Related papers: Exact Spectral Dimension of the Random Surface
The spectral dimension measures the dimensionality of a space as witnessed by a diffusing random walker. Within the causal dynamical triangulations approach to the quantization of gravity, the spectral dimension exhibits novel…
The spectral dimension has been widely used to understand transport properties on regular and fractal lattices. Nevertheless, it has been little studied for complex networks such as scale-free and small world networks. Here we study the…
We evaluate the spectral dimension in causal set quantum gravity by simulating random walks on causal sets. In contrast to other approaches to quantum gravity, we find an increasing spectral dimension at small scales. This observation can…
Consider a square random matrix with independent and identically distributed entries of mean zero and unit variance. We show that as the dimension tends to infinity, the spectral radius is equivalent to the square root of the dimension in…
Consider the long-range percolation model on the integer lattice $\mathbb{Z}^d$ in which all nearest-neighbour edges are present and otherwise $x$ and $y$ are connected with probability $q_{x,y}:=1-\exp(-|x-y|^{-s})$, independently of the…
We introduce two spectral invariants of finite metric spaces, the $q$-spectrum and the transition $q$-spectrum, defined from similarity matrices. These invariants extend the adjacency and Laplacian spectra of graphs to general finite metric…
Quasisymmetry is a well-studied property of homeomorphisms between metric spaces, and Ahlfors regular conformal dimension is a quasisymmetric invariant. In the present paper, we consider the Ahlfors regular conformal dimension of metrics on…
Consider the critical long-range percolation on $\mathbb{Z}$, where an edge connects $i$ and $j$ independently with probability $1-\exp\{-\beta\int_i^{i+1}\int_j^{j+1}|u-v|^{-2}d ud v\}$ for $|i-j|>1$ for some fixed $\beta>0$ and with…
We consider random q-state Potts models for $3\le q \le 8$ on the square lattice where the ferromagnetic couplings take two values $J_1>J_2$ with equal probabilities. For any q the model exhibits a continuous phase transition both in the…
We develop techniques to obtain rigorous bounds on the behaviour of random walks on combs. Using these bounds we calculate exactly the spectral dimension of random combs with infinite teeth at random positions or teeth with random but…
We show that the spectral dimension on non-generic branched polymers with positive susceptibility exponent is given by $d_s=2/(1+\gamma)$. For those models with $\gamma<0$ we find that $d_s=2$.
The spectral dimension $d_s$ of a weighted graph is an exponent associated with the asymptotic behavior of the random walk on the graph. The Ahlfors regular conformal dimension $\dim_\mathrm{ARC}$ of the graph distance is a quasisymmetric…
We show that the spectral dimension on non-generic branched polymer models with susceptibility exponent $\gamma$ is given by $2/(1+\gamma)$. For those models with negative $\gamma$ we find that the spectral dimension is 2.
We show that in-plane-magnetic-field assisted spectroscopy allows extraction of the in-plane orientation and full 3D shape of the quantum mechanical orbitals of a single electron GaAs lateral quantum dot with sub-nm precision. The method is…
Within the causal dynamical triangulations approach to the quantization of gravity, striking evidence has emerged for the dynamical reduction of spacetime dimension on sufficiently small scales. Specifically, the spectral dimension…
We give a new definition of dimension spectrum for non-regular spectral triples and compute the exact (i.e. non only the asymptotics) heat-trace of standard Podles spheres $S^2_q$ for $0<q<1$, study its behavior when $q\to 1$ and fully…
By considering an empirical approximation, and a new class of operators that we will call walking operators, we construct, for any positive ND-toeplitz matrix, an infinite in all dimensions matrix, for which the inverse approximates the…
We present a new solution of the asymmetric two-matrix model in the large $N$ limit which only involves a saddle point analysis. The model can be interpreted as Ising in the presence of a magnetic field, on random dynamical lattices with…
An important probe of quantum geometry is its spectral dimension, defined via a spatial diffusion process. In this work we study the spectral dimension of a ``spatial hypersurface'' in a manifoldlike causal set using the induced spatial…
Fractal dimension (D) is an effective parameter to represent the irregularity and fragmental property of a self-affine surface, which is common in physical vapor deposited thin films. D could be evaluated through the scaling performance of…