Modulus of continuity for polymer fluctuations and weight profiles in Poissonian last passage percolation

@article{Hammond2020ModulusOC,
  title={Modulus of continuity for polymer fluctuations and weight profiles in Poissonian last passage percolation},
  author={Alan Hammond and Sourav Sarkar},
  journal={Electronic Journal of Probability},
  year={2020}
}
In last passage percolation models, the energy of a path is maximized over all directed paths with given endpoints in a random environment, and the maximizing paths are called geodesics. The geodesics and their energy can be scaled so that transformed geodesics cross unit distance and have fluctuations and scaled energy of unit order. Here we consider Poissonian last passage percolation, a model lying in the KPZ universality class, and refer to scaled geodesics as polymers and their scaled… 

Figures from this paper

The geometry of near ground states in Gaussian polymer models

The energy and geometry of maximizing paths in integrable last passage percolation models are governed by the characteristic KPZ scaling exponents of one-third and two-thirds. When represented in

Interlacing and Scaling Exponents for the Geodesic Watermelon in Last Passage Percolation

In discrete planar last passage percolation (LPP), random values are assigned independently to each vertex in $\mathbb Z^2$, and each finite upright path in $\mathbb Z^2$ is ascribed the weight given

Optimal exponent for coalescence of finite geodesics in exponential last passage percolation

In this note, we study the model of directed last passage percolation on $\mathbb{Z}^2$, with i.i.d. exponential weight. We consider the maximum paths from vertices $\left(0,\left\lfloor k^{2/3}

Small deviation estimates and small ball probabilities for geodesics in last passage percolation

For the exactly solvable model of exponential last passage percolation on Z, consider the geodesic Γn joining (0, 0) and (n, n) for large n. It is well known that the transversal fluctuation of Γn

Optimal-order exit point bounds in exponential last-passage percolation via the coupling technique

We develop a new probabilistic method for deriving deviation estimates in directed planar polymer and percolation models. The key estimates are for exit points of geodesics as they cross transversal

Connecting Eigenvalue Rigidity with Polymer Geometry: Diffusive Transversal Fluctuations under Large Deviation.

We consider the exactly solvable model of exponential directed last passage percolation on $\mathbb{Z}^2$ in the large deviation regime. Conditional on the upper tail large deviation event

Longest increasing path within the critical strip

A Poisson point process of unit intensity is placed in the square $[0,n]^2$. An increasing path is a curve connecting $(0,0)$ with $(n,n)$ which is non-decreasing in each coordinate. Its length is

Three-halves variation of geodesics in the directed landscape

We show that geodesics in the directed landscape have $3/2$-variation and that weight functions along the geodesics have cubic variation. We show that the geodesic and its landscape environment

Time evolution of the Kardar-Parisi-Zhang equation

The use of the non-linear SPDEs are inevitable in both physics and applied mathematics since many of the physical phenomena in nature can be effectively modeled in random and non-linear way. The

On the exponent governing the correlation decay of the Airy$_1$ process

We study the decay of the covariance of the Airy1 process, A1, a stationary stochastic process on R that arises as a universal scaling limit in the Kardar-Parisi-Zhang (KPZ) universality class. We

References

SHOWING 1-10 OF 21 REFERENCES

Modulus of continuity of polymer weight profiles in Brownian last passage percolation

  • A. Hammond
  • Mathematics
    The Annals of Probability
  • 2019
In last passage percolation models lying in the KPZ universality class, the energy of long energy-maximizing paths may be studied as a function of the paths' pair of endpoint locations. Scaled

Brownian regularity for the Airy line ensemble, and multi-polymer watermelons in Brownian last passage percolation

  • A. Hammond
  • Mathematics
    Memoirs of the American Mathematical Society
  • 2022
The Airy line ensemble is a positive-integer indexed system of random continuous curves whose finite dimensional distributions are given by the multi-line Airy process. It is a natural object in the

On the probability of several near geodesics with shared endpoints in Brownian last passage percolation, and Brownian bridge regularity for the Airy line ensemble

The Airy line ensemble is a positive-integer indexed system of continuous random curves whose finite dimensional distributions are given by the multi-line Airy process. It is a natural object in the

Last Passage Percolation with a Defect Line and the Solution of the Slow Bond Problem

We address the question of how a localized microscopic defect, especially if it is small with respect to certain dynamic parameters, affects the macroscopic behavior of a system. In particular we

Cube–Root Boundary Fluctuations¶for Droplets in Random Cluster Models

Abstract: For a family of bond percolation models on ℤ2 that includes the Fortuin–Kasteleyn random cluster model, we consider properties of the “droplet” that results, in the percolating regime, from

Phase Separation in Random Cluster Models I: Uniform Upper Bounds on Local Deviation

This is the first in a series of three papers that addresses the behaviour of the droplet that results, in the percolating phase, from conditioning the planar Fortuin-Kasteleyn random cluster model

Phase separation in random cluster models II: the droplet at equilibrium, and local deviation lower bounds

We study the droplet that results from conditioning the subcritical Fortuin-Kasteleyn planar random cluster model on the presence of an open circuit Gamma_0 encircling the origin and enclosing an

Phase Separation in Random Cluster Models III: Circuit Regularity

TLDR
It is shown that, provided that the conditioned circuit is centred at the origin in a natural sense, the set of regeneration sites reaches into all parts of the circuit, with maximal distance from one such site to the next being at most logarithmic in n with high probability.

Transversal fluctuations for increasing subsequences on the plane

Abstract. Consider a realization of a Poisson process in ℝ2 with intensity 1 and take a maximal up/right path from the origin to (N, N) consisting of line segments between the points, where maximal

Lower bounds for boundary roughness for droplets in Bernoulli percolation

Abstract.We consider boundary roughness for the ``droplet'' created when supercritical two-dimensional Bernoulli percolation is conditioned to have an open dual circuit surrounding the origin and