Solvable models in the KPZ class: Approach through periodic and free boundary Schur measures

It is believed that for metric-like models in the KPZ class the following property holds: with probability one, starting from any point, there are at most two semi-infinite geodesics with the same direction that do not coalesce. Until now, such a result was only proved for one model - exponential LPP (Coupier, D., in Electron. Commun. Probab. 16, 517–527, 2011) using its inherent connection to the totally asymmetric exclusion process. We prove that the above property holds for the directed landscape, the universal scaling limit of models in the KPZ class. Our proof reduces the problem to one on line ensembles and therefore paves the way to show similar results for other metric-like models in the KPZ class. Finally, combining our result with the ones in (Bhatia, M., Duality in the directed landscape and its applications to fractal geometry. https://arxiv.org/pdf/2301.07704 , 2023: Busani, O., et al. To appear in Annals of Probability, (2022)) we obtain the full qualitative geometric description of infinite geodesics in the directed landscape.

We review some algebraic and combinatorial structures that underlie models in the KPZ universality class. Emphasis is placed on the Robinson-Schensted-Knuth correspondence and its geometric lifting due to A.N.Kirillov. We present how these combinatorial constructions are used to analyse the structure of solvable models in the KPZ class and lead to computation of their statistics via connecting to representation theoretic objects such as Schur, Macdonald and Whittaker functions, Young tableaux and Gelfand-Tsetlin patterns. We also present how fundamental representation theoretic concepts, such as the Cauchy identity, the Pieri rule and the branching rule, can be used, alongside RSK correspondences, and can be combined with probabilistic ideas, in order to construct integrable stochastic dynamics on two dimensional arrays of Gelfand-Tsetlin type, in ways that couple different one dimensional stochastic processes. For example, interacting particle systems, on the one hand, and processes related to eigenvalues of random matrices, on the other, thus illuminating the emergence of random matrix distributions in interacting stochastic processes. The goal of the notes is to expose some of the overarching principles, which have driven a significant number of developments in the field.

The height fluctuations of the models in the KPZ class are expected to converge to a universal process. The spatial process at equal time is known to converge to the Airy process or its variations. However, the temporal process, or more generally the two-dimensional space-time fluctuation field, is less well understood. We consider this question for the periodic TASEP (totally asymmetric simple exclusion process). For a particular initial condition, we evaluate the multitime and multilocation distribution explicitly in terms of a multiple integral involving a Fredholm determinant. We then evaluate the large-time limit in the so-called relaxation time scale.

We construct a family of stochastic growth models in 2+1 dimensions, that belong to the anisotropic KPZ class. Appropriate projections of these models yield 1+1 dimensional growth models in the KPZ class and random tiling models. We show that correlation functions associated to our models have determinantal structure, and we study large time asymptotics for one of the models. The main asymptotic results are: (1) The growing surface has a limit shape that consists of facets interpolated by a curved piece. (2) The one-point fluctuations of the height function in the curved part are asymptotically normal with variance of order ln(t) for time t>>1. (3) There is a map of the (2+1)-dimensional space-time to the upper half-plane H such that on space-like submanifolds the multi-point fluctuations of the height function are asymptotically equal to those of the pullback of the Gaussian free (massless) field on H.

We study square lattice space group symmetry fractionalization in a family of exactly solvable models with $\mathbb{Z}_2$ topological order in two dimensions. In particular, we have obtained a complete understanding of which distinct types of symmetry fractionalization (symmetry classes) can be realized within this class of models, which are generalizations of Kitaev's $\mathbb{Z}_2$ toric code to arbitrary lattices. This question is motivated by earlier work of A. M. Essin and one of us (M. H.), where the idea of symmetry classification was laid out, and which, for square lattice symmetry, produces 2080 symmetry classes consistent with the fusion rules of $\mathbb{Z}_2$ topological order. This approach does not produce a physical model for each symmetry class, and indeed there are reasons to believe that some symmetry classes may not be realizable in strictly two-dimensional systems, thus raising the question of which classes are in fact possible. While our understanding is limited to a restricted class of models, it is complete in the sense that for each of the 2080 possible symmetry classes, we either prove rigorously that the class cannot be realized in our family of models, or we give an explicit model realizing the class. We thus find that exactly 487 symmetry classes are realized in the family of models considered. With a more restrictive type of symmetry action, where space group operations act trivially in the internal Hilbert space of each spin degree of freedom, we find that exactly 82 symmetry classes are realized. In addition, we present a single model that realizes all $2^6 = 64$ types of symmetry fractionalization allowed for a single anyon species ($\mathbb{Z}_2$ charge excitation), as the parameters in the Hamiltonian are varied. The paper concludes with a summary and a discussion of two results pertaining to more general bosonic models.

We introduce and study a one parameter deformation of the polynuclear growth (PNG) in (1+1)-dimensions, which we call the $t$-PNG model. It is defined by requiring that, when two expanding islands merge, with probability $t$ they sprout another island on top of the merging location. At $t=0$, this becomes the standard (non-deformed) PNG model that, in the droplet geometry, can be reformulated through longest increasing subsequences of uniformly random permutations or through an algorithm known as patience sorting. In terms of the latter, the $t$-PNG model allows errors to occur in the sorting algorithm with probability $t$. We prove that the $t$-PNG model exhibits one-point Tracy–Widom Gaussian Unitary Ensemble asymptotics at large times for any fixed $t\in [0,1)$, and one-point convergence to the narrow wedge solution of the Kardar–Parisi–Zhang equation as $t$ tends to $1$. We further construct distributions for an external source that are likely to induce Baik–Ben Arous–Péché-type phase transitions. The proofs are based on solvable stochastic vertex models and their connection to the determinantal point processes arising from Schur measures on partitions.

Animal migration is a mass biological phenomenon indispensable for comprehension and assessment of food-webs. So far, theoretical models to describe decision-making processes inherent in the animal migration have not been well established, which is the motivation of this research. It is natural to formulate the animal migration based on a stochastic control theory, which can describe system dynamics and its optimization in stochastic environment. To address this issue, a conceptual stochastic control model for the decision-making processes in animal migration is introduced and mathematically analysed. Its novelty is mathematical simplicity and the new theoretical, stochastic control viewpoint. Stochastic differential equations govern the animal population dynamics with gradual and radical migrations from the current habitat toward the next one. The population decides the occurrences, magnitudes, and timings of the migrations, so that a heuristic performance index is maximised. I derive a variational inequality that governs the maximised performance index and is exactly solvable. Its free boundaries govern the gradual and radical migrations. Despite the model simplicity, the exact solution is consistent with the empirical observation results of fish migration, implying its potential applicability to animal migration. The present model can be used for assessing fish migration. References S. Bauer and B. J. Hoye. Migratory animals couple biodiversity and ecosystem functioning worldwide. Science, 344, 2014. Article No. 1242552. N. E. Leonaerd. Multi-agent system dynamics: Bifurcation and behavior of animal groups. Annual Reviews in Control, 38(2):171–183, 2014. A. M. Oberman. Convergent difference schemes for degenerate elliptic and parabolic equations: Hamilton–jacobi equations and free boundary problems. SIAM Journal on Numerical Analysis, 44(2):879–895, 2006. B. \T1\O ksendal. Stochastic Differential Equations. Springer Berlin Heidelberg, 2003. B. \T1\O ksendal and A. Sulem. Applied Stochastic Control of Jump Diffusions. Springer Berlin Heidelberg, 2007. Y. Yaegashi, H. Yoshioka, K. Unami, and M. Fujihara. An optimal management strategy for stochastic population dynamics of released \(plecoglossus\) \(altivelis\) in rivers. International Journal of Modeling, Simulation, and Scientific Computing, 8(2), 207. Article No. 1750039. H. Yoshioka, T. Shirai, and D. Tagami. Viscosity solutions of a mathematical model for upstream migration of potamodromous fish. In Proceedings of 12th SDEWES Conference, October 4-8, 2017, Dubrovnik, Croatia, 2017. In press.

We study a model, introduced initially by Gates and Westcott [11] to describe crystal growth evolution, which belongs to the Anisotropic KPZ universality class [19]. It can be thought of as a $(2+1)$-dimensional generalisation of the well known ($1+1$)-dimensional Polynuclear Growth Model (PNG). We show the full hydrodynamic limit of this process i.e the convergence of the random interface height profile after ballistic space-time scaling to the viscosity solution of a Hamilton-Jacobi PDE: $\\partial _{t}u = v(\\nabla u)$ with $v$ an explicit non-convex speed function. The convergence holds in the strong almost sure sense.

When studying fluctuations of models in the 1D KPZ class including the ASEP and the $q$-TASEP, a standard approach has been to first write down a formula for $q$-deformed moments and constitute their generating function. This works well for the step initial condition, but there is a difficulty for a random initial condition (including the stationary case): in this case only the first few moments are finite and the rest diverge. In a previous work [16], we presented a method dealing directly with the $q$-deformed Laplace transform of an observable, in which the above difficulty does not appear. There the Ramanujan's summation formula and the Cauchy determinant for the theta functions play an important role. In this note, we give an alternative approach for the $q$-TASEP without using them.

The behavior of metal oxide semiconductor field effect transistors (MOSFETs) with small aspect ratio and large doping levels is analyzed using formal perturbation techniques. Formally, we will show that in the limit of small aspect ratio there is a region in the middle of the channel under the control of the gate where the potential is one-dimensional. The influence of interface and internal layers in the one-dimensional potential on the averaged channel conductivity is closely examined in the large doping limit. The interface and internal layers that occur in the one-dimensional potential are resolved in the limit of large doping using the method of matched asymptotic expansions. The asymptotic potential in the middle of the channel is constructed for various classes of variable doping models including a simple doping model for the built-in channel device. Using the asymptotic one-dimensional potential, the asymptotic mobile charge, needed for the derivation of the long-channel I-V curves, is found by using standard techniques in the asymptotic evaluation of integrals. The formal asymptotic approach adopted not only provides a pointwise description of the state variables, but by using the asymptotic mobile charge, the lumped long-channel current-voltage relations, which vary uniformly across the various bias regimes, can be found for various classes of variable doping models. Using the explicit solutions of some free boundary problems solved by Howison and King (1988), the two-dimensional equilibrium potential near the source and drain is constructed asymptotically in strong inversion in the limit of large doping. From the asymptotic potential constructed near the source and drain, a uniform analytical expression for the mobile charge valid throughout the channel is obtained. From this uniform expression for the mobile charge, we will show how it is possible to find the I-V curve in a particular bias regime taking into account the edge effects of the source and drain. In addition, the asymptotic potential for a two-dimensional n+-p junction is constructed.

Reflection equations are used to obtain families of commuting double-row transfer matrices for interaction-round-a-face (IRF) models with fixed and free boundary conditions. We illustrate our methods for the Andrews-Baxter-Forrester (ABF) models which areL-state models associated with the quantum groupU q (su(2)) at a root of unity. We construct elliptic solutions to the reflection equations for the ABF models by a procedure which uses fusion to build the solutions starting from a trivial solution. Presented at the 5th International Colloquium on Quantum Groups: Quantum Groups and Integrable Systems, Prague, 20–22 June 1996.

The logarithmic derivative of the marginal distributions of randomly fluctuating interfaces in one dimension on a large scale evolve according to the Kadomtsev–Petviashvili (KP) equation. This is derived algebraically from a Fredholm determinant obtained in [MQR17] for the Kardar–Parisi–Zhang (KPZ) fixed point as the limit of the transition probabilities of TASEP, a special solvable model in the KPZ universality class. The Tracy–Widom distributions appear as special self-similar solutions of the KP and Korteweg–de Vries equations. In addition, it is noted that several known exact solutions of the KPZ equation also solve the KP equation.

We construct exactly solvable models for a wide class of symmetry enriched topological (SET) phases. Our construction applies to 2D bosonic SET phases with finite unitary onsite symmetry group $G$ and we conjecture that our models realize every phase in this class that can be described by a commuting projector Hamiltonian. Our models are designed so that they have a special property: if we couple them to a dynamical lattice gauge field with gauge group $G$, the resulting gauge theories are equivalent to modified string-net models. This property is what allows us to analyze our models in generality. As an example, we present a model for a phase with the same anyon excitations as the toric code and with a $\mathbb{Z}_2$ symmetry which exchanges the $e$ and $m$ type anyons. We further illustrate our construction with a number of additional examples.

A class of exactly solvable models of the conformally invariant quantum field theory in D dimensions is proposed. It is shown that in any conformal theory of the field φ(x) with the scale dimension d there exists an infinite collection of the tensor fields Ps of the ranks and the dimensions ds=d+s independently of the type of interaction. These tensor fields appear in the product Tµν(x1)φ(x2) operator expansion where Tµν is the energy-momentum tensor. The fields Ps are analogues to the certain superpositions of the secondary fields of D=2 models. The existence of the fields Ps follows from the structure of the Ward identities for the energy-momentum tensor conformally invariant Green functions. Each model of the above-mentioned class is defined by the operator equation Ps(x)=0. The method of solving these models is proposed. Some of the models coincide with the certain lagrangian models. The method allows us to obtain closed differential equations for each Green function of fundamental and composite fields, and also algebraic equations for scale dimensions of the fields. The derivation of all these equations is based on the energy momentum conformal Ward identities’ specific property. Detailed analysis of the Ward identities is given in the paper. It is shown that each D>2-model involves a special scalar field PD−2 with the scale dimension dP=D−2. This field is an analogue of the central charge of D=2 models and becomes the constant coinciding with the central charge when D=2. A new class of D=2 models with broken infinite parametric symmetry is obtained. In each model the closed differential equations for the highest Green functions are derived and the central charge is calculated. The general method of solving the D≥2 models is illustrated on examples of Thirring and Wess-Zumino-Witten models and on a trivial model in four-dimension space.

We provide a comprehensive report on scale-invariant fluctuations of growing interfaces in liquid-crystal turbulence, for which we recently found evidence that they belong to the Kardar-Parisi-Zhang (KPZ) universality class for 1+1 dimensions [Takeuchi and Sano in Phys. Rev. Lett. 104:230601, 2010; Takeuchi et al. in Sci. Rep. 1:34, 2011]. Here we investigate both circular and flat interfaces and report their statistics in detail. First we demonstrate that their fluctuations show not only the KPZ scaling exponents but beyond: they asymptotically share even the precise forms of the distribution function and the spatial correlation function in common with solvable models of the KPZ class, demonstrating also an intimate relation to random matrix theory. We then determine other statistical properties for which no exact theoretical predictions were made, in particular the temporal correlation function and the persistence probabilities. Experimental results on finite-time effects and extreme-value statistics are also presented. Throughout the paper, emphasis is put on how the universal statistical properties depend on the global geometry of the interfaces, i.e., whether the interfaces are circular or flat. We thereby corroborate the powerful yet geometry-dependent universality of the KPZ class, which governs growing interfaces driven out of equilibrium.