Exact Solutions in Multi-Species Exclusion Processes Zeying CHEN ORCID iD: 0000-0003-4989-4300 A thesis submitted in total fulfilment of the requirements of the degree of Doctor of Philosophy in Mathematics School of Mathematics and Statistics The University of Melbourne July, 2019 ii Declaration This thesis is an account of research undertaken between August 2015 and May 2019 at The School of Mathematics and Statistics, Faculty of Science, The University of Melbourne, Melbourne, Australia. Except where acknowledged in the customary manner, the material presented in this thesis is, to the best of my knowledge, original and has not been submitted in whole or part for a degree in any university. The thesis is fewer than 100 000 words in length, exclusive of tables, maps, bibliographies. Zeying CHEN July, 2019 iii iv Acknowledgements First of all, I would like to express my sincere thank to my supervisor Prof. Jan de Gier, for his valuable advice and support, and also for his motivation and patience. Without him I would never finish this thesis. I am very grateful for his motivation in such two interesting research topics, for the opportunities to attend conferences, and also for his encouragement in the talks I gave in conferences. I am also grateful for his financial support which assisted me in traveling and also in the last several months. It is greatly appreciated. I would like to thank Dr. Michael Wheeler for his continuous support of my study and research. I am very grateful for his guidance throughout my research, especially in the first research topics in the multi-species ASEP duality. I would also like to thank Prof. Tomohiro Sasamoto and Iori Hiki in the second research topics in AHR model. Many thanks also goes to my fellow PhD students and the school staffs. Thank you for all the help and support. Also many thanks to Tian Yuan, Xiangnan Shi, Yan Yu, Richard Wang and all my other friends for their endless love and care. I would also like to thank my family: my parents and grandparents, for their considerations, support and love during these years. Thank you all for supporting my overseas life all these years. I wouldn't have made this far without your loves. I also gratefully acknowledge the financial support provided by the Melbourne International Research Scholarship that made this thesis possible. I am also grateful for the financial support from The Australian Research Council (ARC) Centre of Excellence for Mathematical and Statistical Frontiers (ACEMS) and the Matrix program. v vi Abstract The exclusion process has been the default model for the transportation phe- nomenon. One fundamental issue is to compute the exact formulae analytically. Such formulae enable us to obtain the limiting distribution through asymptotics analysis, and they also allow us to uncover relationships between different processes, and even between very different systems. Extensive results have been reported for single-species systems, but few for multi-component systems and mixtures. In this thesis, we focus on multi-species exclusion processes, and propose two approaches for exact solutions. The first one is due to duality, which is defined by a function that co-varies in time with respect to the evolution of two processes. It relates physical quantities, such as the particle flow, in a system with many particles to one with few parti- cles, so that the quantity of interest in the first process can be calculated explicitly via the second one. Historically, published dualities have mostly been found by trial and error. Only very recently have attempts been made to derive these func- tions algebraically. We propose a new method to derive dualities systematically, by exploiting the mathematical structure provided by the deformed quantum Knizhnik- Zamolodchikov equation. With this method, we not only recover the well-known self-duality in single-species asymmetric simple exclusion processes (ASEPs), and also obtain the duality for two-species ASEPs. Solving the master equation is an alternative method. We consider an exclusion process with 2 species particles: the AHR (Arndt-Heinzl-Rittenberg) model and give a full derivation of its Green's function via coordinate Bethe ansatz. Hence using the Green's function, we obtain an integral formula for its joint current dis- tributions, and then study its limiting distribution with step type initial conditions. We show that the long-time behaviour is governed by a product of the Gaussian and the Gaussian unitary ensemble (GUE) Tracy-Widom distributions, which is re- lated to the random matrix theory. Such result agrees with the prediction made by the nonlinear fluctuating hydrodynamic theory (NLFHD). This is the first analytic verification of the prediction of NLFHD in a multi-species system. vii viii Preface The work contained in this thesis was carried out under the supervision of Pro- fessor Jan de Gier at the School of Mathematics and Statistics, The University of Melbourne. The main original contributions of the author are in Chapters 3 and 4. The Chapter 3, a joint work with Jan de Gier and Michael Wheeler, is published as \Integrable Stochastic Dualities and the Deformed Knizhnik-Zamolodchikov Equa- tion" [165], and Chapter 4, a joint work with Jan de Gier, Iori Hiki, and Tomohiro Sasamoto is published as \Exact Confirmation of 1D Nonlinear Fluctuating Hydro- dynamics for a Two-Species Exclusion Process" [35]. ix x Contents Declaration iii Acknowledgementsv Abstract vii Preface ix List of Figures xv 1 Introduction1 1.1 Non equilibrium Systems........................2 1.2 Integrable models.............................3 1.3 KPZ universality class..........................3 1.3.1 Gaussian distribution......................4 1.3.2 Simple exclusion processes....................4 1.4 Notation..................................5 1.5 Outline..................................6 2 Asymmetric Simple Exclusion Process9 2.1 Definition of the model: a Markov chain................9 2.1.1 Dynamics............................. 11 2.2 Master equation............................. 12 2.3 Stationary states............................. 14 2.4 Infinitesimal generator.......................... 15 2.5 Configuration notation.......................... 16 2.5.1 ASEP generators......................... 17 2.5.2 Reverse ASEP generators.................... 18 2.6 ASEP and duality............................ 19 2.6.1 Duality.............................. 20 2.6.2 Self duality in ASEP....................... 21 2.7 Bethe ansatz............................... 25 2.7.1 Periodic condition........................ 25 2.7.2 Infinite system.......................... 31 2.8 Duality vs. Bethe ansatz......................... 38 2.9 Matrix product ansatz.......................... 38 xi xii Contents 3 Stochastic Duality and Deformed KZ Equations 41 3.1 Introduction................................ 41 3.1.1 Background............................ 41 3.1.2 Functional definition of duality................. 43 3.1.3 Matrix definition of duality................... 43 3.1.4 tKZ equations as a source of dualities............. 44 3.1.5 Notation and conventions.................... 45 3.2 Asymmetric simple exclusion process.................. 46 3.2.1 The ASEP generators Li and Mi ................ 47 3.2.2 Divided-difference realization of Li ............... 48 3.2.3 Matrix realization of Mi ..................... 49 3.2.4 Local duality relation...................... 49 3.2.5 Generalization to multi-species ASEP............. 49 3.3 Connection with the t-deformed Knizhnik-Zamolodchikov equations 50 3.3.1 Hecke algebra, ASEP exchange relations and tKZ equations. 51 3.3.2 Non-symmetric Macdonald polynomials............ 53 3.3.3 Reduction............................. 55 3.3.4 Another non-symmetric basis.................. 58 3.3.5 Dualities from reductions of ASEP polynomials........ 60 3.4 Explicit formulae for the ASEP polynomials.............. 62 3.4.1 Matrix product formula for fµ(z; q; t).............. 62 3.4.2 Summation formulae....................... 64 3.5 ASEP dualities.............................. 66 3.5.1 Occupation and position notation................ 67 3.5.2 Reduction from rank-r to rank-one............... 68 3.5.3 Back to the proof of Theorem 3.25............... 72 3.6 Rank-two ASEP dualities........................ 73 3.6.1 Reduction relations between a pair of rank-two sectors.... 74 3.6.2 Rank-two duality functions................... 77 3.7 Duality functions without indicators.................. 80 3.7.1 Rank-one duality functions without indicators......... 80 3.7.2 Rank-one/rank-two duality functions without indicators... 81 3.7.3 Generalization to arbitrary rank................ 83 4 The Green's Function in a Two-Species Exclusion Process 85 4.1 Definition of the model.......................... 85 4.2 Master equation............................. 86 4.3 Transition probability.......................... 87 4.4 Bethe wave functions........................... 92 4.5 Symmetrisation identities........................ 95 4.6 Fredholm determinant.......................... 98 4.7 Joint current distributions........................ 101 4.7.1 Step initial data......................... 101 4.7.2 Bernoulli step data........................ 104 Contents xiii 4.8 Asymptotic analysis........................... 107 4.8.1 NLFHD and scaling limit.................... 110 4.8.2 Asymptotic analysis of the first term.............. 111 4.8.3 Asymptotic analysis of the second term............ 127 5 Conclusion and Outlook 137 5.1 Summary................................. 137 5.1.1