Lucio Russo

On the critical percolation probabilities

SummaryWe prove that the critical probabilities of site percolation on the square lattice satisfy the relation pc+pc/*=1. Furthermore we prove the continuity of the function “percolation probability”.

An Approximate Zero-One Law

SummaryWe prove an approximate zero-one law, which holds for finite Bernoulli schemes. An application to percolation theory is given.

Percolation points and critical point in the Ising model

Rigorous inequalities are proved, which relate percolation probability, mean cluster size and pair connectedness respectively with magnetization, susceptibility and pair correlation function in ferromagnetic Ising models. In two dimensions the critical point is shown to be a percolation point, while in three dimensions this is not true.

Percolation and phase transitions in the Ising model

We give a description of the mechanism of phase transitions in the Ising model, pointing out the connection between the spontaneous magnetization and the existence of infinite clusters of “up” and “down” spins. The picture is more complete in the two-dimensional Ising model, where we can also use a generalized version of a result by Miyamoto.

The infinite cluster method in the two-dimensional Ising model

By studying infinite clusters in the two dimensional ferromagnetic Ising model some new results on the problem of existence of non-translation invariant equilibrium states are obtained. Furthermore a new proof of a theorem by Abraham and Reed is given.

Stable and unstable manifolds of the Hénon mapping

By using a parametric representation of the stable and unstable manifolds, we prove that for some given values of the parameter (in particular in the case first investigated by Hénon) the Hénon mapping has a transversal homoclinic orbit.

On the uniqueness of the infinite cluster in the percolation model

We simplify the recent proof by Aizenman, Kesten and Newman of the uniqueness of the infinite open cluster in the percolation model. Our new proof is more suitable for generalization in the direction of percolation-type processes with dependent site variables.

Markov processes, Bernoulli schemes, and Ising model

We give conditions for the Bernoullicity of the ν-dimensional Markov processes.

A family of codes between some Markov and Bernoulli schemes

We construct a family of almost continuous codes between a mixing one-step Markov process with two symbols and a Bernoulli scheme.

The origin of modern astronomical theories of tides: Chrisogono, de Dominis and their sources

From the Renaissance to the seventeenth century the phenomenon of tidal motion constituted one of the principal arguments of scientific debate. Understanding the times for high and low water was of course often essential for navigation, but local variations (which nowadays are attributed to currents, coastal configurations, prevailing winds, seabed shaping and other geographic characteristics) made an inductive approach impractical and precluded the possibility of constructing a universally valid model for predicting these times. Notwithstanding the complexity of the phenomenon and its practical import, however, the early-modern theory of tidal ebb and flow, as clearly emerges from Duhems analysis, appears to be neither the result of the interpretation of empirical data, nor aimed to their prediction. Rather, the interest in tides was of a theoretical nature and was aroused particularly by their double nature, being at the same time variable and regular, terrestrial and astronomical.