The conjectured limit of last passage percolation is a scale-invariant, independent, stationary increment process with respect to metric composition. We prove this for Brownian last passage percolation. We construct the Airy sheet and characterize it in terms of the Airy line ensemble. We also show that last passage geodesics converge to random functions with Hölder-2/3 − continuous paths. This work completes the construction of the central object in the Kardar-Parisi-Zhang universality class, the directed landscape.
We establish Tracy-Widom asymptotics for the partition function of a random polymer model with gamma-distributed weights recently introduced by Seppäläinen. We show that the partition function of this random polymer can be represented within the framework of the geometric RSK correspondence and consequently its law can be expressed in terms of Whittaker functions. This leads to a representation of the law of the partition function which is amenable to asymptotic analysis. In this model, the partition function plays a role analogous to the smallest eigenvalue in the Laguerre unitary ensemble of random matrix theory.
We obtain exact formulas for moments and generating functions of the height
function of the asymmetric simple exclusion process at one spatial point,
starting from special initial data in which every positive even site is
initially occupied. These complement earlier formulas of E. Lee [J. Stat. Phys.
140 (2010) 635-647] but, unlike those formulas, ours are suitable in principle
for asymptotics. We also explain how our formulas are related to divergent
series formulas for half-flat KPZ of Le Doussal and Calabrese [J. Stat. Mech.
2012 (2012) P06001], which we also recover using the methods of this paper.
These generating functions are given as a series without any apparent Fredholm
determinant or Pfaffian structure. In the long time limit, formal asymptotics
show that the fluctuations are given by the Airy$_{2\to1}$ marginals.Comment: Published at http://dx.doi.org/10.1214/15-AAP1099 in the Annals of
Applied Probability (http://www.imstat.org/aap/) by the Institute of
Mathematical Statistics (http://www.imstat.org
Identifying robust biomarkers of drug response constitutes a key challenge in precision medicine. Patient-derived tumor xenografts (PDX) have emerged as reliable preclinical models that more accurately recapitulate tumor response to chemoand targeted therapies. However, the lack of computational tools makes it difficult to analyze high-throughput molecular and pharmacologic profiles of PDX. We have developed Xenograft Visualization & Analysis (Xeva), an open-source software package for in vivo pharmacogenomic datasets that allows for quantification of variability in gene expression and pathway activity across PDX passages. We found that only a few genes and pathways exhibited passage-specific alterations and were therefore not suitable for biomarker discovery. Using the largest PDX pharmacogenomic dataset to date, we identified 87 pathways that are significantly associated with response to 51 drugs (FDR < 0.05). We found novel biomarkers based on gene expressions, copy number aberrations, and mutations predictive of drug response (concordance index > 0.60; FDR < 0.05). Our study demonstrates that Xeva provides a flexible platform for integrative analysis of preclinical in vivo pharmacogenomics data to identify biomarkers predictive of drug response, representing a major step forward in precision oncology. Significance: A computational platform for PDX data analysis reveals consistent gene and pathway activity across passages and confirms drug response prediction biomarkers in PDX. See related commentary by Meehan, p. 4324
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