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Register a new userHydrodynamical Methods in Last Passage Percolation Models
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Added by
Leandro Pimentel Pinto Rodrigues
Publication date
2011
fields
Physics
and research's language is
English
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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.
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In this paper we consider an equilibrium last-passage percolation model on an environment given by a compound two-dimensional Poisson process. We prove an $LL^2$-formula relating the initial measure with the last-passage percolation time. This formula turns out to be a useful tool to analyze the fluctuations of the last-passage times along non-characteristic directions.
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.
We consider the geodesic of the directed last passage percolation with iid exponential weights. We find the explicit one point distribution of the geodesic location joint with the last passage times, and its limit when the size goes to infinity.
We study first-passage percolation where edges in the left and right half-planes are assigned values according to different distributions. We show that the asymptotic growth of the resulting inhomogeneous first-passage process obeys a shape theorem, and we express the limiting shape in terms of the limiting shapes for the homogeneous processes for the two weight distributions. We further show that there exist pairs of distributions for which the rate of growth in the vertical direction is strictly larger than the rate of growth of the homogeneous process with either of the two distributions, and that this corresponds to the creation of a defect along the vertical axis in the form of a `pyramid.
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.
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