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A forgotten paper


Power-Law Distribution
Benoît Mandelbrot Mathematician
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Again one of these random events, which I was welcoming, because I simply wanted to find something, a mess I could be the first to simplify, These random events came up, and again linked with my uncle. My uncle lived near the observatory in Paris; I lived with my parents very far away, and often went to my uncle to chat. He was trying to improve me, to change me, maybe make me come back on the straight and narrow, and he was trying and trying and trying. And one day when I was leaving I asked him for something to read in the subway, and he leant down to his waste basket and got out a review of a book, which was sent to him by one of his colleagues at Harvard, who was sending him automatically all the reprints, including the review of this book. He had kept the reprints but the review had been thrown away, and he said more or less, "Here is a kind of nonsense which you would be amused by." And that review was amazing. The man's name was Walsh, who was a friend of a man named Zipf, and Zipf had written a book: so it was Walsh reviewing his friend's book. And he was a very good reviewer for a friend. He just covered everything which was terrible in Zipf and emphasised the mystery. And Zipf had come up with a finding, which he expressed at enormous length in his book and which Walsh summarised, which was as follows. He said there is a difference in quality between, on the one hand, randomness in physics, and on the other hand, randomness in social sciences. In physics randomness is ruled by the Gaussian distribution, so-called normal; but in social sciences randomness is ruled by power-law distribution, which was true of Pareto distribution of income, of distribution that became associated with Zipf of word frequencies; firm sizes, city sizes, etc., etc. Zipf put out a big challenge: explain these circumstances and explain the difference. Now, I'm going to go faster perhaps than I should to say how important this finding was for my life. For many years I went on developing power-law distributions and enriching them with quite an apparatus of a theoretical nature on my own, thinking as Zipf thought, as all my friends thought, that this was something specific for those exotic phenomena. And then this merged with turbulence, so the breaking point moved. On the one hand there was obviously the weather and the stock market, very irregular phenomena, which had the power-law distribution; on the other hand there was physics, which didn't have them. "Gaussian distribution was known at that time, but it was considered just" Esoterica. Paul Levy had said a few words in his course about stable distributions, which were due to Goshi, to Polya and to Paul Levy, but very little. He called them exceptional. It was just an exercise. It was something to write exams about, of no significance. It was well-established that these things didn't count, and during that period while preparing my Ph.D. I did take Paul Levy's lectures on other topics which taught me a great deal about probability theory a la Paul Levy, but that was still a bit before this. But in due time, in fact I remember the date almost precisely, I could find it precisely if I wanted, in '71 or '72, I was to give a lecture at a meeting of the American Physical Society near where I worked then at IBM. I was the evening speaker; somebody who was very wise or just simply lucky, for everybody, had thought that what I had to say would be interesting to physicists. At the end of my lecture I went to sit with one of my friends there, Herb Talan, and he was smiling to his ears. He couldn't believe it. "What you say is exactly what's happening in physics. There's a new field of physics opening up. The great names are Karenov, Fisher and Wilson; there are two pre-prints, by Fisher and Wilson which are going to be ground-breaking, and what they do is exactly what you do, but in a different field, and different style." And so I'm going forward fast because I would like to emphasise that in terms of my thinking I was totally alone. I was looking at the most esoteric phenomena in nature and using the most esoteric tools, developing the ideas of what others called scaling, normalisation, fixed points, convergence to fixed points. These ideas were developed in two totally separate environments and styles, by the physicists on their side, - and by me in the story I'm starting to tell you- over a long period of time. I started before them, but that doesn't matter. It was just a matter of a few years.

Benoît Mandelbrot (1924-2010) discovered his ability to think about mathematics in images while working with the French Resistance during the Second World War, and is famous for his work on fractal geometry - the maths of the shapes found in nature.

Listeners: Bernard Sapoval Daniel Zajdenweber

Bernard Sapoval is Research Director at C.N.R.S. Since 1983 his work has focused on the physics of fractals and irregular systems and structures and properties in general. The main themes are the fractal structure of diffusion fronts, the concept of percolation in a gradient, random walks in a probability gradient as a method to calculate the threshold of percolation in two dimensions, the concept of intercalation and invasion noise, observed, for example, in the absorbance of a liquid in a porous substance, prediction of the fractal dimension of certain corrosion figures, the possibility of increasing sharpness in fuzzy images by a numerical analysis using the concept of percolation in a gradient, calculation of the way a fractal model will respond to external stimulus and the correspondence between the electrochemical response of an irregular electrode and the absorbance of a membrane of the same geometry.

Daniel Zajdenweber is a Professor at the College of Economics, University of Paris.

Duration: 5 minutes, 38 seconds

Date story recorded: May 1998

Date story went live: 24 January 2008