We consider N Brownian particles moving on a line starting from initial positions u≡{u1,u2,…uN}\documentclass[12pt]{minimal}
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\begin{document}$$\mathbf{{u}}\equiv \{u_1,u_2,\ldots u_N\}$$\end{document} such that 0<u1<u2<⋯<uN\documentclass[12pt]{minimal}
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\begin{document}$$0<u_1 < u_2 < \cdots < u_N$$\end{document}. Their motion gets stopped at time ts\documentclass[12pt]{minimal}
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\begin{document}$$t_s$$\end{document} when either two of them collide or when the particle closest to the origin hits the origin for the first time. For N=2\documentclass[12pt]{minimal}
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\begin{document}$$N=2$$\end{document}, we study the probability distribution function p1(m|u)\documentclass[12pt]{minimal}
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\begin{document}$$p_1(m|\mathbf{{u}})$$\end{document} and p2(m|u)\documentclass[12pt]{minimal}
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\begin{document}$$p_2(m|\mathbf{{u}})$$\end{document} of the maximal distance travelled by the 1st\documentclass[12pt]{minimal}
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\begin{document}$$1^{\text {st}}$$\end{document} and 2nd\documentclass[12pt]{minimal}
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\begin{document}$$2^{\text {nd}}$$\end{document} walker till ts\documentclass[12pt]{minimal}
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\begin{document}$$t_s$$\end{document}. For general N particles with identical diffusion constants D\documentclass[12pt]{minimal}
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\begin{document}$$D$$\end{document}, we show that the probability distribution pN(m|u)\documentclass[12pt]{minimal}
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\begin{document}$$p_N(m|\mathbf{u})$$\end{document} of the global maximum mN\documentclass[12pt]{minimal}
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\begin{document}$$m_N$$\end{document}, has a power law tail pi(m|u)∼N2BNFN(u)/mνN\documentclass[12pt]{minimal}
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\begin{document}$$p_i(m|\mathbf{{u}}) \sim {N^2B_N\mathcal {F}_{N}(\mathbf{u})}/{m^{\nu _N}}$$\end{document} with exponent νN=N2+1\documentclass[12pt]{minimal}
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\begin{document}$$\nu _N =N^2+1$$\end{document}. We obtain explicit expressions of the function FN(u)\documentclass[12pt]{minimal}
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\begin{document}$$\mathcal {F}_{N}(\mathbf{u})$$\end{document} and of the N dependent amplitude BN\documentclass[12pt]{minimal}
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\begin{document}$$B_N$$\end{document} which we also analyze for large N using techniques from random matrix theory. We verify our analytical results through direct numerical simulations.
