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

Hydrodynamics and perfect fluids: uniform description of soft observables in Au+Au collisions at RHIC

It is argued that the use of the initial Gaussian energy density profile for hydrodynamics leads to much better uniform description of the RHIC heavy-ion data than the use of the standard initial condition obtained from the Glauber model. With the modified Gaussian initial conditions we successfully reproduce the transverse-momentum spectra, v2, and the pionic HBT radii (including their azimuthal dependence). The emerging consistent picture of hadron production hints that a solution of the long standing RHIC HBT puzzle has been found.Comment: Talk presented by WF at the XXXVIII International Symposium on Multiparticle Physic

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

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

Semileptonic Lambda_b decay to excited Lambda_c baryons at order Lambda_{QCD}/m_Q

Exclusive semileptonic Lambda_b decays to excited charmed Lambda_c baryons are investigated at order Lambda_{QCD}/m_Q in the heavy quark effective theory. The differential decay rates are analyzed for the J^\pi=1/2^- Lambda_c(2593) and the J^\pi=3/2^- \Lambda_c(2625). They receive 1/m_{c,b} corrections at zero recoil that are determined by mass splittings and the leading order Isgur-Wise function. With some assumptions, we find that the branching fraction for Lambda_b decays to these states is 2.5-3.3%. The decay rate to the helicity \pm 3/2 states, which vanishes for m_Q \to \infty, remains small at order Lambda_{QCD}/m_Q since 1/m_c corrections do not contribute. Matrix elements of weak currents between a Lambda_b and other excited Lambda_c states are analyzed at zero-recoil to order Lambda_{QCD}/m_Q. Applications to baryonic heavy quark sum-rules are explored.Comment: 27 pages, 1 fig., minor changes, version to appear in Phys.Rev.

The Cuntz splice does not preserve ∗*-isomorphism of Leavitt path algebras over Z\mathbb{Z}

We show that the Leavitt path algebras $L_{2,\mathbb{Z}}$ and $L_{2-,\mathbb{Z}}$ are not isomorphic as $*$-algebras. There are two key ingredients in the proof. One is a partial algebraic translation of Matsumoto and Matui's result on diagonal preserving isomorphisms of Cuntz--Krieger algebras. The other is a complete description of the projections in $L_{\mathbb{Z}}(E)$ for $E$ a finite graph. This description is based on a generalization, due to Chris Smith, of the description of the unitaries in $L_{2,\mathbb{Z}}$ given by Brownlowe and the second named author. The techniques generalize to a slightly larger class of rings than just $\mathbb{Z}$.Comment: 17 pages. Since version 2 we extended the arguments from Z to more general ring