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Register a new userLast passage percolation in an exponential environment with discontinuous rates
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We prove a strong law of large numbers for directed last passage times in an independent but inhomogeneous exponential environment. Rates for the exponential random variables are obtained from a discretisation of a speed function that may be discontinuous on a locally finite set of discontinuity curves. The limiting shape is cast as a variational formula that maximises a certain functional over a set of weakly increasing curves. Using this result, we present two examples that allow for partial analytical tractability and show that the shape function may not be strictly concave, and it may exhibit points of non-differentiability, flat segments, and non-uniqueness of the optimisers of the variational formula. Finally, in a specific example, we analyse further the macroscopic optimisers and uncover a phase transition for their behaviour.
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A well-known question in the planar first-passage percolation model concerns the convergence of the empirical distribution along geodesics. We demonstrate this convergence for an explicit model, directed last-passage percolation on $mathbb{Z}^2$ with i.i.d. exponential weights, and provide explicit formulae for the limiting distributions, which depend on the asymptotic direction. For example, for geodesics in the direction of the diagonal, the limiting weight distribution has density $(1/4+x/2+x^2/8)e^{-x}$, and so is a mixture of Gamma($1,1$), Gamma($2,1$) and Gamma($3,1$) distributions with weights $1/4$, $1/2$, and $1/4$ respectively. More generally, we study the local environment as seen from vertices along the geodesics (including information about the shape of the path and about the weights on and off the path in a local neighborhood). We consider finite geodesics from $(0,0)$ to $nboldsymbol{rho}$ for some vector $boldsymbol{rho}$ in the first quadrant, in the limit as $ntoinfty$, as well as the semi-infinite geodesic in direction $boldsymbol{rho}$. We show almost sure convergence of the empirical distributions along the geodesic, as well as convergence of the distribution around a typical point, and we give an explicit description of the limiting distribution. We make extensive use of a correspondence with TASEP as seen from a single second-class particle for which we prove new results concerning ergodicity and convergence to equilibrium. Our analysis relies on geometric arguments involving estimates for the last-passage time, available from the integrable probability literature.
These lecture notes are written as reference material for the Advanced Course Hydrodynamical Methods in Last Passage Percolation Models, given at the 28th Coloquio Brasileiro de Matematica at IMPA, Rio de Janeiro, July 2011.
In this note we investigate the last passage percolation model in the presence of macroscopic inhomogeneity. We analyze how this affects the scaling limit of the passage time, leading to a variational problem that provides an ODE for the deterministic limiting shape of the maximal path. We obtain a sufficient analytical condition for uniqueness of the solution for the variational problem. Consequences for the totally asymmetric simple exclusion process are discussed.
The aim of this article is to study the forest composed by point-to-line geodesics in the last-passage percolation model with exponential weights. We will show that the location of the root can be described in terms of the maxima of a random walk, whose distribution will depend on the geometry of the substrate (line). For flat substrates, we will get power law behaviour of the height function, study its scaling limit, and describe it in terms of variational problems involving the Airy process.
In this paper we study stationary last passage percolation (LPP) in half-space geometry. We determine the limiting distribution of the last passage time in a critical window close to the origin. The result is a new two-parameter family of distributions: one parameter for the strength of the diagonal bounding the half-space (strength of the source at the origin in the equivalent TASEP language) and the other for the distance of the point of observation from the origin. It should be compared with the one-parameter family giving the Baik--Rains distributions for full-space geometry. We finally show that far enough away from the characteristic line, our distributions indeed converge to the Baik--Rains family. We derive our results using a related integrable model having Pfaffian structure together with careful analytic continuation and steepest descent analysis.
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