Disjoint optimizers and the directed landscape

We study maximal length collections of disjoint paths, or `disjoint optimizers', in the directed landscape. We show that disjoint optimizers always exist, and that their lengths can be used to construct an extended directed landscape. The extended directed landscape can be built from an independent collection of extended Airy sheets, which we define from the Airy line ensemble. We show that the extended directed landscape and disjoint optimizers are scaling limits of the corresponding objects in Brownian last passage percolation (LPP). As two consequences of this work, we show that one direction of the Robinson-Schensted-Knuth bijection passes to the KPZ limit, and we find a criterion for geodesic disjointness in the directed landscape that uses only a single Airy line ensemble. The proofs rely on a new notion of multi-point LPP across the Airy line ensemble, combinatorial properties of multi-point LPP, and probabilistic resampling ideas

Fractal geometry of the space-time difference profile in the directed landscape via construction of geodesic local times

The Directed Landscape, a random directed metric on the plane (where the first and the second coordinates are termed spatial and temporal respectively), was constructed in the breakthrough work of Dauvergne, Ortmann, and Vir\'ag, and has since been shown to be the scaling limit of various integrable models of Last Passage percolation, a central member of the Kardar-Parisi-Zhang universality class. It exhibits several scale invariance properties making it a natural source of rich fractal behavior. Such a study was initiated in Basu-Ganguly-Hammond, where the difference profile i.e., the difference of passage times from two fixed points (say $(\pm 1,0)$), was considered. Owing to geodesic geometry, it turns out that this difference process is almost surely locally constant. The set of non-constancy is connected to disjointness of geodesics and inherits remarkable fractal properties. In particular, it has been established that when only the spatial coordinate is varied, the set of non-constancy of the difference profile has Hausdorff dimension $1/2$, and bears a rather strong resemblance to the zero set of Brownian motion. The arguments crucially rely on a monotonicity property, which is absent when the temporal structure of the process is probed, necessitating the development of new methods. In this paper, we put forth several new ideas, and show that the set of non-constancy of the 2D difference profile and the 1D temporal process (when the spatial coordinate is fixed and the temporal coordinate is varied) have Hausdorff dimensions $5/3$ and $2/3$ respectively. A particularly crucial ingredient in our analysis is the novel construction of a local time process for the geodesic akin to Brownian local time, supported on the "zero set" of the geodesic. Further, we show that the latter has Hausdorff dimension $1/3$ in contrast to the zero set of Brownian motion which has dimension $1/2.$Comment: 89 pages, 23 figure

Pearcey universality at cusps of polygonal lozenge tiling

We study uniformly random lozenge tilings of general simply connected polygons. Under a technical assumption that is presumably generic with respect to polygon shapes, we show that the local statistics around a cusp point of the arctic curve converge to the Pearcey process. This verifies the widely predicted universality of edge statistics in the cusp case. Together with the smooth and tangent cases proved in Aggarwal-Huang and Aggarwal-Gorin, these are believed to be the three types of edge statistics that can arise in a generic polygon. Our proof is via a local coupling of the random tiling with non-intersecting Bernoulli random walks (NBRW). To leverage this coupling, we establish an optimal concentration estimate for the tiling height function around the cusp. As another step and also a result of potential independent interest, we show that the local statistics of NBRW around a cusp converge to the Pearcey process when the initial configuration consists of two parts with proper density growth, via careful asymptotic analysis of the determinantal formula.Comment: 59 pages, 9 figure

Brownian bridge limit of path measures in the upper tail of KPZ models

For models in the KPZ universality class, such as the zero temperature model of planar last passage-percolation (LPP) and the positive temperature model of directed polymers, its upper tail behavior has been a topic of recent interest, with particular focus on the associated path measures (i.e., geodesics or polymers). For Exponential LPP, diffusive fluctuation had been established in Basu-Ganguly. In the directed landscape, the continuum limit of LPP, the limiting Gaussianity at one point, as well as of related finite-dimensional distributions of the KPZ fixed point, were established, using exact formulas in Liu and Wang-Liu. It was further conjectured in these works that the limit of the corresponding geodesic should be a Brownian bridge. We prove it in both zero and positive temperatures; for the latter, neither the one-point limit nor the scale of fluctuations was previously known. Instead of relying on formulas (which are still missing in the positive temperature literature), our arguments are geometric and probabilistic, using the results on the shape of the weight and free energy profiles under the upper tail from Ganguly-Hegde as a starting point. Another key ingredient involves novel coalescence estimates, developed using the recently discovered shift-invariance Borodin-Gorin-Wheeler in these models. Finally, our proof also yields insight into the structure of the polymer measure under the upper tail conditioning, establishing a quenched localization exponent around a random backbone.Comment: 76 pages, 6 figure

Infinite order phase transition in the slow bond TASEP

In the slow bond problem the rate of a single edge in the Totally Asymmetric Simple Exclusion Process (TASEP) is reduced from 1 to $1-\varepsilon$ for some small $\varepsilon>0$. 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 $\varepsilon$ is 0. This was ultimately resolved rigorously in Basu-Sidoravicius-Sly which established that $\varepsilon_c=0$. Here we study the effect of the current as $\varepsilon$ 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 $\varepsilon$ 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.Comment: 33 pages, 4 figures. Revised according to referee reports. Comm. Pure Appl. Math. to appea

Istraživanje simultane lokalizacije, kalibracije i kartiranja umreženim robotskim sustavima

In a network robot system, a robot and a sensor network are integrated smoothly to develop their advantages and benefit from each other. Robot localization, sensor network calibration and environment mapping are three coupled issues to be solved once network robot system is introduced into a service environment. In this article, the problem of simultaneous localization, calibration and mapping is raised in order to improve their precision. The coupled relations among localization, calibration and mapping are denoted as a joint conditional distribution and then decomposed into three separate analytic terms according to Bayesian and Markov properties. The framework of Rao-Blackwellized particle filtering is used to solve the three analytic terms, in which extended particle filter is used for localization and unscented Kalman filter is used for both calibration and mapping. Simulations have been performed to demonstrate the validity and efficiency of the proposed solutions.U umreženom robotskom sustavu, robot i senzorska mreža su međusobno integrirani i povezani na način da i jedan i drugi iskoriste svoje prednosti, te da imaju koristi jedan od drugoga. Kako bi umreženi robotski sustav mogao djelovati u radnom okruženju potrebno je riješiti tri međusobno povezana problema: lokalizaciju, kalibraciju senzorske mreže i kartiranje prostora. U ovom radu razmatraju se problemi istodobne lokalizacije, kalibracije i kartiranja te se razmatraju mogućnosti poboljšanja njihove preciznosti. Povezanost lokalizacije, kartiranja i kalibracije predstavljena je pomoću zajedničke uvjetne razdiobe i zatim rastavljena u tri razdvojena analitička izraza korištenjem Bayesovih i Markovljevih svojstava. Za rješavanje svih triju analitičkih izraza koristi se Rao-Blackwell čestično filtriranje, pri čemu se prošireni čestični filtar koristi kod lokalizacije a nederivirajući Kalmanov filtar za kalibraciju i kartiranje. Ispravnost i efikasnost predloženog pristupa pokazana je kroz provedene simulacije