Strong law of large numbers for loop lengths

Consider a random loop configuration in a box $\Lambda$ of side length $L$, with loop lengths $\ell_j(\omega)$ ordered decreasingly. Let $m(\beta)$ denote the limiting density of points belonging to long loops. Strong law conjecture. There exists $m(\beta)$ such that, as $L\to\infty$, for almost every realization $\omega$,

$$\lim_{K\to\infty}\lim_{L\to\infty}\sum_{j=1}^{K}\frac{\ell_j(\omega)}{|\Lambda|}=m(\beta),$$

and

$$\lim_{k\to\infty}\lim_{L\to\infty}\sum_{j:\,\ell_j\leq k}\frac{\ell_j(\omega)}{|\Lambda|}=1-m(\beta).$$

Here $m(\beta)$ is the mass of points in long loops: it is expected to vanish for small $\beta$ and become positive beyond the transition point. The conjecture asserts that the density in intermediate, mesoscopic loops vanishes; the supplied text gives no resolution.