Numerical simulations on the total mass, the numbers of bonds on the hull, external perimeter, singly connected bonds and gates into large fjords of the Fortuin-Kasteleyn clusters for two-dimensional q-state Potts models at criticality are presented. The data are found consistent with the...

The scaling properties of financial prices raise many questions. To provide background - appropriately so in the first issue of a new journal! - this paper, part I (sections 1 to 3), is largely a survey of the present form of some material that is well known yet repeatedly rediscovered. It...

Benoit B Mandelbrot comments on the paper by Blake LeBaron, on page 621 of this issue, by tracing the merits and pitfalls of power-law scaling models from antiquity to the present.

In the Brownian model, even the largest of N successive daily price increments contributes negligibly to the overall sample variance. The resulting 'absent' concentration justifies the role of variance in measuring Brownian volatility. Mandelbrot introduced in 1963 an alternative 'mesofractal...

This article describes a versatile family of functions that are increasingly roughened by successive interpolations. They reproduce, in the simplest way possible, the main features of financial prices: continually varying volatility, discontinuity or concentration, and the fact that many changes...

This is a direct continuation of the preceding paper, with which it shares the front material and the numbering of the sections. A little repetition makes it possible to read this paper, part II, by itself. It describes the progression of the formalism from the financial model the author...

We showed in an earlier paper (1995a) that negatively correlated fractional Brownian motion (FBM) can be generated as a fractal sum of one kind of micropulses (FSM). That is, FBM of exponent is the limit (in the sense of finite-dimensional distributions) of a certain sequence of processes...

We begin with stochastic processes obtained as sums of "up-and-down" pulses with random moments of birth [tau] and random lifetime w determined by a Poisson random measure. When the pulse amplitude [var epsilon] -- 0, while the pulse density [delta] increases to infinity, one obtains a process...

This short paper advances and defends a strong statement concerning financial modeling. It argues that, even when the present fractal models become superseded, fractal tools are bound to remain central to finance. The reasons are that the main feature of price records is roughness and that the...