## DIMACS TR: 2006-07

## Large Deviations for a Point Process of Bounded Variability

### Authors: S. Goldstein, J. L. Lebowitz and E. R. Speer

**
ABSTRACT
**

We consider a one-dimensional translation invariant point
process of density one with uniformly bounded variance of the number $N_I$
of particles in any interval $I$. Despite this suppression of fluctuations
we obtain a large deviation principle with rate function
$\F(\rho)\simeq-L^{-1}\log\Prob(\rho)$ for observing a macroscopic density
profile $\rho(x)$, $x\in[0,1]$, corresponding to the coarse-grained and
rescaled density of the points of the original process in an interval of
length $L$ in the limit $L\to\infty$. $\F(\rho)$ is not convex and is
discontinuous at $\rho\equiv1$, the typical profile.

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