Subleading-order theory for condensation transitions in large deviations of sums of independent and identically distributed random variables

Abstract

We study the full distribution $P_{N}\left(A\right)$ of sums $A = \sum_{i=1}^N$ where $x_1, \dots, x_N$ are $N \gg 1$ independent and identically distributed random variables each sampled from a given distribution $p(x)$ with a subexponential $x \to \infty$ tail. We consider two particular cases: (I) the one-sided stretched exponential distribution $p(x) \propto e^{-x^\alpha}$ where $0 < x < \infty$, (II) the two-sided stretched exponential distribution $p(x) \propto e^{-|x|^\alpha}$ where $-\infty < x < \infty$. We assume $0 < \alpha < 1$ (in both cases). As follows immediately from known theorems, for both cases (i) typical fluctuations of $\Delta A = A - \left\langle A\right\rangle $ are described by the central-limit theorem, (ii) the tail $A \to \infty$ is described by the big-jump principle $P_{N}\left(A\right) \simeq N p\left(A\right)$, and (iii) in between these two regimes there is a nontrivial intermediate regime which displays anomalous scaling $P_{N}\left(A\right) \sim e^{-N^\beta f(\Delta A/N^\gamma)}$ with anomalous exponents $\beta,\gamma \in (0,1)$ and large-deviation function $f(y)$ that are all exactly known. In practice, although these theoretical predictions of $P_{N}\left(A\right)$ work very well in regimes (i) and (ii), they often perform quite poorly in the intermediate regime (ii), with errors of several orders of magnitude for $N$ as large as $10^4$. We calculate subleading order corrections to the theoretical predictions in the intermediate regime. We find that for $0 < \alpha < \alpha_c$, these corrections scale as power laws in $N$, while for $\alpha_c < \alpha < 1$ they scale as stretched exponentials, where the threshold value is $\alpha_c = 1/2$ in case (I) and $\alpha_c = 2/3$ in case (II). This difference between the two cases is a result of the mirror symmetry $p(x) = p(-x)$ which holds only in the latter case.