The statistical geometry of scale-free random trees

The properties of scale-free random trees are investigated using both preconditioning on non-extinction and fixed size averages, in order to study the thermodynamic limit. The scaling form of volume probability is found, the connectivity dimensions are determined and compared with other exponents which describe the growth. The (local) spectral dimension is also determined, through the study of the massless limit of the Gaussian model on such trees.Comment: 21 pages, 2 figures, revtex4, minor changes (published version

Fractional variational calculus of variable order

We study the fundamental problem of the calculus of variations with variable order fractional operators. Fractional integrals are considered in the sense of Riemann-Liouville while derivatives are of Caputo type.Comment: Submitted 26-Sept-2011; accepted 18-Oct-2011; withdrawn by the authors 21-Dec-2011; resubmitted 27-Dec-2011; revised 20-March-2012; accepted 13-April-2012; to 'Advances in Harmonic Analysis and Operator Theory', The Stefan Samko Anniversary Volume (Eds: A. Almeida, L. Castro, F.-O. Speck), Operator Theory: Advances and Applications, Birkh\"auser Verlag (http://www.springer.com/series/4850

Phase separation and suppression of critical dynamics at quantum transitions of itinerant magnets: MnSi and (Sr1−x_{1-x}Cax_{x})RuO3_{3}

Quantum phase transitions (QPTs) have been studied extensively in correlated electron systems. Characterization of magnetism at QPTs has, however, been limited by the volume-integrated feature of neutron and magnetization measurements and by pressure uncertainties in NMR studies using powderized specimens. Overcoming these limitations, we performed muon spin relaxation ($\mu$SR) measurements which have a unique sensitivity to volume fractions of magnetically ordered and paramagnetic regions, and studied QPTs from itinerant heli/ferro magnet to paramagnet in MnSi (single-crystal; varying pressure) and (Sr$_{1-x}$Ca$_{x}$)RuO$_{3}$ (ceramic specimens; varying $x$). Our results provide the first clear evidence that both cases are associated with spontaneous phase separation and suppression of dynamic critical behavior, revealed a slow but dynamic character of the ``partial order'' diffuse spin correlations in MnSi above the critical pressure, and, combined with other known results in heavy-fermion and cuprate systems, suggest a possibility that a majority of QPTs involve first-order transitions and/or phase separation.Comment: 11 pages, 4 figures, 21 authors, to appear in Nature Physic

The holomorphy conjecture for nondegenerate surface singularities

The holomorphy conjecture states roughly that Igusa's zeta function associated to a hypersurface and a character is holomorphic on $\mathbb{C}$ whenever the order of the character does not divide the order of any eigenvalue of the local monodromy of the hypersurface. In this article we prove the holomorphy conjecture for surface singularities which are nondegenerate over $\mathbb{C}$ with respect to their Newton polyhedron. In order to provide relevant eigenvalues of monodromy, we first show a relation between the normalized volume (which appears in the formula of Varchenko for the zeta function of monodromy) of faces in a simplex in arbitrary dimension. We then study some specific character sums that show up when dealing with false poles. In contrast with the context of the trivial character, we here need to show fakeness of certain poles in addition to the candidate poles contributed by $B_1$-facets.Comment: 21 pages, 3 figure

Three particles in a finite volume: The breakdown of spherical symmetry

Lattice simulations of light nuclei necessarily take place in finite volumes, thus affecting their infrared properties. These effects can be addressed in a model-independent manner using Effective Field Theories. We study the model case of three identical bosons (mass m) with resonant two-body interactions in a cubic box with periodic boundary conditions, which can also be generalized to the three-nucleon system in a straightforward manner. Our results allow for the removal of finite volume effects from lattice results as well as the determination of infinite volume scattering parameters from the volume dependence of the spectrum. We study the volume dependence of several states below the break-up threshold, spanning one order of magnitude in the binding energy in the infinite volume, for box side lengths L between the two-body scattering length a and L = 0.25a. For example, a state with a three-body energy of -3/(ma^2) in the infinite volume has been shifted to -10/(ma^2) at L = a. Special emphasis is put on the consequences of the breakdown of spherical symmetry and several ways to perturbatively treat the ensuing partial wave admixtures. We find their contributions to be on the sub-percent level compared to the strong volume dependence of the S-wave component. For shallow bound states, we find a transition to boson-diboson scattering behavior when decreasing the size of the finite volume.Comment: 21 pages, 4 figures, 2 table