# Particle-number distribution in large fluctuations at the tip of branching random walks

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• معلومة اضافية
• Publication Date:
2019
• Collection:
Condensed Matter
High Energy Physics - Phenomenology
• Abstract:
We investigate properties of the particle distribution near the tip of one-dimensional branching random walks at large times $t$, focusing on unusual realizations in which the rightmost lead particle is very far ahead of its expected position -- but still within a distance smaller than the diffusion radius $\sim\sqrt{t}$. Our approach consists in a study of the generating function $G_{\Delta x}(\lambda)=\sum_n \lambda^n p_n(\Delta x)$ for the probabilities $p_n(\Delta x)$ of observing $n$ particles in an interval of given size $\Delta x$ from the lead particle to its left, fixing the position of the latter. This generating function can be expressed with the help of functions solving the Fisher-Kolmogorov-Petrovsky-Piscounov (FKPP) equation with suitable initial conditions. In the infinite-time and large-$\Delta x$ limits, we find that the mean number of particles in the interval grows exponentially with $\Delta x$, and that the generating function obeys a nontrivial scaling law, depending on $\Delta x$ and $\lambda$ through the combined variable $[\Delta x-f(\lambda)]^{3}/\Delta x^2$, where $f(\lambda)\equiv -\ln(1-\lambda)-\ln[-\ln(1-\lambda)]$. From this property, one may conjecture that the growth of the typical particle number with the size of the interval is slower than exponential, but, surprisingly enough, only by a subleading factor at large $\Delta x$. The scaling we argue is consistent with results from a numerical integration of the FKPP equation.
Comment: 24 pages, 4 figures
• Accession Number:
edsarx.1910.06382
• Citations
• ABNT:
MUELLER, A. H.; MUNIER, S. Particle-number distribution in large fluctuations at the tip of branching random walks. [s. l.], 2019. Disponível em: http://search.ebscohost.com/login.aspx?direct=true&site=eds-live&db=edsarx&AN=edsarx.1910.06382&custid=s8280428. Acesso em: 21 jan. 2020.
• AMA:
Mueller AH, Munier S. Particle-number distribution in large fluctuations at the tip of branching random walks. 2019. http://search.ebscohost.com/login.aspx?direct=true&site=eds-live&db=edsarx&AN=edsarx.1910.06382&custid=s8280428. Accessed January 21, 2020.
• APA:
Mueller, A. H., & Munier, S. (2019). Particle-number distribution in large fluctuations at the tip of branching random walks. Retrieved from http://search.ebscohost.com/login.aspx?direct=true&site=eds-live&db=edsarx&AN=edsarx.1910.06382&custid=s8280428
• Chicago/Turabian: Author-Date:
Mueller, A. H., and S. Munier. 2019. “Particle-Number Distribution in Large Fluctuations at the Tip of Branching Random Walks.” http://search.ebscohost.com/login.aspx?direct=true&site=eds-live&db=edsarx&AN=edsarx.1910.06382&custid=s8280428.
• Harvard:
Mueller, A. H. and Munier, S. (2019) ‘Particle-number distribution in large fluctuations at the tip of branching random walks’. Available at: http://search.ebscohost.com/login.aspx?direct=true&site=eds-live&db=edsarx&AN=edsarx.1910.06382&custid=s8280428 (Accessed: 21 January 2020).
• Harvard: Australian:
Mueller, AH & Munier, S 2019, ‘Particle-number distribution in large fluctuations at the tip of branching random walks’, viewed 21 January 2020, .
• MLA:
Mueller, A. H., and S. Munier. Particle-Number Distribution in Large Fluctuations at the Tip of Branching Random Walks. 2019. EBSCOhost, search.ebscohost.com/login.aspx?direct=true&site=eds-live&db=edsarx&AN=edsarx.1910.06382&custid=s8280428.
• Chicago/Turabian: Humanities:
Mueller, A. H., and S. Munier. “Particle-Number Distribution in Large Fluctuations at the Tip of Branching Random Walks,” 2019. http://search.ebscohost.com/login.aspx?direct=true&site=eds-live&db=edsarx&AN=edsarx.1910.06382&custid=s8280428.
• Vancouver/ICMJE:
Mueller AH, Munier S. Particle-number distribution in large fluctuations at the tip of branching random walks. 2019 [cited 2020 Jan 21]; Available from: http://search.ebscohost.com/login.aspx?direct=true&site=eds-live&db=edsarx&AN=edsarx.1910.06382&custid=s8280428