Hello!

I am trying to understand the paper of Dembo-Vershik-Zeitouni, Large deviations for integer partitions. I am only interested in Theorem 2, which deals with the case of the uniform distribution. It describes a Large Deviation Principle LDP with a rate function $I:\widehat{DF}\to[0,\infty]$ of a form I expected to see, and speed $\sqrt n$. (Here, $\widehat{DF}$ is a space of functions that includes the ones describing the rims of partitions.) The LDP seems to imply that, for example, if I take the set $S_\varepsilon$ of partitions $\lambda$ such that $I(\lambda)\geq \varepsilon$ then $$\mathbb P(S_\varepsilon)\approx \exp\left(-\frac1{\sqrt n}\varepsilon\right).$$ Since the limit shape is the only function for which $I$ vanishes, I expected the probability of $S_\varepsilon$ to decrease as $n\to\infty$, because ---as I had understood this limit shape business--- a greater proportion would be closer to the limit shape every time (and here we are keeping partitions far from the limit shape by requiring the value of $I$ to be greater than $\varepsilon$). However, in the asymptotics above, it seems that $\mathbb P(S_\varepsilon)\to 1$ as $n\to\infty$.

What is going on?!

Thank you so much in advance!!