Uniform splitting conjecture for macroscopic cycles

Let $T$ satisfy $\nu_\infty(T)>0$. For a macroscopic cycle $\gamma^{(i)}$ of length $L^{(i)}$, let $R_i(\pi)$ be its splitting rate and, for $a\in[0,1]$, define the cumulative distribution function of the split length by

$$\theta_i^{(a)}(\pi)=\frac{1}{R_i(\pi)}\sum_{x\in\gamma^{(i)}}\sum_{k=K}^{aL^{(i)}}P\left(\pi,\pi\circ\tau_{x,\pi^k(x)}\right).$$

Uniform splitting conjecture. For every $\varepsilon>0$,

$$\lim_{|\Lambda|\to\infty}\mathbb{P}_{\Lambda}\left(\left|\theta_i^{(a)}(\pi)-a\right|>\varepsilon\right)=0.$$

This asserts that, asymptotically, a macroscopic cycle is split uniformly with respect to the normalized split length. It is presented as an additional conjecture supporting the effective split-merge description of the cycle dynamics.