The Kardar-Parisi-Zhang equation and universality class

Brownian motion is a continuum scaling limit for a wide class of random processes, and there has been great success in developing a theory for its properties (such as distribution functions or regularity) and expanding the breadth of its universality class. Over the past twenty five years a new universality class has emerged to describe a host of important physical and probabilistic models (including one dimensional interface growth processes, interacting particle systems and polymers in random environments) which display characteristic, though unusual, scalings and new statistics. This class is called the Kardar-Parisi-Zhang (KPZ) universality class and underlying it is, again, a continuum object -- a non-linear stochastic partial differential equation -- known as the KPZ equation. The purpose of this survey is to explain the context for, as well as the content of a number of mathematical breakthroughs which have culminated in the derivation of the exact formula for the distribution function of the KPZ equation started with {\it narrow wedge} initial data. In particular we emphasize three topics: (1) The approximation of the KPZ equation through the weakly asymmetric simple exclusion process; (2) The derivation of the exact one-point distribution of the solution to the KPZ equation with narrow wedge initial data; (3) Connections with directed polymers in random media. As the purpose of this article is to survey and review, we make precise statements but provide only heuristic arguments with indications of the technical complexities necessary to make such arguments mathematically rigorous.Comment: 57 pages, survey article, 7 figures, addition physics ref. added and typo's correcte

Macdonald processes, quantum integrable systems and the Kardar-Parisi-Zhang universality class

Integrable probability has emerged as an active area of research at the interface of probability/mathematical physics/statistical mechanics on the one hand, and representation theory/integrable systems on the other. Informally, integrable probabilistic systems have two properties: 1) It is possible to write down concise and exact formulas for expectations of a variety of interesting observables (or functions) of the system. 2) Asymptotics of the system and associated exact formulas provide access to exact descriptions of the properties and statistics of large universality classes and universal scaling limits for disordered systems. We focus here on examples of integrable probabilistic systems related to the Kardar-Parisi-Zhang (KPZ) universality class and explain how their integrability stems from connections with symmetric function theory and quantum integrable systems.Comment: Proceedings of the ICM, 31 pages, 10 figure

Economic Implications of Business Boundary Laws

Den här avhandlingen belyser hur avverkning och markberedning påverkar markfloran i den svenska barrskogen. Dessutom utvärderas två inventeringsmetoder som används inom växtekologin. Vid arbetet har både rikstäckande inventeringsdata och fältförsök använts och de likartade resultaten tyder på att rikstäckande inventeringar är en underutnyttjad resurs i forskningen. Ju större andel av träden som avverkas desto större blir förändringen av markflorans sammansättning. Vissa arter, som lingon, ljung, etc., verkar dock inte påverkas i nämnvärd omfattning, medan andra, som blåbär, minskar i relation till hur mycket som avverkats. Gräs och mjölkört ökar efter avverkning, dock visar sig vissa gräs och mjölkört inte reagera om inte avverkningen överskrider ett tröskelvärde på ca 80 %. Avverkning har en liten, men signifikant, effekt på antalet arter, medan artomsättning, d.v.s. arters etablering på och/eller försvinnande från provytorna, framförallt påverkas av andel gran innan avverkning, markens produktionsförmåga och först därefter av hur stor andel av träden som avverkas. Det var också uppenbart att markberedning har en stark effekt som skiljer sig från avverkning. Framförallt gynnas björnmossor av markberedning men även vårfryle, kruståtel och mjölkört. Arter som missgynnas av markberedning var bl.a., en levermossa, lingon, väggmossa och kråkbär. I växtekologi är visuell täckningsbedömning, d.v.s. hur stor del av en provyta som täcks av en växtart, och registrering av förekomst/icke förekomst, d.v.s. finns en växtart på en provyta eller inte, de två vanligaste metoderna vid vegetationsinventering. Vid registrering av förekomst/icke förekomst missas upp till en tredjedel av förekomsterna, vanligaste orsaken till missade registreringar verkar vara att man inte upptäcker arten snarare än att den inte kan identifieras. Det var stora variationer mellan arter, där arter med få exemplar på provytan missas oftare. Både den visuella täckningsbedömningen och förekomst/icke förekomst visar sig ha personberoende fel, d.v.s. att olika personer genomgående ger högre eller lägre värden än andra. Trots det personberoende felet visar sig täckningsbedömningar ha ett större informationsvärde än registrering av förekomst/icke förekomst när det gäller att särskilja olika typer av vegetation. Erfarenhet har en förvånansvärt liten effekt på kvaliteten av täckningsbedömningar.This thesis has two main focuses; first, the response of forest ground layer flora on forestry, mainly harvesting and secondly, the quality of the vegetation assessment methods, cover estimates by eye and presence/absence data. The effect of harvesting intensity was evaluated with survey data from permanent plots as well as vegetation data from a field trial fourteen years after harvesting. Both data sets confirmed that response of ground layer flora increased with increasing logging intensity. Thereby, indicating that survey data is possible to use in research. From the survey data set, existence of a time lag was evident for several species and also a threshold level was evident in cutting intensity needed to affect a number of species. Logging had a modest, but significant positive effect on the change in species number per plot. Species turnover was influenced by the proportion of Picea abies in the tree canopy; site productivity; and logging intensity. In the field trial scarification had a strong effect that was different from the one created by cutting. In plant ecology cover estimate by eye and presence/absence recording are the two most frequent methods used. The methods were evaluated with survey data and a field trial. In the first data set vegetation was recorded independently by two observers in 342 permanent 100-m2 plots. Overall, one third of each occurrence was missed by one of the two observers, but with large differences among species. Species occurring at low abundance tended to be frequently overlooked. Observer-explained variance in cover estimates was <10% in 15 of 17 species. In the second data set, 10 observers independently estimated cover in sixteen 100-m2 plots in two different vegetation types. The bias connected to observer varied substantially between species. The estimates of missing field and bottom layer had the highest bias, indicating that missing layers are problematic to use in analysis of change. Experience had a surprisingly small impact on the bias connected to observer. Analyses revealed that for the statistical power, cover estimates by eye carries a higher information value than do presence/absence data when distinguishing between vegetation types, differences between observers is negligible, and using more than one observer had little effect