Aggregation rates in one-dimensional stochastic gas model with finite polynomial moments of particle speeds

We consider one-dimensional system of auto-gravitating sticky particles with random initial speeds and describe the process of aggregation in terms of the largest cluster size $L_n$ at any fixed time prior to the critical time. We study the asymptotic behavior of $L_n$ for the warm gas, i.e., for a system of particles with nonzero initial speeds $v_i(0)=u_i$, where $(u_i)$ is a family of i.i.d. random variables with mean zero, unit variance and finite $p$-th moment $E(|u_i|^p)<\infty$, $p\ge 2$, and obtain sharp lower and upper bounds for $L_n(t)$.