arXiv subjects

Chiyu Zhou

Publications and source records attributed to Chiyu Zhou.

8 recordsLinked to original sources

arXiv2026

Volume Doubling and Geometric Inequalities on Graphs with Bakry-Émery Curvature

We consider irreducible reversible Markov chains whose nonzero off-diagonal transition probabilities are bounded below by $ε>0$ and whose reversible measure satisfies $m_-\le m(x)\leκm_-$. Under the Bakry--Émery curvature condition $\mathrm{CD}(K,\infty)$, with $K\in\mathbb{R}$, we establish Wang's log-Harnack inequality $$ P_t(\log f)(x)\le\log P_tf(y) +\frac{C\mathrm{dist}(x,y)^2}{\int_0^t e^{2Ks}\mathrm{d} s}, $$ for bounded positive $f$ bounded away from zero and all $x,y\in V$, where $C$ is a constant dependent only on $\varepsilon$. Building upon this inequality, we further prove volume doubling property: let $K_-:=\max\{-K,0\}$ $$ \mathrm{Vol}(B(o,2R))\le C_0\exp\!\bigl(C_1\sqrt{K_-}\,R\bigr) \mathrm{Vol}(B(o,R)),\qquad R>0, $$ where $C_0,C_1$ depend only on $ε,κ$. If $d$ is the maximum combinatorial degree of the support graph, we further prove under $\mathrm{CD}(0,\infty)$ the sharp volume growth bound $\mathrm{Vol}(B(o,R))\le m_-CR^{\lfloor d/2\rfloor}$ for all $o$ and $R\geq 1$.

math.PR↗

arXiv2026

Entropy estimates and volume rigidity for graphs with positive Bakry-Émery curvature

We establish a volume bound for weighted graphs with positive Bakry--Émery curvature through entropy estimates for the heat flow and characterize the equality case. As a consequence, if a connected unweighted graph $G=(V,E)$ satisfies the Bakry--Émery curvature--dimension condition $CD(2,\infty)$ with respect to the unnormalized Laplacian, then $$ |V|\le 2^d, $$ where $d$ denotes the maximum degree. Equality holds if and only if $G$ is a hypercube. We also establish an analogous volume rigidity result for the normalized Laplacian.

math.DG↗

arXiv2026

Sharp Rogers--Shephard Inequalities for Rényi Entropy

We establish sharp weighted Rogers--Shephard inequalities for Rényi differential entropy. For independent copies $X,X'$ of a log-concave random vector in $\R^n$, we prove $h_p(aX-bX')\le h_p(X)+n\log(a+b)$ for all $a,b>0$ and $1\le p\le\infty$, where orders $1$ and $\infty$ correspond to Shannon entropy and min-entropy, respectively. The Shannon case settles the entropic Rogers--Shephard conjecture of Madiman and Kontoyiannis and its higher-dimensional formulation. For $1\le p<\infty$, equality holds precisely for invertible affine images of random vectors with independent exponential coordinates. At $p=\infty$, the larger equality class is characterized by possibly extended-valued convex potentials that are positively homogeneous of degree one. For every $0<p<1$, uniform laws on sufficiently high-dimensional simplices violate the unweighted bound. Thus $p=1$ is the critical order for this bound to hold in every dimension. The proof combines weighted convex intersections with an exponential lift to a log-affine probability density. At the Shannon endpoint, this lift yields an exact deficit decomposition with a conditional mutual information term for an additive channel with asymmetric Laplace noise.

math.PR↗

arXiv2026

Structure theorems for Lichnerowicz-sharp graphs

Hypercube graphs are fundamental model spaces of positive curvature in discrete comparison geometry. Let $G$ be a finite, connected, simple, unweighted graph with Bakry--Émery curvature bounded below by $K$. We call $G$ Lichnerowicz-sharp if its first non-zero non-normalized Laplacian eigenvalue $λ_1=K$. We prove that, after removing a canonical collection of edges on which every $K$-eigenfunction is constant, the resulting graph has a canonical bundle structure. Its fibers are regular, have similar structure with hypercubes, and are Laplacian-cospectral with hypercubes, although they need not themselves be hypercubes. If the base graph is nontrivial, then it satisfies $\mathrm{CD}(K,\infty)$ and has first eigenvalue strictly greater than $K$. As a consequence, if the vertex degree in $G$ is constant along each canonical fiber, then every fiber is a hypercube and $G$ is a hypercube bundle. Conversely, for every $d\geq 4$, we construct Lichnerowicz-sharp graphs with non-hypercube canonical fibers of degree $d$.

math.DG↗

arXiv2026

Regular Lichnerowicz-sharp graphs are hypercube bundles

Hypercube graphs are fundamental model spaces of positive curvature in discrete comparison geometry. Let $G$ be a finite, connected, simple, unweighted graph with Bakry--Émery curvature bounded below by $K$. We call $G$ Lichnerowicz-sharp if its first non-zero non-normalized Laplacian eigenvalue $λ_1=K$. We prove that all regular Lichnerowicz-sharp graphs are hypercube bundles with constant Bakry-Émery curvature $2$. A hypercube bundle is a graph bundle whose fiber graphs are hypercubes. As applications, we show that any $d$-regular Lichnerowicz-sharp graph with the multiplicity $m_K(G)$ of $λ_1=K$ at least $d/2$ can split off a hypercube of certain dimension. Moreover, we characterize all $d$-regular Lichnerowicz-sharp graphs with $m_{K}(G)\geq d-3$. Interestingly, our result leads to the following spectral rigidity theorem of hypercubes. For a graph $G$ with maximum degree $Δ$, if the multiplicity $m_K(G)\geq Δ-1$, then $G$ is a $Δ$-dimensional hypercube. This improves, in the unweighted setting, the multiplicity condition $m_K(G)\geq Δ$ appearing in the hypercube rigidity theorem of Liu, Münch, and Peyerimhoff. This improvement is optimal.

math.DG↗

arXiv2026

Pointwise subexponential growth and near-diffusive displacement on bounded-degree graphs with non-negative Ollivier--Ricci curvature

Let $G=(V,E)$ be a possibly infinite, locally finite graph with non-negative Ollivier--Ricci curvature and degrees bounded by $d<\infty$. We prove that there exists a constant $C_d$ such that the continuous-time random walk displacement and log-volume growth satisfy \[ \mathbb{E}_x \mathrm{dist}(x,X_t)^2 \le t \exp\left[C_d \sqrt{\log t \log\log t}\right], \] \[ \log \mathrm{Vol}(B(x,r)) \le \exp\left[C_d \sqrt{\log r \log\log r}\right], \] for every $x\in V$ and all $r,t \ge e^e$.

math.DG↗

arXiv2025

Quantitative Obata's theorem in discrete setting

Under mild assumptions, we show that a connected weighted graph $G$ with lower Ricci curvature bound $K>0$ in the sense of Bakry-Émery and the $d$-th non-zero Laplacian eigenvalue $λ_d$ close to $K$, with $d$ being the maximal combinatorial vertex degree of $G$, has an underlying combinatorial structure of the $d$-dimensional hypercube graph. Moreover, such a graph $G$ is close in terms of Frobenius distance to a properly weighted hypercube graph. Furthermore, we establish their closeness in terms of eigenfunctions. Our results can be viewed as discrete analogies of the almost rigidity theorem and quantitative Obata's theorem on Rimennian manifolds.

math.DG↗

arXiv2024

Design of ANF/MXene/SSG sandwich structure with electromagnetic shielding performance and impact resistance

Since entering the information era, electronic devices gradually play an important role in daily lives. However, the abuse of electronic devices leads to corresponding electromagnetic EM wave pollution. The complex external environment causes the potential for physical impact. In this work, an ANF MXene SSG flexible sandwich structure was fabricated according to methods of vacuum filtration, directional freeze-casting solidification, and polyurethane encapsulation. Apart from its excellent protection function, the sandwich structure also acts as a human body movement sensor.

physics.app-ph↗