124 research outputs found
Scaling limits of random plane partitions and six-vertex models
No full textThesis: Ph. D., Massachusetts Institute of Technology, Department of Mathematics, 2018.Cataloged from PDF version of thesis.Includes bibliographical references (pages 235-239).We present a collection of results about the scaling limits of several models from integrable probability. Our first result concerns the asymptotic behavior of the bottom slice of a Hall-Littlewood random plane partition. We show the latter concentrates around a limit shape and in two different scaling regimes identify the fluctuations around this shape with the GUE Tracy- Widom distribution and the narrow wedge initial data solution to the Kardar-Parisi-Zhang (KPZ) equation. The second result concerns the limiting behavior of a class of six-vertex models in the quadrant, and we obtain the GUE-corners process as a scaling limit for this class near the boundary. Our final result, joint with Ivan Corwin, demonstrates the (long predicted) transversal 2/3 critical exponent for the height functions of the stochastic sixvertex model and asymmetric simple exclusion process (ASEP). The algebraic parts of our arguments involve the construction and use of degenerations and modifications of the Macdonald difference operators to obtain rich families of observables for the models we consider. These formulas are in terms of multiple contour integrals and provide a direct access to quantities of interest. The analytic parts of our arguments include the detailed asymptotic analysis of Fredholm determinants and contour integrals through steepest descent methods. An important aspect of our approach, is the combination of exact formulas with more probabilistic arguments, based on various Gibbs properties enjoyed by the models we study.by Evgeni Dimitrov.Ph. D
Characterization of -Brownian Gibbsian line ensembles
Full text linkIn this paper we show that an -Brownian Gibbsian line ensemble is
completely characterized by the finite-dimensional marginals of its lowest
indexed curve for a large class of interaction Hamiltonians . A particular
consequence of our result is that the KPZ line ensemble is the unique line
ensemble that satisfies the -Brownian Gibbs property with
and whose lowest indexed curve is equal to the Hopf-Cole solution to the narrow
wedge KPZ equation.Comment: 47 pages, 6 figures. Fixed some typos in the previous version of the
paper. arXiv admin note: text overlap with arXiv:2002.0068
Six-vertex Models and the GUE-corners Process
No full textAbstract
We consider a class of probability distributions on the six-vertex model, which originates from the higher spin vertex models of [13]. We define operators, inspired by the Macdonald difference operators, which extract various correlation functions, measuring the probability of observing different arrow configurations. For the class of models we consider, the correlation functions can be expressed in terms of multiple contour integrals, which are suitable for asymptotic analysis. For a particular choice of parameters we analyze the limit of the correlation functions through the steepest descent method. Combining this asymptotic statement with some new results about Gibbs measures on Gelfand–Tsetlin cones and patterns, we show that the asymptotic behavior of our six-vertex model near the boundary is described by the Gaussian Unitary Ensemble-corners process.</jats:p
Airy wanderer line ensembles
Full text linkIn (J. Stat. Phys. 132, 275-290, 2008) Borodin and P\'ech\'e introduced a
generalization of the extended Airy kernel based on two infinite sets of
parameters. For an arbitrary choice of parameters we construct determinantal
point processes on for these generalized kernels. In addition,
for a subset of the parameter space we show that the point processes can be
lifted to line ensembles on , which satisfy the Brownian Gibbs
property. Our ensembles generalize the wanderer line ensembles introduced by
Corwin and Hammond in (Invent. Math. 195, 441-508, 2014).Comment: 76 pages, 8 figure
Log-gases on quadratic lattices via discrete loop equations and q-boxed plane partitions
No full textTransversal Fluctuations of the ASEP, Stochastic Six Vertex Model, and Hall-Littlewood Gibbsian Line Ensembles
No full textApplication of Preformed Metal Crowns on Class II Carious Lesions on Primary Molars /// Приложение на преформирани метални коронки при втори клас кариозни лезии на временни молари
No full textThe aim of this thesis is to represent the results of the study of the approximal caries prevalence on primary molars in the primary and the early mixed dentition, as well as the possibilities for treatment and restoration with preformed metal crowns. According to the clinical study the intensity of the caries in Varna is significantly high. The relative share of the aproximal caries in the age of 5-7 years significantly prevails the occlusal caries. The average distribution of the approximal caries is 3 per child. The caries treatment of is a main problem among children which imposes finding more atraumatic, concerning the psyche, methods of caries management. Parents’ requirements about the aesthetics of the restorations, as well as the properties of the filling materials and children’s behavior to the dental treatment, put some limits concerning the proper choice of obturation material. Nowadays the most commonly used filling material for primary molars restorations are GIC. They have a very good biological activity (fluor releasing potential) to hard tooth tissues, provide chemical bond with the tooth’s structures. Thus these materials do not require any additional retentions in the prepared cavities which preserve the healthy dental structures. In addition, they are quite tolerant to moisture. One of the main disadvantages is the low mechanical durability. More than 1/3 of the GIC fail in 2 years after their application. The Hall technique is well accepted by children and their parents. It does not traumatize children’s psyche as the method avoids the usage of rotary instruments and local anaesthesia. Most of the children feel insignificant discomfort while the Hall teqchnique is performed.ЦЕЛ: Да се представят резултати в проучване на разпространението на апроксималния кариес на временни молари, във временно и ранно смесено съзъбие при деца на възраст 5-7 години, както и възможностите за лечение и възстановяване с помощта на преформирани метални коронки. Според направеното клинично изследване интензитетът на кариеса в гр. Варна е значително висок. Относителният дял на апроксималния кариес в тази възраст (5-7 години), значително преобладава над оклузалния. Средно на всяко дете се падат по 3 апроксимални кариозни лезии. Лечението му е основен проблем при децата, което налага да се търсят по-атравматични, по отношение на психиката, методи за управление на кариозния процес. Изискванията на родителите по отношение на естетиката на възстановяванията, както и свойствата на материалите и поведението на малките пациенти към денталното лечение относително ограничават избора на подходящ материал за възстановяване. В днешно време най-често прилаганото обтурационно средство за възстановяване на временни молари са ГЙЦ. Те притежават изключителни биологично активни качества (флуороизлъчващ потенциал) и осъществяват химическа връзка с твърдите зъбни структури. Това позволява да не се изработва специфична кавитетна форма и изрязване на здрави зъбни тъкани. Освен това са относително „толерантни“ към влагата. Основният им недостатък е механичната им здравина. Повече от 1/3 от глас йономерните възстановявания търпят неуспех до 2 години след апликацията им. Техниката на Hall се приема изключително добре от децата и техните родители. Тя не травмира психиката на детето поради факта, че се избягва употребата на ротационни инструменти и локална анестезия. По-голяма част от малките пациенти усещат незначителен дискомфорт по време на прилагането на техниката
Multi-level loop equations for -corners processes
Full text linkThe goal of the paper is to introduce a new set of tools for the study of
discrete and continuous -corners processes. In the continuous setting,
our work provides a multi-level extension of the loop equations (also called
Schwinger-Dyson equations) for -log gases obtained by Borot and Guionnet
in (Commun. Math. Phys. 317, 447-483, 2013). In the discrete setting, our work
provides a multi-level extension of the loop equations (also called Nekrasov
equations) for discrete -ensembles obtained by Borodin, Gorin and
Guionnet in (Publications math{\' e}matiques de l'IH{\' E}S 125, 1-78, 2017).Comment: 46 pages, 2 figures. Fixed a few typos and added a few reference
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