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Hello, bonjour, gamardjoba, jo napo,

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welcome to the fourth homework session of Statistical Mechanics: Algorithms and Computations

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from the Physics Department of Ecole Normale supérieure.

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In this week's lecture and tutorial, we really understood

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the  connection between sampling and integration.

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Let us now use what we learned to calculate difficult,

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high-dimensional integrals,

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a field where Monte Carlo methods just cannot be beaten.

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The central object this week was the unit hypersphere,

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let's compute its volume using a Markov-chain Monte Carlo random walk in high dimensions.

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These algorithms are extremely efficient,

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as you will find out for yourselves

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and they allow for beautiful calculations.

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On the Monte Carlo beach, it is impossible to compute the area of the circle,

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that is to say of a two-dimensional sphere, in a direct way.

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Instead, what we can determine is the ratio between the area of the circle and the area of the square.

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In this week's homework, you will face a similar problem:

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it is impossible to determine directly the volume of sphere in d dimensions.

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Instead, what you will write is a program that allows you to determine the ratio

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between the volumes of spheres in successive dimensions: d, d+1.

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All in all, by accumulating results, you will be able to determine the volume of a sphere

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not in 3 dimensions, but in 200 dimensions

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with a tight control on the error.

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You will get a beautiful result, which illustrates the power of statistical mechanics in the design of algorithms.