Стохастические интегрируемые процессы и представления группы кос тема диссертации и автореферата по ВАК РФ 01.01.03, кандидат наук Трофимова Анастасия Алексеевна

• Трофимова Анастасия Алексеевна
• кандидат наук
• 2022

Оглавление диссертации кандидат наук Трофимова Анастасия Алексеевна

Contents

Introduction

1 Representations of finite-dimensional quotients of C[B3]

1.1 Braid group B3 and its quotients

1.2 Spectra of the central element and of generators

1.3 Low dimensional representations of QX

1.4 Semisimplicity criteria

2 q-boson zero range process

2.1 The model and its observables

2.1.1 Stationary state and scaled current cumulants

2.1.2 Interface growth and KPZ-EW universality

2.2 Exact formulae for scaled current cumulants

2.2.1 Bethe ansatz and T-Q equation

2.2.2 First order calculation

2.2.3 Second order calculation

2.3 Asymptotic analysis

2.3.1 Saddle point approximation

2.3.2 KPZ regime |q| =

2.3.3 KPZ - EW crossover

Conclusion

References

Введение диссертации (часть автореферата) на тему «Стохастические интегрируемые процессы и представления группы кос»

Abstract

Integrability is an important property of models of quantum mechanics and statistical physics that makes an exact solution possible. One of the main methods that is used to study integrable systems is the Bethe ansatz approach based on the Yang-Baxter equation. The study and classification of existing R-matrices as well as the search of the new ones become a problem of mathematical physics. One of the major sources of R-matrix representations is the representations of the braid group on three strings and its finite-dimensional quotient algebras. New results were obtained in two directions.

The first direction is the classification of low dimensional irreducible representations of the braid group on three strings. We consider families of the finite dimensional quotient algebras of the group algebra of the braid group on three strings by a p-th order generic monic polynomial relation on the elementary braids (also known as Artin generators). These are the cases of power p = 2, 3,4, 5 polynomial relation with the corresponding dimensions of quotient algebras equal to 6, 24, 96, and 600, respectively. We construct a series of representations of dimension < 6 (Proposition 1.3.1) and find the conditions under which they are irreducible (Proposition 1.4.1). For the considered quotient algebras we formulate semisimplicity criteria, and if those criteria are satisfied we give a complete classification of irreducible representations (Theorem 1.4.1). Our classification complements the list of all the irreducible representations of the braid group on three strings of dimensions < 5 found by I. Tuba and H. Wenzl (2001) by adding irreducible representations of dimension 6. The study of these 6-dimensional representations brings new criteria.

Another subject of the research is stochastic integrable particle systems. Our interest concerns the statistics of a particle flow in the q-boson zero range process (ZRP) on a ring. With the use of Bethe ansatz and TQ-relation methods we calculate the first two cumulants of the particle current. The exact formula for the second cumulant is obtained in the form of an infinite sum of double contour integrals (Theorem 2.1.1). This representation allows us to perform the asymptotic analysis of the large system size limit. In this limit we find that at generic values of parameter controlling interparticle interaction the second cumulant reproduces the scaling expected for the models in the Kardar-Parisi-Zhang universality class (Theorem 2.1.1). Another result is the universal scaling function describing the crossover between the Kardar-Parisi-Zhang and Edwards-Wilkinson universality classes. It is obtained from the exact formula for the second cumulant in a scaling limit corresponding to the KPZ-EW transition (Theorem 2.1.2). It agrees with the scaling function first found for the asymmetric simple exclusion process and conjectured to be universal.

Acknowledgments

The author is deeply grateful to her scientific advisors Alexander Povolotsky and Pavel Pyatov for stating the problems, for fruitful discussions and ideas, for guidance, patience and support.

Author is grateful to Higher School of Economics and Skolkovo Institute of Science for the organisation of education procedure that gave a lot of space for research and for supporting trips to scientific conferences.

Author is also grateful to her family for endless support.

Заключение диссертации по теме «Математическая физика», Трофимова Анастасия Алексеевна

Conclusion

To summarize, for all finite-dimensional quotient algebras of the group algebra C[B3] by p-order generic polynomial relations on elementary braids we have classified all their irreducible representations and found the semisimplicity criteria. Even though the representations of dimension < 5 were already found [11] the choice of diagonal basis for the first elementary generator allows us to find also 6-dimensional irreducible representations and obtain semisimplisity criteria including the new ones. Our result contributes to the list of low-dimensional irreducible representations of the braid group B3. We believe that the problem of construction of a new integrable stochastic R-matrix representations of the braid group leads to the new integrable models and is perspective for future research.

As for the examination of stochastic integrable particle systems we studied a particle flow in the q-boson zero range process with the use of Bethe ansatz and T-Q-relation methods. These methods were effective for the search of the exact expressions for the first two cumulants of the particle current. We reproduced the scaling N3/2 for q = 1, which is expected for the models in the Kardar-Parisi-Zhang universality class. In the limit of q = exp(- ) ^ 1 we found the universal scaling function describing crossover between the Kardar-Parisi-Zhang and Edwards-Wilkinson universality classes. It agrees with the universal scaling crossover function initially found for the asymmetric simple exclusion process and conjectured to be universal.

The exact expressions for the first two cumulants can be considered as a first step in the studies of the large time behaviour of q-boson type models in confined geometry. The next goal could be a construction of the higher cumulants of the model. In particular, it would be interesting to study KPZ-EW crossover not only for particular current cumulants, but also for the whole large deviation function.

The other perspective direction is a study of the q-boson-like models with more parameters, for example, q-Hahn(chipping model) or q,^, v-ZRP. Some of the models are good candidates for research of transitions between KPZ and other than EW types of behaviour. One of the examples of such models is the TASEP with generalized update studied recently in [80]. In that case the Bethe ansatz solution simplifies greatly delivering the whole scaled cumulant generating function by the Derrida-Lebowitz method. The model experience crossover from the KPZ specific behaviour to Gaussian deterministic aggregation regime. The other example is the chipping model [88] with parameter q = 0. In this case the method of Derrida-Lebowitz is inapplicable. The appropriate tool would then be the T-Q relation approach developed in [76] adopted for the q-boson model.

One more intriguing model is the asymmetric avalanche process [89, 79]. It is an ASEP-like 1-dimensional particle model with totally asymmetric avalanche dynamics. At a low (subcritical) particle density the particle flow is intermittent, while at a high particle density, it becomes continuous [69] due to presence of giant avalanches. The subcritical large deviation function of the particle current in AAP in a periodic system obtained in [92] has a universal scaling form specific for KPZ universality class. Increase of density leads to transition of the model to another universality class. In the article recently accepted for publication [87] we also applied the Bethe ansatz and the T-Q relation approach to obtain the exact expressions for the first two scaled cumulants of the particle current in the large time limit. The asymptotic analysis in the large system size limit N ^ ro differs significantly from the q-boson ZRP. It results in interesting scaling behaviour both for the particle current and diffusion coefficient. The approach proved its effectiveness delivering

scaling exponents at three different model regimes and giving crossover functions from the KPZ specific behaviour to to the tilted interface universality class [90, 91].

Finally, one may consider the similar problem for q-boson ZRP on the open segment. By now the current-density diagram was obtained for this model using the matrix product representation of the stationary state, however the further current cumulants and the large deviation function are not known. The generalization of the T-Q approach to this problem is a challenging. However, it is expected that these results match with the corresponding solution of ASEP with open boundaries [72, 41]. This is a matter of further studies.

Список литературы диссертационного исследования кандидат наук Трофимова Анастасия Алексеевна, 2022 год

Publications

The main results of the thesis are presented in two papers:

1. Pyatov P., Trofimova A., Representations of finite-dimensional quotient algebras of the 3-string braid group, Moscow Math. J., 2021, 21, 427-442.

2. A. Trofimova, A. Povolotsky, Current statistics in the q-boson zero range process, J. Phys. A: Math. Theor. 2020, 53, 283003-1-283003-35.

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