arXiv subjects

Yanjin Xiang

Publications and source records attributed to Yanjin Xiang.

9 recordsLinked to original sources

arXiv2026

Fourth-Moment Strong Universality for Finite-Type Transpose-Correlated Random Matrices

We prove a strong-universality theorem at the exact fourth-moment threshold for finite families of non-Hermitian random matrices assembled from independent unordered-pair vector atoms. Within one atom, the matrix colors and the two endpoint orientations may have arbitrary joint real covariance, subject to reversal consistency across ordered type pairs and endpoint exchangeability in same-type blocks; the law may also depend on finitely many endpoint types. The matrices may be adjoined to an arbitrary deterministic tuple that converges jointly strongly with the type projections. For every fixed matrix amplification and fixed noncommutative $*$-polynomial, the resulting tuple converges strongly to an explicit covariance-matched free Gaussian family. Cross-type blocks are described by masked circular variables, whereas same-type blocks split into independent endpoint-symmetric and endpoint-antisymmetric semicircular sectors. Only a finite radial fourth moment is assumed off the diagonal, and a finite second moment suffices on the diagonal. The mode of convergence depends on the coupling: corners of one infinite array converge almost surely on a common event, while fixed-law nonnested triangular arrays converge in probability for each fixed test. As an application, we obtain exact-fourth-moment strong limits for finite-separable continuous left/right profiles, including separately weighted literal-transpose terms. For the nested almost-sure formulation, the fourth-moment threshold is sharp already on the Wigner subfamily; at this threshold arbitrary fresh rows admit no coupling-invariant almost-sure upgrade of our in-probability conclusion.

math.ST↗

arXiv2026

Optimal State-Space Order for Spectral Gaps of Sliding-Window Occupation Counts

Let $P$ be an irreducible reversible Markov kernel on a $m$-state space $Ω$, and denote its right spectral gap $γ=1-λ_2(P)$. From a stationary trajectory, let $K_t$ be the occupation-count vector of the length-$n$ window beginning at time $t$. The stationary pair $(K_0,K_1)$ defines a reversible projected count kernel $\widetilde P_n$. For every $m\ge2$, let \[ c_m^\star= \inf_{\substack{ P,\; n \ge 2}} \frac{n\Gap(\widetilde P_n)}{\Gap(P)}. \] We prove \[ \frac1{1080m}\le c_m^\star\le q_{m-2}, \qquad q_0=\frac14,\quad q_{r+1}=q_r(1-q_r). \] We also show $q_{m-2}=(m+\log m+O(1))^{-1}$, which implies that $c_m^\star=Θ(m^{-1})$. Thus, the optimal comparison coefficient has order $m$, although its exact value remains open. The lower bound also holds for $n=1$ and is uniform in $P$, including sparse and periodic kernels. Its proof combines short-window decorrelation with an averaged anchor-excursion decomposition, a Green-kernel hitting estimate, and a stopped Carleson--Hardy inequality. A nested rare-state construction produces finite $m$-state witnesses whose normalized Rayleigh quotients approach $q_{m-2}$ through an ordered sequence of limits. For every fixed finite irreducible reversible aperiodic kernel on at least two states, $\Gap(\widetilde P_n)=Θ_P(n^{-1})$.

math.ST↗

arXiv2026

Conditionally Resampled Sliding-Window Count Kernels: Spectral-Gap Bounds and Poincaré Inequalities

We study the conditionally resampled sliding-window count kernel associated with the empirical counts of length-$n$ windows from a stationary finite-state reversible Markov chain. Although the resulting count process is generally not Markov, its stationary one-step conditional law defines a genuine Markov kernel. For every fixed strictly positive reversible kernel $P$ on a finite state space, we present a Poincaré inequality for the induced count kernel $\tP_n$ of length $n$. In other words, we derive the lower bound of the spectral gap $\Gap(\tP_n)$ of $\tP_n$ as \[ \Gap(\tP_n)\ge \frac{c(P)}{n}, \] where $c(P)>0$ depends only on $P$. The proof combines a martingale oscillation inequality for the stationary path law with a direct comparison of coordinate oscillations to the Dirichlet form of the count kernel. A linear statistic of the count vector gives the matching $O(1/n)$ upper bound, so for every fixed strictly positive reversible $P$ one has $\Gap(\tP_n)=Θ_P(1/n)$. The resulting count-space Poincaré inequality yields a local-to-global variance bound for finite-window count statistics and, together with a general matrix-concentration principle, operator-norm concentration for matrix-valued empirical averages.

math.ST↗

arXiv2026

Strong Convergence for a General Class of Random Matrix Models

Let $X_{1,n},\ldots,X_{d,n}$ be $n\times n$ random matrices built from independent i.i.d. entry arrays, with centered entries, normalized by $n^{-1/2}$. We prove that, if every entry law has finite fourth moment, then this tuple converges almost surely strongly in $*$-distribution to a free circular family with the matching variances. Equivalently, normalized traces and operator norms converge for every fixed noncommutative $*$-polynomial, including polynomials with fixed matrix coefficients. No assumption is imposed on the pseudo-variances of the complex entries. The bounded-entry argument applies the spectrum and moment universality estimates of Brailovskaya and van Handel to all self-adjoint linear pencils. The matching Gaussian pencils are reduced to independent Wigner matrices and identified by Anderson's strong convergence theorem. A fixed-level centered truncation, followed by the Bai--Yin norm bound, transfers the result to finite fourth moments.

math.PR↗

arXiv2026

Layerwise Terminal Discrepancy in Chen's Reverse-Heat Coupling on the Boolean Cube

Recently, Chen \cite{Chen2026} proved that Talagrand's Boolean convolution conjecture holds up to the dimension-free factor $(\log\logη)^{3/2}$, namely for every fixed $τ>0$, \[ μ\{P_τf>η\|f\|_1\} \le C_τ \frac{(\log\logη)^{3/2}}{η\sqrt{\logη}}, \qquad η>e^3. \] We revisit the terminal testing-discrepancy step in Chen's perturbed reverse-heat coupling. Chen estimates this discrepancy globally in terms of the remaining gap to the terminal level. We keep the same coupling and the same reverse-heat formulations, but localize the terminal discrepancy on each remaining-gap layer before summing the layers. This changes the fixed-time anti-concentration cost from order $(\log L)^{3/2}/\sqrt L$ to order $(\log L)/\sqrt L$, where $L=\logη$. Consequently, we obtain a $(\log\logη)^{1/2}$ improvement as \[ μ\{P_τf>η\|f\|_1\} \le C_τ \frac{\log\logη}{η\sqrt{\logη}}, \qquad η>e^3. \]

math.FA↗

arXiv2026

Universality for Products of Random Matrices with i.i.d. Entries and the Fuss--Catalan Number

Let $(w_{ij})_{i,j\ge1}$ be a single infinite array of independent identically distributed real- or complex-valued entries of mean zero, variance $σ^2$, and finite fourth moment. Set $W_n=(w_{ij})_{1\le i,j\le n}$ and $X_n=n^{-1/2}W_n$. For every fixed $k\ge1$, we identify the almost sure limiting operator norm of several fixed products built from this family. Define the $k$-th freeness coefficient by \[ γ_k:=\sqrt{\frac{(k+1)^{k+1}}{k^k}}. \] Then we prove \[ \|X_n^k\|\toσ^kγ_k \qquad \text{almost surely}. \] The same limit holds for products sampled with replacement from any fixed finite pool of independent copies of $X_n$; in particular, it holds for the product of $k$ independent copies. Thus, the freeness coefficient captures the non-commuting characteristic between large random matrices %powers and independent or fixed-pool sampled products under the finite fourth moment assumption. The improvement of the classical Bai--Yin-type power estimate from the scale $σ^k(k{+}1)$ to $σ^k \sqrt{k{+}1}$ is a direct corollary of our result. The main technical challenge is to prove the upper bound using a high-moment expansion of %the upper bound is proved by a high-moment expansion of $\E\Tr((X_n^kX_n^{*k})^m)$. The leading zero-defect trace words are tree-like and are counted by the Fuss--Catalan number \[ F_{k,m}= \frac1{km+1}\binom{(k+1)m}{m}. \] The combinatorial tool helps to devise a defect-sensitive global enumeration: if $L=km$ and \[ r=(L+1-v)+(L-q), \] then the number of admissible word classes with defect $r$ is at most $F_{k,m}(Cm)^{Dr}$. This polynomial-in-$m$ loss, with degree proportional to the defect, is summable in the logarithmic moment range.

math.ST↗

arXiv2026

Finite-Iteration Local Dynamics and Warm Starts for Alternating Power Iteration in Spiked Tensor PCA

We study simultaneous alternating power iteration for fixed-order asymmetric rank-one spiked tensor models. Our main contribution is a finite-iteration local theory that is independent of any particular initialization. Once the iterates enter a sufficiently small neighborhood of the planted rank-one direction, their error decomposes into a geometrically decaying transient and an intrinsic noise floor caused by fixed orthogonal noise contractions at the planted point. The deterministic finite-sample conditions are stated explicitly, but under a coarse fixed-order multilinear noise event they reduce to a conservative high-signal regime for fixed or slowly expanding local radii. We then separate the warm-start mechanism from any specific spectral construction. A generic one-sweep principle shows that, if a sign-compatible initializer has correlation $γ_N$, first-sweep noise level $a_N$, and $a_N/(γ_N^{d-1}ω_{N,d})\to0$, then one can choose an expanding radius $r_N=o(ω_{N,d})$ for which the first sweep enters the local basin. After entry, the local affine contraction yields convergence to the unique informative local fixed point in that basin. For centered-Gram initialization, we verify the required correlation and same-sample first-sweep noise bound under i.i.d. finite-fourth-moment noise by a signal-preserving noise-only leave-one comparison and an averaged leave-one slice-contraction estimate, which we call a pressed-back estimate. The leave-one comparison keeps the spike fixed and averages over the deleted coordinate, so planted coordinates enter through $\ell_2$-weighted sums rather than worst-case incoherence bounds.

stat.ML↗

arXiv2026

Universality of General Spiked Tensor Models

We study asymmetric rank-one spiked tensor models in the high-dimensional regime, where the noise entries are independent and identically distributed with zero mean, unit variance, and finite fourth moment. This extends the classical Gaussian framework to a substantially broader class of noise distributions. We analyze the maximum-likelihood estimator associated with the best rank-one approximation of an order-$d$ tensor, for $d\ge 3$. Our approach is formulated along an informative, spectrally separated branch of stationary points of the non-convex maximum-likelihood landscape. In the core order-three asymmetric model, we verify locally in the high-signal regime that such an informative branch exists and remains separated from the bulk. Under this branch-selection framework, we show that the empirical spectral distribution of a suitable block-wise tensor contraction converges almost surely to the same deterministic limit as in the Gaussian case. As a consequence, the asymptotic singular value and the mode-wise alignments between the estimated and planted spike directions admit the same explicit characterizations as under Gaussian noise. These results establish a universality principle for asymmetric spiked tensor models: the high-dimensional spectral behavior and statistical limits of the selected maximum-likelihood stationary point are robust beyond the Gaussian setting. Our proof combines resolvent methods from random matrix theory, cumulant expansions under finite fourth-moment assumptions, and Efron--Stein-type variance bounds. A main technical difficulty is to control the statistical dependence between the estimator and the noise, including the associated cross terms in the non-Gaussian setting.

math.ST↗

arXiv2026

UniMatSim: A High-Throughput Materials Simulation Automation Framework Based on Universal Machine Learning Potentials

Universal machine learning interatomic potentials (UMLIPs) offer accuracy close to first-principles calculations at a fraction of the cost, showing significant potential for large-scale material simulations. However, the fragmented UMLIPs ecosystem lacks unified interface standards and integration frameworks, hindering their automated deployment in high-throughput workflows. To address this, we present UniMatSim, a modular Python framework. It systematically integrates various UMLIPs (e.g., CHGNet, M3GNet, MACE) and automates workflows from structural optimization to stability verification. The framework enables seamless model switching via abstracted interfaces, incorporates task orchestration, and provides standardized modules for key properties (elasticity, phonons, molecular dynamics), including automated handling for low-dimensional materials. As a test case, using the 2D Lieb lattice system, we constructed a multi-stage high-throughput screening workflow covering structural optimization, elastic stability, and phonon spectrum calculations. Starting from 1,176 candidate compositions, a four-model consensus pipeline yields 393 stable structures. These are refined by magnetic-state screening and DFT band-structure calculations to 59 Lieb-lattice candidates with staggered-magnetic-band characteristics. Results show UniMatSim significantly improves computational efficiency and reproducibility, providing a reliable infrastructure for data-driven materials discovery and design.

cond-mat.mtrl-sci↗