every fixed $λ>0$ and all sufficiently large $n$, any $z_1,\dots,z_n\in\C$ with $|z_j|\geq1$ satisfy $\max_{2\leq k\leq n+1}|\sum_j z_j^k|>e^{-λn}$. Consequently, the $n$th root of the optimal maximum tends to $1$, so no constant $C>1$ in Erdős 973 can exist. The proof combines a truncated exponential factorization with overconvergence on an open set outside the unit disk and a normal-family obstruction for Cauchy transforms.
We consider the nearest-neighbor Edwards--Anderson Ising model on cubic boxes with random perturbations supported at the boundary. We prove that any perturbation admitting an energy envelope that is negligible compared with the volume, both in expectation and almost surely, leaves the limiting quenched specific free energy unchanged. For independent finite-variance bulk disorder in dimension at least two, an Efron--Stein estimate also yields almost-sure self-averaging along the full sequence of boxes. The hypotheses are verified for i.i.d. scalar surface fields, fixed random exterior spins, and periodic wrap-around bonds, with an $ O(L^{-1}) $ comparison of quenched means. The result concerns the specific free energy and does not assert convergence of finite-volume Gibbs measures.
The boundary correction to the free energy of a disordered Ising model depends on the boundary condition, the normalization, and the finite-volume sequence. We give a one-dimensional i.i.d. counterexample showing that the surface free-energy density need not be independent of the van Hove sequence and that the corresponding random correction need not converge in probability. For bounded couplings in the Dobrushin uniqueness regime on cubic boxes in dimensions $d\geq2$, we prove convergence of the normalized free-to-fixed boundary free-energy difference in expectation, almost surely, and in every $L^p$, $1\leq p<\infty$. For Gaussian boundary couplings, we derive an exact finite-volume interpolation identity and explain why it does not by itself imply a low-temperature surface limit. For a seam-flip free-energy difference $D_L$ across a set $S_L$ of independent symmetric bonds of variance $v$, we prove $\Var(D_L)\leq4v\abs{S_L}$; one-dimensional examples show that symmetry and finite moments alone do not determine a stiffness exponent, and the low-temperature Gaussian Edwards--Anderson problem remains open.
For the nearest-neighbour Edwards--Anderson Ising model on the discrete torus, we prove existence of the quenched thermodynamic limit and almost-sure self-averaging of the free-energy density. The only moment assumption in the main argument is $\E|J|<\infty$, and no symmetry or centering of the coupling law is needed. The proof avoids periodic subadditivity: the wrap-around bonds form a surface-order perturbation, while a tiling argument and the strong law of large numbers give the free-boundary limit. We also obtain the quantitative bounds $(O(L^{-1})$ for the disorder-averaged finite-volume correction. A precise common probability space is specified, since an almost-sure statement across volumes is otherwise not well defined.