A Note on Edwards' Hypothesis for Zero-Temperature Ising Dynamics

We give a simple criterion for checking the so called Edwards' hypothesis in certain zero-temperature, ferromagnetic spin-flip dynamics and use it to invalidate the hypothesis in various examples in dimension one and higher.Comment: 11 pages, 4 figure

Towards Conformal Invariance and a Geometric Representation of the 2D Ising Magnetization Field

We study the continuum scaling limit of the critical Ising magnetization in two dimensions. We prove the existence of subsequential limits, discuss connections with the scaling limit of critical FK clusters, and describe work in progress of the author with C. Garban and C.M. Newman.Comment: 20 pages, 1 figure, presented at the workshop "Inhomogeneous Random Systems" held at IHP (Paris) on January 26-27, 201

At the crossroads of different traditions. Social and cultural dynamics in Roman Thrace through the epigraphic practice

Il presente articolo studia i processi di integrazione della Tracia nel mondo romano attraverso l'analisi di tre fenomeni tra loro correlati: diffusione della lingua latina; diffusione della cittadinanza romana; diffusione dei nomi romani. Per illustrare questi fenomeni il contributo prende in considerazione la produzione epigrafica di tre centri urbani della 'provincia Thracia': Maroneia, Perinthos e Philippolis. Data la loro posizione geografica (rispettivamente nella Tracia egea, nel Chersoneso tracico e nell'entroterra), questi tre centri possono fornire un quadro indicativo delle dinamiche socio-culturali indotte dalla presenza romana nel territorio trace.This paper deals with the integration of Thrace into the Roman world through the analysis of three interrelated phenomena: the diffusion of the Latin language; the diffusion of Roman citizenship; the diffusion of Roman names. To highlight these phenomena the present contribution analyses the epigraphic production of three urban centres of the 'provincia Thracia': Maroneia, Perinthos, and Philippolis. Due to their geographical position (in Aegean Thrace, Thracian Chersonesos, and mainland Thrace respectively), these three cities can provide an indicative picture of the social and cultural dynamics induced by the Roman presence in the Thracian territory

Universal Behavior of Connectivity Properties in Fractal Percolation Models

Partially motivated by the desire to better understand the connectivity phase transition in fractal percolation, we introduce and study a class of continuum fractal percolation models in dimension d greater than or equal to 2. These include a scale invariant version of the classical (Poisson) Boolean model of stochastic geometry and (for d=2) the Brownian loop soup introduced by Lawler and Werner. The models lead to random fractal sets whose connectivity properties depend on a parameter lambda. In this paper we mainly study the transition between a phase where the random fractal sets are totally disconnected and a phase where they contain connected components larger than one point. In particular, we show that there are connected components larger than one point at the unique value of lambda that separates the two phases (called the critical point). We prove that such a behavior occurs also in Mandelbrot's fractal percolation in all dimensions d greater than or equal to 2. Our results show that it is a generic feature, independent of the dimension or the precise definition of the model, and is essentially a consequence of scale invariance alone. Furthermore, for d=2 we prove that the presence of connected components larger than one point implies the presence of a unique, unbounded, connected component.Comment: 29 pages, 4 figure

Critical Percolation Exploration Path and SLE(6): a Proof of Convergence

It was argued by Schramm and Smirnov that the critical site percolation exploration path on the triangular lattice converges in distribution to the trace of chordal SLE(6). We provide here a detailed proof, which relies on Smirnov's theorem that crossing probabilities have a conformally invariant scaling limit (given by Cardy's formula). The version of convergence to SLE(6) that we prove suffices for the Smirnov-Werner derivation of certain critical percolation crossing exponents and for our analysis of the critical percolation full scaling limit as a process of continuum nonsimple loops.Comment: 45 pages, 14 figures; revised version following the comments of a refere

Continuum Nonsimple Loops and 2D Critical Percolation

Substantial progress has been made in recent years on the 2D critical percolation scaling limit and its conformal invariance properties. In particular, chordal SLE6 (the Stochastic Loewner Evolution with parameter k=6) was, in the work of Schramm and of Smirnov, identified as the scaling limit of the critical percolation ``exploration process.'' In this paper we use that and other results to construct what we argue is the full scaling limit of the collection of all closed contours surrounding the critical percolation clusters on the 2D triangular lattice. This random process or gas of continuum nonsimple loops in the plane is constructed inductively by repeated use of chordal SLE6. These loops do not cross but do touch each other -- indeed, any two loops are connected by a finite ``path'' of touching loops.Comment: 16 pages, 3 figure