21 research outputs found
Existence of a Meromorphic Extension of Spectral Zeta Functions on Fractals
We investigate the existence of the meromorphic extension of the spectral
zeta function of the Laplacian on self-similar fractals using the classical
results of Kigami and Lapidus (based on the renewal theory) and new results of
Hambly and Kajino based on the heat kernel estimates and other probabilistic
techniques. We also formulate conjectures which hold true in the examples that
have been analyzed in the existing literature
Weak uncertainty principle for fractals, graphs and metric measure spaces
We develop a new approach to formulate and prove the weak uncertainty
inequality which was recently introduced by Okoudjou and Strichartz. We assume
either an appropriate measure growth condition with respect to the effective
resistance metric, or, in the absence of such a metric, we assume the
Poincare inequality and reverse volume doubling property. We also consider
the weak uncertainty inequality in the context of Nash-type inequalities. Our
results can be applied to a wide variety of metric measure spaces, including
graphs, fractals and manifolds
Brownian motion on the Sierpinski carpet
We prove that, up to scalar multiples, there exists only one local regular
Dirichlet form on a generalized Sierpinski carpet that is invariant with
respect to the local symmetries of the carpet. Consequently for each such
fractal the law of Brownian motion is uniquely determined and the Laplacian is
well defined
Heat Kernels and Zeta Functions on Fractals
On fractals, spectral functions such as heat kernels and zeta functions
exhibit novel features, very different from their behaviour on regular smooth
manifolds, and these can have important physical consequences for both
classical and quantum physics in systems having fractal properties.Comment: 19 pages, invited contribution for JPhysA Special Issue in honour of
J. S. Dowke
The Alexander-Orbach conjecture holds in high dimensions
We examine the incipient infinite cluster (IIC) of critical percolation in
regimes where mean-field behavior has been established, namely when the
dimension d is large enough or when d>6 and the lattice is sufficiently spread
out. We find that random walk on the IIC exhibits anomalous diffusion with the
spectral dimension d_s=4/3, that is, p_t(x,x)= t^{-2/3+o(1)}. This establishes
a conjecture of Alexander and Orbach. En route we calculate the one-arm
exponent with respect to the intrinsic distance.Comment: 25 pages, 2 figures. To appear in Inventiones Mathematica
Dirac and magnetic Schr\"odinger operators on fractals
In this paper we define (local) Dirac operators and magnetic Schr\"odinger
Hamiltonians on fractals and prove their (essential) self-adjointness. To do so
we use the concept of 1-forms and derivations associated with Dirichlet forms
as introduced by Cipriani and Sauvageot, and further studied by the authors
jointly with R\"ockner, Ionescu and Rogers. For simplicity our definitions and
results are formulated for the Sierpinski gasket with its standard self-similar
energy form. We point out how they may be generalized to other spaces, such as
the classical Sierpinski carpet