  In [A dozen de {F}inetti-style results in search of a theory, Ann. Inst. H.
Poincar\'{e} Probab. Statist. 23(2)(1987), 397--423], Diaconis and Freedman
studied low-dimensional projections of random vectors from the Euclidean unit
sphere and the simplex in high dimensions, noting that the individual
coordinates of these random vectors look like Gaussian and exponential random
variables respectively. In subsequent works, Rachev and R\"uschendorf and Naor
and Romik unified these results by establishing a connection between $\ell_p^N$
balls and a $p$-generalized Gaussian distribution. In this paper, we study
similar questions in a significantly generalized and unifying setting, looking
at low-dimensional projections of random vectors uniformly distributed on sets
of the form \[B_{\phi,t}^N := \Big\{(s_1,\ldots,s_N)\in\mathbb{R}^N : \sum_{ i
=1}^N\phi(s_i)\leq t N\Big\},\] where $\phi:\mathbb{R}\to [0,\infty]$ is a
potential (including the case of Orlicz functions). Our method is different
from both Rachev-R\"uschendorf and Naor-Romik, based on a large deviation
perspective in the form of quantitative versions of Cram\'er's theorem and the
Gibbs conditioning principle, providing a natural framework beyond the
$p$-generalized Gaussian distribution while simultaneously unraveling the role
this distribution plays in relation to the geometry of $\ell_p^N$ balls. We
find that there is a critical parameter $t_{\mathrm{crit}}$ at which there is a
phase transition in the behaviour of the projections: for $t >
t_{\mathrm{crit}}$ the coordinates of random points sampled from $B_{\phi,t}^N$
behave like uniform random variables, but for $t \leq t_{\mathrm{crit}}$ the
Gibbs conditioning principle comes into play, and here there is a parameter
$\beta_t>0$ (the inverse temperature) such that the coordinates are
approximately distributed according to a density proportional to $e^{
-\beta_t\phi(s)}$.
