We give a finite $n$ enumerative derivation of the local surplus marked point process form of Aldous's critical window limit for the Erd\H{o}s-R\'enyi random graph. For $p_n=n^{-1}+\lambda n^{-4/3}$, let $\Xi_n$ place an atom at the rescaled size and surplus of each component. Exact component enumeration yields the limiting factorial correlation densities and the uniform bound \[ \mathbb E[(\Xi_n(K))_q]\le C_K^q e^{-c_Kq^3}, \qquad q\ge1, \] for every compact marked window $K$, uniformly in admissible $n$. The cubic order is optimal for the limiting factorial measures, and the estimate persists after summing over all surpluses on compact size intervals. It yields overcrowding and local exponential moment bounds, quantitative truncation of the finite $n$ Laplace functional expansion, and local point process convergence. Using the Janson-Spencer Palm description and a classical all excess estimate, we also recover ordered $\ell^2$ component size convergence.
Let $W=n^{-1/2}\sum_{i=1}^n X_i$, where the $X_i$ are independent centered random vectors in ${\mathbb R}^p$ with $|X_{ij}|\le B$ almost surely. Suppose that $\text{Cov}(W)$ has unit diagonal and smallest eigenvalue at least $b^2>0$. We prove that the distance between $W$ and a Gaussian vector with the same covariance,...
We establish a central limit theorem in R\'enyi divergence for independent and identically distributed lattice random variables $X_1, \cdots, X_n$ with zero mean, unit variance, and maximal span $h>0$. Let $S_n=(X_1+\cdots+X_n)/\sqrt n$. Let $Z_n$ denote the standard Gaussian distribution quantized on the support latti...
We establish two complementary lower-bound mechanisms for individual eigenvalues of the Dirichlet Laplacian. First, energy orthogonality yields a frequency-dependent cap on the Fourier density of a finite spectral projection. Combining this cap with the standard $L^2$ Bessel estimate and a radial-capacity bathtub princ...
We study the eigenvalues and eigenvectors of the adjacency matrix $A$ of the Erd\H{o}s-R\'enyi graph $\mathbb G(N,p)$ in the regime $Np \gg (\log N)^2$. We establish optimal isotropic delocalization for the bulk eigenvectors $\boldsymbol u$, meaning that $\langle \boldsymbol v, \boldsymbol u\rangle^2 \leq \frac{C \log...
Let $A\in\mathbb{R}^{m\times n}$ have columns of Euclidean norm at most one. We prove that $\operatorname{disc}(A)\le2395\left(1+\log_+\frac n9\right)^{1/4}+2\sqrt2$. Here $\log_+t=\max\{0,\log t\}$. Building on Bansal and Jiang's affine spectral independence framework, we remove the $(\log\log n)^{7/4}$ factor from th...
Let $W_N$ be an exact length-$N$ block from a strictly positive i.i.d. source $\mathbf p$ on a fixed finite alphabet. We prove that the canonical T-complexity $c_T$ satisfies \[ \frac{c_T(W_N)}{e^{-\gamma}h(\mathbf p) N/\log N}\longrightarrow1 \] in probability and in $L^r$ for every fixed $1\le r<\infty$, where $h(\ma...
Thomas Schürmann· 0 citations
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