arXiv subjects

B. Scoppola

Publications and source records attributed to B. Scoppola.

5 recordsLinked to original sources

arXiv2015

On Diffusion Limited Deposition

We propose a simple model of columnar growth through {\it diffusion limited aggregation} (DLA). Consider a graph $G_N\times\N$, where the basis has $N$ vertices $G_N:=\{1,\dots,N\}$, and two vertices $(x,h)$ and $(x',h')$ are adjacent if $|h-h'|\le 1$. Consider there a simple random walk {\it coming from infinity} which {\it deposits} on a growing cluster as follows: the cluster is a collection of columns, and the height of the column first hit by the walk immediately grows by one unit. Thus, columns do not grow laterally. We prove that there is a critical time scale $N/\log(N)$ for the maximal height of the piles, i.e., there exist constants $α<β$ such that the maximal pile height at time $αN/\log(N)$ is of order $\log(N)$, while at time $βN/\log(N)$ is larger than $N^χ$. This suggests that a \emph{monopolistic regime} starts at such a time and only the highest pile goes on growing. If we rather consider a walk whose height-component goes down deterministically, the resulting \emph{ballistic deposition} has maximal height of order $\log(N)$ at time $N$. These two deposition models, diffusive and ballistic, are also compared with uniform random allocation and Polya's urn.

math.PR↗

arXiv2009

Queueing systems with pre-scheduled random arrivals

We consider a point process $i+ξ_i$, where $i\in \bZ$ and the $ξ_{i}$'s are i.i.d. random variables with variance $σ^{2}$. This process, with a suitable rescaling of the distribution of $ξ_i$'s, converges to the Poisson process in total variation for large $σ$. We then study a simple queueing system with our process as arrival process, and we provide a complete analytical description of the system. Although the arrival process is very similar to the Poisson process, due to negative autocorrelation the resulting queue is very different from the Poisson case. We found interesting connections of this model with the statistical mechanics of Fermi particles. This model is motivated by air traffic systems.

math.PR↗

arXiv2008

The Global Renormalization Group Trajectory in a Critical Supersymmetric Field Theory on the Lattice Z^3

We consider an Euclidean supersymmetric field theory in $Z^3$ given by a supersymmetric $Φ^4$ perturbation of an underlying massless Gaussian measure on scalar bosonic and Grassmann fields with covariance the Green's function of a (stable) Lévy random walk in $Z^3$. The Green's function depends on the Lévy-Khintchine parameter $α={3+ε\over 2}$ with $0<α<2$. For $α={3\over 2}$ the $Φ^{4}$ interaction is marginal. We prove for $α-{3\over 2}={ε\over 2}>0$ sufficiently small and initial parameters held in an appropriate domain the existence of a global renormalization group trajectory uniformly bounded on all renormalization group scales and therefore on lattices which become arbitrarily fine. At the same time we establish the existence of the critical (stable) manifold. The interactions are uniformly bounded away from zero on all scales and therefore we are constructing a non-Gaussian supersymmetric field theory on all scales. The interest of this theory comes from the easily established fact that the Green's function of a (weakly) self-avoiding Lévy walk in $Z^3$ is a second moment (two point correlation function) of the supersymmetric measure governing this model. The control of the renormalization group trajectory is a preparation for the study of the asymptotics of this Green's function. The rigorous control of the critical renormalization group trajectory is a preparation for the study of the critical exponents of the (weakly) self-avoiding Lévy walk in $Z^3$.

math-ph↗

arXiv2002

CRITICAL (Phi^{4}_{3,ε})

The Euclidean $(ϕ^{4})_{3,ε$ model in $R^3$ corresponds to a perturbation by a $ϕ^4$ interaction of a Gaussian measure on scalar fields with a covariance depending on a real parameter $ε$ in the range $0\le ε\le 1$. For $ε=1$ one recovers the covariance of a massless scalar field in $R^3$. For $ε=0$ $ϕ^{4}$ is a marginal interaction. For $0\le ε< 1$ the covariance continues to be Osterwalder-Schrader and pointwise positive. After introducing cutoffs we prove that for $ε> 0$, sufficiently small, there exists a non-gaussian fixed point (with one unstable direction) of the Renormalization Group iterations. These iterations converge to the fixed point on its stable (critical) manifold which is constructed.

hep-th↗

arXiv1999

Renormalization group approach to interacting polymerised manifolds

We propose to study the infrared behaviour of polymerised (or tethered) random manifolds of dimension D interacting via an exclusion condition with a fixed impurity in d-dimensional Euclidean space in which the manifold is embedded. We prove rigorously, via methods of Wilson's renormalization group, the convergence to a non Gaussian fixed point for suitably chosen physical parameters.

hep-th↗