Passage Percolation Research Articles

Almost sure convergence of Liouville first passage percolation

Almost sure convergence of Liouville first passage percolation

First passage percolation with recovery

First passage percolation with recovery is a process aimed at modeling the spread of epidemics. On a graph G place a red particle at a reference vertex o and colorless particles (seeds) at all other vertices. The red particle starts spreading a red first passage percolation of rate 1, while all seeds are dormant. As soon as a seed is reached by the process, it turns red and starts spreading red first passage percolation. All vertices are equipped with independent exponential clocks ringing at rate γ>0, when a clock rings the corresponding red vertex turns black. For t≥0, let Ht and Mt denote the size of the longest red path and of the largest red cluster present at time t. If G is the semi-line, then for all γ>0 almost surely lim suptHtloglogtlogt=1 and lim inftHt=0. In contrast, if G is an infinite Galton–Watson tree with offspring mean m>1 then, for all γ>0, almost surely lim inftHtlogtt≥m−1 and lim inftMtloglogtt≥m−1, while lim suptMtect≤1, for all c>m−1. Also, almost surely as t→∞, for all γ>0Ht is of order at most t. Furthermore, if we restrict our attention to bounded-degree graphs, then for any ɛ>0 there is a critical value γc>0 so that for all γ>γc, almost surely lim suptMtt≤ɛ.

Diffusive scaling limit of the Busemann process in last passage percolation

Diffusive scaling limit of the Busemann process in last passage percolation

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

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

Strict monotonicity for first passage percolation on graphs of polynomial growth and quasi-trees

Strict monotonicity for first passage percolation on graphs of polynomial growth and quasi-trees

A non-oriented first passage percolation model and statistical invariance by time reversal

We introduce and study a non-oriented first passage percolation model having a property of statistical invariance by time reversal. This model is defined in a graph having directed edges and the passage times associated with each set of outgoing edges from a given vertex are distributed according to a generalized Bernoulli–Exponential law and i.i.d. among vertices. We derive the statistical invariance property by time reversal through a zero-temperature limit of the random walk in Dirichlet environment model.

Stability and chaos in dynamical last passage percolation

Many complex disordered systems in statistical mechanics are characterized by intricate energy landscapes. The ground state, the configuration with lowest energy, lies at the base of the deepest valley. In important examples, such as Gaussian polymers and spin glass models, the landscape has many valleys and the abundance of near-ground states (at the base of valleys) indicates the phenomenon of chaos, under which the ground state alters profoundly when the disorder of the model is slightly perturbed. In this article, we compute the critical exponent that governs the onset of chaos in a dynamic manifestation of a canonical model in the Kardar-Parisi-Zhang [KPZ] universality class, Brownian last passage percolation [LPP]. In this model in its static form, semidiscrete polymers advance through Brownian noise, their energy given by the integral of the white noise encountered along their journey. A ground state is a geodesic, of extremal energy given its endpoints. We perturb Brownian LPP by evolving the disorder under an Ornstein-Uhlenbeck flow. We prove that, for polymers of length n n , a sharp phase transition marking the onset of chaos is witnessed at the critical time n − 1 / 3 n^{-1/3} . Indeed, the overlap between the geodesics at times zero and t > 0 t > 0 that travel a given distance of order n n will be shown to be of order n n when t ≪ n − 1 / 3 t\ll n^{-1/3} ; and to be of smaller order when t ≫ n − 1 / 3 t\gg n^{-1/3} . We expect this exponent to be universal across a wide range of interface models. The present work thus sheds light on the dynamical aspect of the KPZ class; it builds on several recent advances. These include Chatterjee’s harmonic analytic theory [Superconcentration and related topics, Springer, Cham, 2014] of equivalence of superconcentration and chaos in Gaussian spaces; a refined understanding of the static landscape geometry of Brownian LPP developed in the companion paper (see S. Ganguly and A. Hammond [Electron. J. Probab. 28 (2023), 80 pp.]); and, underlying the latter, strong comparison estimates of the geodesic energy profile to Brownian motion (see J. Calvert, A. Hammond, and M. Hegde [Astérisque 441 (2023), pp. v+119]).

Discrete and continuous Muttalib–Borodin processes: The hard edge

In this note, we study a natural measure on plane partitions giving rise to a certain discrete-time Muttalib–Borodin process (MBP): each time slice is a discrete version of a Muttalib–Borodin ensemble (MBE). The process is determinantal with explicit-time-dependent correlation kernel. Moreover, in the q \rightarrow 1 limit, it converges to a continuous Jacobi-like MBP with Muttalib–Borodin marginals supported on the unit interval. This continuous process is also determinantal with explicit correlation kernel. We study its hard-edge scaling limit (around 0) to obtain a discrete-time-dependent generalization of the classical continuous Bessel kernel of random matrix theory (and, in fact, of the Meijer G -kernel as well). We lastly discuss two related applications: random sampling from such processes and their interpretations as models of directed last passage percolation (LPP). In doing so, we introduce a corner growth model naturally associated to Jacobi processes, a version of which is the “usual” corner growth of Forrester–Rains in logarithmic coordinates.

Duality in the Directed Landscape and Its Applications to Fractal Geometry

Abstract Geodesic coalescence, or the tendency of geodesics to merge together, is a hallmark phenomenon observed in a variety of planar random geometries involving a random distortion of the Euclidean metric. As a result of this, the union of interiors of all geodesics going to a fixed point tends to form a tree-like structure that is supported on a vanishing fraction of the space. Such geodesic trees exhibit intricate fractal behaviour; for instance, while almost every point in the space has only one geodesic going to the fixed point, there exist atypical points that admit two such geodesics. In this paper, we consider the directed landscape, the recently constructed [ 18] scaling limit of exponential last passage percolation (LPP), with the aim of developing tools to analyse the fractal aspects of the tree of semi-infinite geodesics in a given direction. We use the duality [ 39] between the geodesic tree and the interleaving competition interfaces in exponential LPP to obtain a duality between the geodesic tree and the corresponding dual tree in the landscape. Using this, we show that problems concerning the fractal behaviour of sets of atypical points for the geodesic tree can be transformed into corresponding problems for the dual tree, which might turn out to be easier. In particular, we use this method to show that the set of points admitting two semi-infinite geodesics in a fixed direction a.s. has Hausdorff dimension $4/3$, thereby answering a question posed in [ 12]. We also show that the set of points admitting three semi-infinite geodesics in a fixed direction is a.s. countable.

First passage percolation with long-range correlations and applications to random Schrödinger operators

First passage percolation with long-range correlations and applications to random Schrödinger operators

Time-time covariance for last passage percolation in half-space

Time-time covariance for last passage percolation in half-space

The second class particle process at shocks

We consider the totally asymmetric simple exclusion process (TASEP) starting with a shock discontinuity at the origin, with asymptotic densities λ to the left of the origin and ρ to the right of it and λ<ρ. We find an exact identity for the distribution of a second class particle starting at the origin. Then we determine the limiting joint distributions of the second class particle. Bypassing the last passage percolation model, we work directly in TASEP, allowing us to extend previous one-point distribution results via a more direct and shorter ansatz.

Superconcentration for minimal surfaces in first passage percolation and disordered Ising ferromagnets

We consider the standard first passage percolation model on Zd\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\mathbb {Z}}^ d$$\\end{document} with a distribution G taking two values 0<a<b\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$0<a<b$$\\end{document}. We study the maximal flow through the cylinder [0,n]d-1×[0,hn]\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$[0,n]^ {d-1}\ imes [0,hn]$$\\end{document} between its top and bottom as well as its associated minimal surface(s). We prove that the variance of the maximal flow is superconcentrated, i.e. in O(nd-1logn)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$O(\\frac{n^{d-1}}{\\log n})$$\\end{document}, for h≥h0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$h\\ge h_0$$\\end{document} (for a large enough constant h0=h0(a,b)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$h_0=h_0(a,b)$$\\end{document}). Equivalently, we obtain that the ground state energy of a disordered Ising ferromagnet in a cylinder [0,n]d-1×[0,hn]\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$[0,n]^ {d-1}\ imes [0,hn]$$\\end{document} is superconcentrated when opposite boundary conditions are applied at the top and bottom faces and for a large enough constant h≥h0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$h\\ge h_0$$\\end{document} (which depends on the law of the coupling constants). Our proof is inspired by the proof of Benjamini–Kalai–Schramm (Ann Probab 31:1970–1978, 2003). Yet, one major difficulty in this setting is to control the influence of the edges since the averaging trick used in Benjamini et al. (Ann Probab 31:1970–1978, 2003) fails for surfaces. Of independent interest, we prove that minimal surfaces (in the present discrete setting) cannot have long thin chimneys.

Fluctuations around the diagonal in Bernoulli-Exponential first passage percolation

Fluctuations around the diagonal in Bernoulli-Exponential first passage percolation

Last passage percolation and limit theorems in Barak-Erdős directed random graphs and related models

Last passage percolation and limit theorems in Barak-Erdős directed random graphs and related models

Uniform fluctuation and wandering bounds in first passage percolation

Uniform fluctuation and wandering bounds in first passage percolation

First passage percolation in hostile environment is not monotone

First passage percolation in hostile environment is not monotone

First passage percolation for weakly correlated fields

First passage percolation for weakly correlated fields

Francis Comets’ Gumbel last passage percolation

In 2015, Francis Comets shared with me a clever way to relate a model of directed last passage percolation with i.i.d. Gumbel edge weights to a special case of the log-gamma directed polymer model. To my knowledge, he never wrote this down. In the wake of his recent passing I am recording Francis’ observation along with some associated asymptotics and discussion. This note is dedicated in memory of Francis whose work in the study of directed polymers defined and refined the field immensely.

Infinite order phase transition in the slow bond TASEP

AbstractIn the slow bond problem the rate of a single edge in the Totally Asymmetric Simple Exclusion Process (TASEP) is reduced from 1 to for some small . Janowsky and Lebowitz posed the well‐known question of whether such very small perturbations could affect the macroscopic current. Different groups of physicists, using a range of heuristics and numerical simulations reached opposing conclusions on whether the critical value of is 0. This was ultimately resolved rigorously in Basu‐Sidoravicius‐Sly which established that .Here we study the effect of the current as tends to 0 and in doing so explain why it was so challenging to predict on the basis of numerical simulations. In particular we show that the current has an infinite order phase transition at 0, with the effect of the perturbation tending to 0 faster than any polynomial. Our proof focuses on the Last Passage Percolation formulation of TASEP where a slow bond corresponds to reinforcing the diagonal. We give a multiscale analysis to show that when is small the effect of reinforcement remains small compared to the difference between optimal and near optimal geodesics. Since geodesics can be perturbed on many different scales, we inductively bound the tails of the effect of reinforcement by controlling the number of near optimal geodesics and giving new tail estimates for the local time of (near) geodesics along the diagonal.