arXiv subjects

Xiufan Yang

Publications and source records attributed to Xiufan Yang.

7 recordsLinked to original sources

arXiv2026

A Law of Fractional Logarithm for Nested Complex Sample Covariance Matrices

We prove a law of fractional logarithm for the largest eigenvalue along a northwest-nested path of complex sample covariance matrices from one infinite array. The entries are independent and centered, with unit variance, vanishing complex second moment, and uniformly bounded moments of every fixed order. The row dimension is nondecreasing, has bounded increments, and has a positive limiting aspect ratio. After finite-size edge centering and scaling, the almost-sure limsup on the $(\log N)^{2/3}$ scale is $(1/4)^{2/3}$, and the liminf on the $(\log N)^{1/3}$ scale is $-4^{1/3}$. The corresponding cluster sets in $\mathbb{R}$ are $[0,(1/4)^{2/3}]$ and $[-4^{1/3},\infty)$. The proof compares the Laplace transform of a single smoothed count over a growing grid of full nested matrices with its Gaussian counterpart. Gaussian count concentration gives block occurrences with probability tending to one. Dyadic tail bounds yield the endpoints, and deterministic interpolation gives the cluster sets.

math.PR↗

arXiv2026

Cellular Maximal-Density Factoring: Joint Shadings, Stable Refinements, and a Conditional Kakeya Application

Maximal-density factoring must often make one parent object carry several forms of information at once: child mass, two multiplicity levels, a density estimate, and the ability to survive later geometric refinements. These properties are not stable under arbitrary deletion. We introduce a cellular factoring method that places positive child mass on a weighted bipartite graph of labelled parents and half-open spatial cells. After regularizing edge weights and cell degrees, one edge set defines both the child shading and the parent shading. Compatibility and pointwise multiplicity bounds are then exact, while any later selection of whole parent-cell incidences lifts quantitatively back to the children. The combinatorial core does not require Euclidean geometry: it extends to finite measurable atomic partitions with comparable atom measures and admits an explicit finite-iteration loss. We combine this joint construction with a weighted planar incidence estimate, a resolution-limited cellwise angular selection, a labelled slab estimate, and a global synchronization of tubelet and fine-tube mass. The resulting rectangular parent estimate has shading exponent $2-σ$, which is sharp uniformly over the class considered here. With the stated GWZ factoring data and corresponding $K_F(β)$ or $K_{KT}(β)$, we obtain the two small-middle interfaces used in the reduction. For the very-non-sticky branch, assuming both $K_{KT}(β)$ and $K_F(β)$, the stated GWZ Lemma 9.1 hypotheses together with the source-uniformity and refinement interface (I1)--(I4) yield the terminal gain for the represented fine-tube input; this conditional application supplies the corresponding input to the surrounding reduction, while the sharp-convex-parent problem lies outside its scope.

math.CA↗

arXiv2026

Profile-Stable Buffered Multiplicity Factoring in Four Dimensions

Multiplicity factoring is usually formulated for child families at comparable scales. For children of mixed geometry, thickening at the shortest parent scale produces nonuniform inflation ratios, and a single worst-case replacement does not preserve the natural density normalization. We prove a multiplicity-factoring theorem for finite indexed convex parent--child families in $\mathbb{R}^4$ that accommodates arbitrary child shapes, scales, orientations, aspect ratios, and repetitions. The local geometry of each assigned family is encoded by a thickening-weighted Frostman coefficient and a mean-normalized inflation efficiency, both determined by the base family before any shading refinement. The resulting coarse density satisfies $\mathcal{L}^{-A_\varepsilon}(w_1/w_4)^\varepsilon \mathfrak{P}_\varepsilonλ^{K_\varepsilon}$, up to the stated parameter-dependent constant, where $\mathfrak{P}_\varepsilon$ is an explicit profile of the parent loads and local efficiencies. The proof thickens arbitrary measurable shadings, projects along a shortest parent direction, establishes an indexed three-dimensional convex-union estimate in the presence of collisions, and lifts the resulting density response back to four dimensions. A weighted Hölder inequality then assembles the nonuniform parent data, while a common cellular refinement regularizes the fine and coarse multiplicities and yields the stated parentwise multiplicity-product estimate. Under a relative convex Frostman hypothesis, the profile is expressed explicitly in the minimum, mean, and maximum inflation ratios. The comparable-scale regime follows as a specialization after a preliminary load selection. Thus the theorem supplies a structural factoring input for four-dimensional overlap arguments; deriving new Kakeya maximal or Hausdorff-dimension estimates would require additional analytic ingredients.

math.CA↗

arXiv2026

Simple and distinct zeros in a prime-modulus Dirichlet family from near-microscopic to polylogarithmic heights

Fix $η>0$ and $A_0>0$. Let $q$ tend to infinity through odd primes, put $Q=\log q$, and let $T=T(q)$ satisfy \[ \frac{(\log Q)^{1+η}}{Q}\le T\le Q^{A_0}. \] Set $I=(T,2T]$, and sum without weights over the $q-2$ nonprincipal characters modulo $q$. Let $\mathcal N_q$ count nontrivial zeros in $I$ with multiplicity, let $\mathcal N^s_{0,q}$ and $\mathcal N^*_{0,q}$ count simple and distinct zeros on the critical line, and let $\mathcal N_{d,q}$ count all distinct zeros in $I$. Uniformly in this height range, we prove unconditionally \[ \mathcal N_q=\frac{qT\log q}{2π}\{1+o_{η,A_0}(1)\}, \quad \frac{\mathcal N^s_{0,q}}{\mathcal N_q}, \frac{\mathcal N^*_{0,q}}{\mathcal N_q} \ge C_{\mathrm{MT}}-o_{η,A_0}(1), \quad \frac{\mathcal N_{d,q}}{\mathcal N_q}\ge C_d-o_{η,A_0}(1), \] where \[ C_{\mathrm{MT}}=\frac32-\frac1{\sqrt2}\cot\!\left(\frac1{\sqrt2}\right) =0.672500703679\ldots, \qquad C_d=\frac{1+C_{\mathrm{MT}}}{2}=0.836250351839\ldots. \] The proof combines Selberg's family-averaged argument estimate and zero-density deletion with a finite Gevrey Gabor compression of Weil's Hermitian form. Quantitative Fourier--Laplace estimates control exterior zeros down to the lower endpoint, while a local--remote shell decomposition gives uniform control throughout the polylogarithmic upper range. Matrix moment estimates and an inertia-based rank--trace inequality then yield the counting bounds. No form of the generalized Riemann hypothesis is assumed.

math.NT↗

arXiv2026

Gevrey localization for simple and distinct zeros in a prime-modulus Dirichlet family

Let $q$ tend to infinity through odd primes, let $T=T(q)$, and put $\ell=\log(qT/2π)$. Suppose that $\ell^s=o(T)$ for some fixed $s>1$ and that \[ λ_{\rm bw}:=\min\left\{1, \liminf_{\substack{q\to\infty\\q\ \mathrm{prime}}} \frac{\log(q-1)}{\ell}\right\}>0. \] For the unweighted family of the $q-2$ nonprincipal characters modulo $q$, we prove unconditional lower bounds, relative to the total zero multiplicity in $(T,2T]$, of $2-1/c_{λ_{\rm bw}}^*$ for both simple and distinct critical-line zeros and of $\tfrac12(3-1/c_{λ_{\rm bw}}^*)$ for all distinct zeros, where \[ c_λ^*=\frac{\sqrt2\tan(λ/\sqrt2)} {1+(λ/\sqrt2)\tan(λ/\sqrt2)}. \] When $\log T=o(\log q)$, this gives respectively $0.6725007036\ldots$ and $0.8362503518\ldots$; in particular it covers every fixed $T=(\log q)^A$ with $A>1$. As the growing-height companion to the authors' public mesoscopic shrinking-height theorem, this article is organized around the different localization technology required when $T$ grows. An abstract theorem converts complex-strip Gevrey decay, sampling density, and a local multiset count into stretched-exponential localization in trace and nuclear norm. In the Dirichlet specialization the strip growth is $X^{1/4}$, where $X=\exp(λ\ell)$, and a buffer $D_0=(K\ell)^s$ absorbs the remote-zero contribution uniformly in the character. Quantitative finite-sampling end-effect bounds, a finite explicit-formula matrix, character-averaged first and second traces, hyperbolic functional-equation blocks, and a rank--trace inequality then yield the three zero statistics. The shared finite-inertia mechanism and the Montgomery--Taylor constant are not claimed as new. No form of GRH is assumed.

math.NT↗

arXiv2026

One-Sided Product-Scale Upper Bounds for Nested Complex Wishart Extremes

I study the largest eigenvalue $Λ_j$ of $X^{(j)}(X^{(j)})^*$, where $X^{(j)}$ is the $M_j\times j$ northwest rectangle of one infinite complex Gaussian array. For two upper-tail events and, separately, for two lower-tail events at the largest-eigenvalue soft edge, I prove finite-$N$ one-sided product-scale upper bounds for their joint probabilities. The logarithmic walls have polynomially small marginals, so an additive $o(1)$ covariance estimate may be much larger than the vanishing product that must be controlled. Uniformly for levels in a short macroscopic window, bounded deterministic shifts, and separations at least $N^{2/3+ε}$, each joint probability is at most $(1+o(1))$ times the product of its marginals. The classical $N^{2/3}$ correlation window corresponds to order-one extended-Airy time; the theorem works at supercritical separation. Exact Laguerre operators along admissible row--column paths provide the finite-dimensional input. The upper-tail proof uses occupancy counts and cross-block trace estimates, whereas the lower-tail proof uses gap determinants, diagonal resolvents, and a Schur complement. I transfer the result between the half-integer Laguerre and physical rectangular normalizations and deduce intrinsic first- and second-moment estimates for separated sparse grids under the rarity, count-growth, spacing, and macroscopic-window budget.

math.PR↗

arXiv2026

Extreme least singular values of random row submatrices with bounded-density subgaussian entries

Let $ξ$ be a centered real subgaussian random variable with positive variance and a bounded Lebesgue density, and let $A_m\in\mathbb{R}^{N_m\times m}$ have independent entries distributed as $ξ$, where $N_m/m\toγ>1$. For each set $I\subset[N_m]$ with $|I|=m$, let $(A_m)_I$ denote the row submatrix indexed by $I$, and define $M_m(A_m):=\min_{I\subset[N_m],\,|I|=m}σ_{\min}((A_m)_I)$. We determine its exponential scale: $\frac{1}{m}\log M_m(A_m)\xrightarrow{\mathbb{P}}-h(γ)$, where $h(γ):=γ\logγ-(γ-1)\log(γ-1)$. This extends the corresponding real Gaussian result. The main new ingredient is an upper-tail argument that avoids uniform control over exponentially many random hyperplanes. We combine a density-level local central limit theorem for delocalized directions, an averaged delocalization estimate for hyperplane normals, an exponential bound for nearly parallel pairs, and amplification using a linear number of independent probe rows. For every fixed $\varepsilon\in(0,h(γ))$, the probability of an $\varepsilon$-deviation is at most $C\exp(-c\sqrt{m})$ for all sufficiently large $m$. Under the canonical coupling induced by a single infinite i.i.d. array, this summable deviation estimate yields a uniform almost-sure exponential law over every compact range of aspect ratios. In particular, at the real phase-retrieval threshold $N_m=2m-1$, the Balan--Wang stability parameter has exponential base $1/4$ in probability and, under this coupling, almost surely.

math.PR↗