We study the normalized half moment $\alpha_{\mathbb R}^{(\lambda)}(N)$ of the size-$N$ Laguerre orthogonal ensemble for real shape $\lambda\ge0$. At integer shape this is the expected average singular value of an $N\times(N+\lambda)$ real Gaussian matrix. The square mean increases with the dimension, whereas every real shape $\lambda\ge1$ decreases. Between these two regimes the decrement is strictly increasing in $\lambda$, and hence has a unique zero in $(0,1)$ for every $N$. There is also a unique crossing of the Marchenko--Pastur limit. Both thresholds converge to $\lambda_*=1-\pi/4$, and their first corrections show that they separate on the scale $(\log N)/N$. The proof starts from an exact decomposition of the real half moment into its complex counterpart and a positive orthogonal correction. Recent unitary estimates take care of the complex term. An Abel completion, together with a Laguerre connection formula, turns the orthogonal correction into a positive diagonal series; the square case, the regime $\lambda\ge1$, and the transition can then all be read from this same series.
Let $W_{N,N+\lambda}$ have the Laguerre unitary distribution with size $N$ and real shape $\lambda\ge0$. For $s>0$, we consider the normalized moment $$C_{s,\lambda}(N)=N^{-s-1}\mathbb{E}[\operatorname{Tr}W_{N,N+\lambda}^{s}]$$ and its dimension decrement $\Gamma_{N,s,\lambda}=C_{s,\lambda}(N)-C_{s,\lambda}(N+1)$. Iter...
We study the smallest nonzero eigenvalue of the sample correlation matrix $\mathbf{R}_n$ formed from a $p_n \times n$ data matrix with i.i.d. real entries $\xi$ of mean zero and unit variance, in the high-dimensional regime $p_n / n \to \phi \in (0, \infty) \setminus \{1\}$. In the tall regime $\phi>1$, we prove almost...
Ze-Qin Lin, Guamgming Pan, Hao-Zhu Zhao et al.· 1 citation
For a centered, variance-one random variable $X$ with finite fourth moment, let $x$ be the vector of square-free degree-$d$ monomials in $n$ independent copies of $X$. At the critical scale $d^2/n \to \lambda \in (0,\infty)$, the normalized squared length of $x$ converges to the lognormal variable $R = \exp(\sqrt{\lamb...
We provide limit theory for the trace of the squared sample correlation matrix $\mathbf R$, constructed from $n$ observations of a $p$-dimensional random vector with iid components. If the entries have finite fourth moment and $p$ and $n$ grow proportionally, it is known that $\operatorname{tr}({\mathbf R}^2)$ satisfie...
We study the maximizers of the generalized volume product \[ \gamma_\sigma^n(A)\,\gamma_\sigma^n(A^\circ) \] among all measurable subsets $A\subset\mathbb{R}^n$, where $A^\circ$ denotes the polar set of $A$, and where $\gamma_\sigma^n$ denotes the centered Gaussian probability measure on $\mathbb{R}^n$ with covariance...
S. Artstein-Avidan, M. Fradelizi, K. Wyczesany· 0 citations
We study forward orbits in $\mathbb Z^2$ generated by the expanding affine maps $(x,y)\mapsto(mx+i,ny+j)$, where $m\ge n\ge2$ are integers and $(i,j)$ ranges over a nonempty digit set $\Lambda\subseteq\{0,\ldots,m-1\}\times\{0,\ldots,n-1\}$. We obtain explicit formulae for the mass, Beurling, discrete packing, and Asso...
Jun-Jie Miao, Ming-Hui Xu· 0 citations
We use cookies to run the site and, with your consent, for analytics and to show ads.
See our Cookie Policy.
