Maximal distance travelled by N vicious walkers till their survival

We consider $N$ Brownian particles moving on a line starting from initial positions $\mathbf{u}\equiv \{u_1,u_2,\dots u_N\}$ such that $0<u_1 < u_2 < \cdots < u_N$\mathrm{.}Theirmotiongetsstoppedattime$t_s$ when either two of them collide or when the particle closest to the origin hits the origin for the first time. For $N=2$, we study the probability distribution function $p_1(m\mathrm{\mid }\mathbf{u})$ and $p_2(m\mathrm{\mid }\mathbf{u})$ of the maximal distance travelled by the $1^{\text{st}}$ and $2^{\text{nd}}$ walker till $t_s$. For general $N$ particles with identical diffusion constants $D$, we show that the probability distribution $p_N(m\mathrm{\mid }\mathbf{u})$ of the global maximum $m_N$, has a power law tail $p_i(m\mathrm{\mid }\mathbf{u})\sim N^2{B}_N{\mathcal{F}}_N(\mathbf{u})\mathrm{/}{m}^{{\nu }_N}$ with exponent ${\nu }_N=N^2+1$. We obtain explicit expressions of the function ${\mathcal{F}}_N(\mathbf{u})$ and of the $N$ dependent amplitude $B_N$ which we also analyze for large $N$ using techniques from random matrix theory. We verify our analytical results through direct numerical simulations.