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

We study the full distribution P N ( A ) of sums A = ∑ i = 1 N where x 1 , … , x N are N ≫ 1 independent and identically distributed random variables each sampled from a given distribution p(x) with a subexponential x → ∞ tail. We consider two particular cases: (I) the one-sided stretched exponential distribution p ( x ) ∝ e − x α where 0 < x < ∞ , (II) the two-sided stretched exponential distribution p ( x ) ∝ e − | x | α where − ∞ < x < ∞ . We assume 0 < α < 1 (in both cases). As follows immediately from known theorems, for both cases (i) typical fluctuations of Δ A = A − ⟨ A ⟩ are described by the central-limit theorem, (ii) the tail A → ∞ is described by the big-jump principle P N ( A ) ≃ N p ( A ) , and (iii) in between these two regimes there is a nontrivial intermediate regime which displays anomalous scaling P N ( A ) ∼ e − N β f ( Δ A / N γ ) with anomalous exponents β , γ ∈ ( 0 , 1 ) and large-deviation function f(y) that are all exactly known. In practice, although these theoretical predictions of P N ( A ) work very well in regimes (i) and (ii), they often perform poorly in the intermediate regime (ii), with errors of several orders of magnitude for N as large as 10⁴. We calculate subleading order corrections to the theoretical predictions in the intermediate regime. We find that for 0 < α < α c , these corrections scale as power laws in N, while for α c < α < 1 they scale as stretched exponentials, where the threshold value is α c = 1 / 2 in case (I) and α c = 2 / 3 in case (II). The difference between the two cases is a result of mirror symmetry p ( x ) = p ( − x ) which holds only in the latter case.