45,546 research outputs found

### Degenerate elliptic operators: capacity, flux and separation

Let $S=\{S_t\}_{t\geq0}$ be the semigroup generated on L_2(\Ri^d) by a
self-adjoint, second-order, divergence-form, elliptic operator $H$ with
Lipschitz continuous coefficients. Further let $\Omega$ be an open subset of
\Ri^d with Lipschitz continuous boundary $\partial\Omega$. We prove that $S$
leaves $L_2(\Omega)$ invariant if, and only if, the capacity of the boundary
with respect to $H$ is zero or if, and only if, the energy flux across the
boundary is zero. The global result is based on an analogous local result.Comment: 18 page

### Crystallization of random matrix orbits

Three operations on eigenvalues of real/complex/quaternion (corresponding to
$\beta=1,2,4$) matrices, obtained from cutting out principal corners, adding,
and multiplying matrices can be extrapolated to general values of $\beta>0$
through associated special functions.
We show that $\beta\to\infty$ limit for these operations leads to the finite
free projection, additive convolution, and multiplicative convolution,
respectively.
The limit is the most transparent for cutting out the corners, where the
joint distribution of the eigenvalues of principal corners of a
uniformly-random general $\beta$ self-adjoint matrix with fixed eigenvalues is
known as $\beta$-corners process. We show that as $\beta\to\infty$ these
eigenvalues crystallize on the irregular lattice of all the roots of
derivatives of a single polynomial. In the second order, we observe a version
of the discrete Gaussian Free Field (dGFF) put on top of this lattice, which
provides a new explanation of why the (continuous) Gaussian Free Field governs
the global asymptotics of random matrix ensembles.Comment: 25 pages. v2: misprints corrected, to appear in IMR

### A rectangular additive convolution for polynomials

We define the rectangular additive convolution of polynomials with
nonnegative real roots as a generalization of the asymmetric additive
convolution introduced by Marcus, Spielman and Srivastava. We then prove a
sliding bound on the largest root of this convolution. The main tool used in
the analysis is a differential operator derived from the "rectangular Cauchy
transform" introduced by Benaych-Georges. The proof is inductive, with the base
case requiring a new nonasymptotic bound on the Cauchy transform of Gegenbauer
polynomials which may be of independent interest

### Degenerate elliptic operators in one dimension

Let $H$ be the symmetric second-order differential operator on L_2(\Ri)
with domain C_c^\infty(\Ri) and action $H\varphi=-(c \varphi')'$ where c\in
W^{1,2}_{\rm loc}(\Ri) is a real function which is strictly positive on
\Ri\backslash\{0\} but with $c(0)=0$. We give a complete characterization of
the self-adjoint extensions and the submarkovian extensions of $H$. In
particular if $\nu=\nu_+\vee\nu_-$ where $\nu_\pm(x)=\pm\int^{\pm 1}_{\pm x}
c^{-1}$ then $H$ has a unique self-adjoint extension if and only if $\nu\not\in
L_2(0,1)$ and a unique submarkovian extension if and only if $\nu\not\in
L_\infty(0,1)$. In both cases the corresponding semigroup leaves
$L_2(0,\infty)$ and $L_2(-\infty,0)$ invariant.
In addition we prove that for a general non-negative c\in W^{1,\infty}_{\rm
loc}(\Ri) the corresponding operator $H$ has a unique submarkovian extension.Comment: 28 page

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