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

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

The q-PushASEP: A New Integrable Model for Traffic in 1+1 Dimension

We introduce a new interacting (stochastic) particle system q-PushASEP which interpolates between the q-TASEP introduced by Borodin and Corwin (see arXiv:1111.4408, and also arXiv:1207.5035; arXiv:1305.2972; arXiv:1212.6716) and the q-PushTASEP introduced recently by Borodin and Petrov (arXiv:1305.5501). In the q-PushASEP, particles can jump to the left or to the right, and there is a certain partially asymmetric pushing mechanism present. This particle system has a nice interpretation as a model of traffic on a one-lane highway in which cars are able to accelerate or slow down. Using the quantum many body system approach, we explicitly compute the expectations of a large family of observables for this system in terms of nested contour integrals. We also discuss relevant Fredholm determinantal formulas for the distribution of the location of each particle, and connections of the model with a certain two-sided version of Macdonald processes and with the semi-discrete stochastic heat equation.Comment: 22 pages; 4 figures. v2: minor improvements of presentation and discussions. To appear in Journal of Statistical Physic

Unrestricted State Complexity of Binary Operations on Regular and Ideal Languages

We study the state complexity of binary operations on regular languages over different alphabets. It is known that if $L'_m$ and $L_n$ are languages of state complexities $m$ and $n$, respectively, and restricted to the same alphabet, the state complexity of any binary boolean operation on $L'_m$ and $L_n$ is $mn$, and that of product (concatenation) is $m 2^n - 2^{n-1}$. In contrast to this, we show that if $L'_m$ and $L_n$ are over different alphabets, the state complexity of union and symmetric difference is $(m+1)(n+1)$, that of difference is $mn+m$, that of intersection is $mn$, and that of product is $m2^n+2^{n-1}$. We also study unrestricted complexity of binary operations in the classes of regular right, left, and two-sided ideals, and derive tight upper bounds. The bounds for product of the unrestricted cases (with the bounds for the restricted cases in parentheses) are as follows: right ideals $m+2^{n-2}+2^{n-1}$ ($m+2^{n-2}$); left ideals $mn+m+n$ ($m+n-1$); two-sided ideals $m+2n$ ($m+n-1$). The state complexities of boolean operations on all three types of ideals are the same as those of arbitrary regular languages, whereas that is not the case if the alphabets of the arguments are the same. Finally, we update the known results about most complex regular, right-ideal, left-ideal, and two-sided-ideal languages to include the unrestricted cases.Comment: 30 pages, 15 figures. This paper is a revised and expanded version of the DCFS 2016 conference paper, also posted previously as arXiv:1602.01387v3. The expanded version has appeared in J. Autom. Lang. Comb. 22 (1-3), 29-59, 2017, the issue of selected papers from DCFS 2016. This version corrects the proof of distinguishability of states in the difference operation on p. 12 in arXiv:1609.04439v

Uncertainty in the Design Stage of Two-Stage Bayesian Propensity Score Analysis

The two-stage process of propensity score analysis (PSA) includes a design stage where propensity scores are estimated and implemented to approximate a randomized experiment and an analysis stage where treatment effects are estimated conditional upon the design. This paper considers how uncertainty associated with the design stage impacts estimation of causal effects in the analysis stage. Such design uncertainty can derive from the fact that the propensity score itself is an estimated quantity, but also from other features of the design stage tied to choice of propensity score implementation. This paper offers a procedure for obtaining the posterior distribution of causal effects after marginalizing over a distribution of design-stage outputs, lending a degree of formality to Bayesian methods for PSA (BPSA) that have gained attention in recent literature. Formulation of a probability distribution for the design-stage output depends on how the propensity score is implemented in the design stage, and propagation of uncertainty into causal estimates depends on how the treatment effect is estimated in the analysis stage. We explore these differences within a sample of commonly-used propensity score implementations (quantile stratification, nearest-neighbor matching, caliper matching, inverse probability of treatment weighting, and doubly robust estimation) and investigate in a simulation study the impact of statistician choice in PS model and implementation on the degree of between- and within-design variability in the estimated treatment effect. The methods are then deployed in an investigation of the association between levels of fine particulate air pollution and elevated exposure to emissions from coal-fired power plants