How quickly does randomness appear?

This was the [slightly off-key]  title of the math column in the November issue of La Recherche, in any case intriguing enough for me to buy this general public science magazine on the metro platform and to read it immediately while waiting for an uncertain train, thanks to the nth strike of the year on my train line… But this was the occasion for an exposition of the Metropolis algorithm in a general public journal! The column actually originated from a recently published paper by Persi Diaconis, Gilles Lebeaux, and Laurent Michel,  Geometric analysis for the Metropolis algorithm on Lipschitz domain, in Inventiones Mathematicae [one of the top pure math journals]. The column in La Recherche described the Metropolis algorithm (labelled there a random walk on Markov chains!), alluded to the use of MCMC methods in statistics, told the genesis of the paper [namely the  long-term invitation of Persi Diaconis in Nice a few years ago] and briefly explained the convergence result, namely the convergence of the Metropolis algorithm to the stationary measure at a geometric rate, with an application to the non-overlapping balls problem.

If you take a look at the paper, you will see it is a beautiful piece of mathematics, establishing a spectral gap on the Markov operator associated with the Metropolis algorithm and deducing a uniformly geometric convergence [in total variation] for most regular-and-bounded-support distributions. A far from trivial and fairly general result. La Recherche however fails to mention the whole corpus of MCMC convergence results obtained in the 1990’s and 2000’s, by many authors, incl. Richard Tweedie, Gareth Roberts, Jeff Rosenthal, Eric Moulines, Gersende Fort, Randal Douc, Kerrie Mengersen, and others…

3 Responses to “How quickly does randomness appear?”

  1. […] than of the underlying (pseudo-)random generator. (It had been a while since I had a go at this randomness controvery!) […]

  2. Despite your negative ending, I am amazed to see a popular science magazine write about an Inventiones paper — I don’t think you’ll find anything like this in Germany…

    • Yes, this is a good thing! I think it is due to the fact that the field columns of La Recherche are done by scientists from those fields, rather than regular journalists. For instance, the mathematician Jean-Michel Ghidaglia is the scientific director of the journal and I know of several mathematicians who regularly contribute… (Sorry for being negative, as often!, I just reacted to the hyperbole in the journal statement that it was “the” convergence result…)

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