# A variational formula for large deviations in First-passage percolation under tail estimates

@inproceedings{Cosco2021AVF, title={A variational formula for large deviations in First-passage percolation under tail estimates}, author={Cl{\'e}ment Cosco and Shuta Nakajima}, year={2021} }

Consider first passage percolation with identical and independent weight distributions and first passage time T. In this paper, we study the upper tail large deviations P(T(0, nx) > n(μ + ξ)), for ξ > 0 and x 6= 0 with a time constant μ and a dimension d, for weights that satisfy a tail assumption β1 exp (−αt) ≤ P(τe > t) ≤ β2 exp (−αt). When r ≤ 1 (this includes the wellknown Eden growth model), we show that the upper tail large deviation decays as exp (−(2dξ + o(1))n). When 1 < r ≤ d, we find…

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## References

SHOWING 1-10 OF 27 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 the chemical distance in supercritical Bernoulli percolation

- Mathematics
- 2007

The chemical distance D(x, y) is the length of the shortest open path between two points x and y in an infinite Bernoulli percolation cluster. In this work, we study the asymptotic behavior of this…

### 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…

### Exponential inequalities for martingales and asymptotic properties of the free energy of directed polymers in a random environment

- Mathematics
- 2009

### First-Passage Percolation

- Mathematics
- 2003

First-passage percolation was introduced by Hammersley and Welsh in 1965 (see [13]), partly as a generalization of ordinary percolation. (For later surveys see [36], [21] and [22].) It can be thought…

### 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…

### Maximal edge-traversal time in First-passage percolation

- MathematicsElectronic Journal of Probability
- 2022

In this paper, we study the maximal edge-traversal time (simply we call maximal weight hereafter) on the optimal paths in the first passage percolation for several edge distributions, including the…

### The shape theorem for the frog model

- Mathematics
- 2001

In this work we prove a shape theorem for a growing set of Simple Random Walks (SRWs) on Z d , known as frog model. The dynam-ics of this process is described as follows: There are active particles,…

### Probability on Trees and Networks

- Mathematics
- 2017

Starting around the late 1950s, several research communities began relating the geometry of graphs to stochastic processes on these graphs. This book, twenty years in the making, ties together…