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The Kac model describes the local evolution of a gas of $N$ particles withthree dimensional velocities by a random walk in which the steps correspond tobinary collisions that conserve momentum as well as energy. The state space ofthis walk is a sphere of dimension $3N - 4$. The Kac conjecture concerns thespectral gap in the one step transition operator $Q$ for this walk. In thispaper, we compute the exact spectral gap.  As in previous work by Carlen, Carvalho and Loss where a lower bound on thespectral gap was proved, we use a method that relates the spectral propertiesof $Q$ to the spectral properties of a simpler operator $P$, which is simply anaverage of certain non commuting projections. The new feature is that we showhow to use a knowledge of certain eigenfunctions and eigenvalues of $P$ todetermine spectral properties of $Q$, instead of simply using the spectral gapfor $P$ to bound the spectral gap for $Q$, inductively in $N$, as in previouswork. The methods developed here can be applied to many other high--dimensionalstochastic process, as we shall explain.  We also use some deep results on Jacobi polynomials to obtain the requiredspectral information on $P$, and we show how the identity through which Jacobipolynomials enter our problem may be used to obtain new bounds on Jacobipolynomials.

Determination of the spectral gap in the Kac model for physical momentum and energy conserving collisions