Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Determination of the spectral gap in the Kac model for physical momentum
  and energy conserving collisions   The Kac model describes the local evolution of a gas of $N$ particles with
three dimensional velocities by a random walk in which the steps correspond to
binary collisions that conserve momentum as well as energy. The state space of
this walk is a sphere of dimension $3N - 4$. The Kac conjecture concerns the
spectral gap in the one step transition operator $Q$ for this walk. In this
paper, we compute the exact spectral gap.
  As in previous work by Carlen, Carvalho and Loss where a lower bound on the
spectral gap was proved, we use a method that relates the spectral properties
of $Q$ to the spectral properties of a simpler operator $P$, which is simply an
average of certain non commuting projections. The new feature is that we show
how to use a knowledge of certain eigenfunctions and eigenvalues of $P$ to
determine spectral properties of $Q$, instead of simply using the spectral gap
for $P$ to bound the spectral gap for $Q$, inductively in $N$, as in previous
work. The methods developed here can be applied to many other high--dimensional
stochastic process, as we shall explain.
  We also use some deep results on Jacobi polynomials to obtain the required
spectral information on $P$, and we show how the identity through which Jacobi
polynomials enter our problem may be used to obtain new bounds on Jacobi
polynomials.

8
The categories of this paper are: hep-ex
0
The categories of this paper are: hep-ph
11
The categories of this paper are: hep-lat
36
The categories of this paper are: cmp-lg
25
The categories of this paper are: patt-sol
23
The categories of this paper are: solv-int
3
The categories of this paper are: physics
29
The categories of this paper are: funct-an
27
The categories of this paper are: comp-gas
4
The categories of this paper are: cond-mat
7
The categories of this paper are: hep-th
28
The categories of this paper are: alg-geom
10
The categories of this paper are: q-bio
24
The categories of this paper are: dg-ga
30
The categories of this paper are: q-alg
31
The categories of this paper are: ao-sci