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تسجيل مستخدم جديدUpper tail large deviations for a class of distributions in First-passage percolation
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تأليف
Shuta Nakajima
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In this paper we consider the first passage percolation with identical and independent exponentially distributions, called the Eden growth model, and we study the upper tail large deviations for the first passage time ${rm T}$. Our main results prove that for any $xi>0$ and $x eq 0$, $mathbb{P}({rm T}(0,nx)>n(mu+xi))$ decays as $exp{(-(2dxi +o(1))n)}$ with a time constant $mu$ and a dimension $d$. Moreover, we extend the result to stretched exponential distributions. On the contrary, we construct a continuous distribution with a finite exponential moment where the rate function does not exist.
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Consider first passage percolation with identical and independent weight distributions and first passage time ${rm T}$. In this paper, we study the upper tail large deviations $mathbb{P}({rm T}(0,nx)>n(mu+xi))$, for $xi>0$ and $x eq 0$ with a time co
nstant $mu$ and a dimension $d$, for weights that satisfy a tail assumption $ beta_1exp{(-alpha t^r)}leq mathbb P(tau_e>t)leq beta_2exp{(-alpha t^r)}.$ When $rleq 1$ (this includes the well-known Eden growth model), we show that the upper tail large deviation decays as $exp{(-(2dxi +o(1))n)}$. When $1< rleq d$, we find that the rate function can be naturally described by a variational formula, called the discrete p-Capacity, and we study its asymptotics. For $r<d$, we show that the large deviation event ${rm T}(0,nx)>n(mu+xi)$ is described by a localization of high weights around the origin. The picture changes for $rgeq d$ where the configuration is not anymore localized.
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