# Upper tail large deviations for a class of distributions in First-passage percolation

@article{Nakajima2019UpperTL, title={Upper tail large deviations for a class of distributions in First-passage percolation}, author={Shuta Nakajima}, journal={arXiv: Probability}, year={2019} }

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 $x\neq 0$, $\mathbb{P}({\rm T}(0,nx)>n(\mu+\xi))$ decays as $\exp{(-(2d\xi +o(1))n)}$ with a time constant $\mu$ and a dimension $d$. Moreover, we extend the result to stretched exponential distributions. On the contrary…

## One Citation

### Upper Tail Large Deviations in First Passage Percolation

- MathematicsCommunications on Pure and Applied Mathematics
- 2021

For first passage percolation on ℤ2 with i.i.d. bounded edge weights, we consider the upper tail large deviation event, i.e., the rare situation where the first passage time between two points at…

## References

SHOWING 1-10 OF 10 REFERENCES

### Upper Tail Large Deviations in First Passage Percolation

- MathematicsCommunications on Pure and Applied Mathematics
- 2021

For first passage percolation on ℤ2 with i.i.d. bounded edge weights, we consider the upper tail large deviation event, i.e., the rare situation where the first passage time between two points at…

### Large deviations for weighted sums of stretched exponential random variables

- Mathematics
- 2014

We consider the probability that a weighted sum of n i.i.d. random variables $X_j, j = 1,\ldots,n$, with stretched exponential tails is larger than its expectation and determine the rate of its…

### 50 years of first passage percolation

- Mathematics
- 2015

We celebrate the 50th anniversary of one the most classical models in probability theory. In this survey, we describe the main results of first passage percolation, paying special attention to the…

### On Large Deviation Regimes For Random Media Models

- Mathematics
- 2009

The focus of this article is on the different behavior of large deviations of random subadditive functionals above the mean versus large deviations below the mean in two random media models. We…

### First-Passage Percolation, Subadditive Processes, Stochastic Networks, and Generalized Renewal Theory

- Mathematics
- 1965

In 1957, Broadbent and Hammersley gave a mathematical formulation of percolation theory. Since then much work has been done in this field and has now led to first-passage percolation problems. In the…

### Dynamic scaling of growing interfaces.

- PhysicsPhysical review letters
- 1986

A model is proposed for the evolution of the profile of a growing interface that exhibits nontrivial relaxation patterns, and the exact dynamic scaling form obtained for a one-dimensional interface is in excellent agreement with previous numerical simulations.

### A Two-dimensional Growth Process

- Biology
- 1961

Examination of properties of populations of cells, in particular, properties relating to the architecture of cell colonies finds little evidence available but it is to be anticipated that, on a cellular level, there would be little more correlation between twins than between any other members of a given species population.