Independence properties for tree beta models
This paper is devoted to introduce new probabilistic tree-based models complementing already known Matsumoto-Yor and Hamza-Vallois tree models with two tree beta-type models.
math.PR↗
arXiv subjects
Makiko Sasada
Publications and source records attributed to Makiko Sasada.
Stationary fluctuations for an exclusion process with mass and energy conservation
We introduce a novel exclusion process with two conservation laws, mass and energy, designed to mimic the essential features of continuous systems like interacting oscillators within the framework of interacting particle systems. This distinguishes our model from conventional multi-species processes where only particle numbers are conserved. As a basis for our fluctuation analysis, we first show that applying nonlinear fluctuating hydrodynamics (NFH) to this model reveals a wide variety of universality classes depending on the parameter choices. The main objective of this work is to study the stationary fluctuations of these conserved quantities. For a suitable choice of parameters, we rigorously show that the fluctuation fields converge to uncoupled stochastic Burgers equations (SBE) in the scaling limit. The proof relies on the second-order Boltzmann-Gibbs principle that we establish for this model, along with the spectral gap estimate and the equivalence of ensembles. Of independent interest is our general proof of the diagonalizability of the Jacobian matrix for the macroscopic current with distinct real eigenvalues. While this property is often taken as given in the physics literature, we establish it rigorously for multi-component systems even when the eigenvectors cannot be explicitly computed, offering a firm mathematical foundation for a broad class of models.
math.PR↗
Relationships between two linearizations of the box-ball system : Kerov-Kirillov-Reshetikhin bijection and slot configuration
The box-ball system (BBS), which was introduced by Takahashi and Satsuma in 1990, is a soliton cellular automaton. Its dynamics can be linearized by a few methods, among which the best known is the Kerov-Kirillov-Reshetikhin (KKR) bijection using rigged partitions. Recently a new linearization method in terms of "slot configurations" was introduced by Ferrari-Nguyen-Rolla-Wang, but its relations to existing ones have not been clarified. In this paper we investigate this issue and clarify the relation between the two linearizations. For this we introduce a novel way of describing the BBS dynamics using a carrier with seat numbers. We show that the seat number configuration also linearizes the BBS and reveals explicit relations between the KKR bijection and the slot configuration. In addition, by using these explicit relations, we also show that even in case of finite carrier capacity the BBS can be linearized via the slot configuration.
math.CO↗
Degree-preserving conservative processes and a unified approach for their hydrodynamics
We investigate a broad class of large-scale one-dimensional interacting systems characterized by a single conservation law and satisfying the "degree-preserving property". Under mild and natural assumptions, we establish a unified framework for the analysis of both invariant measures and hydrodynamic limits. In particular, we prove that when the generator preserves the degree of polynomials of the state variables up to order two, the marginals of any product invariant measure must belong to a family of six specific distributions. This classification is shown to be consistent with a classical result on univariate natural exponential families due to C.N. Morris, which we apply here for the first time in the context of microscopic stochastic systems. As a consequence, we construct a new interacting particle system whose invariant measure is given by the generalized hyperbolic secant distribution. Furthermore, we prove that, despite the generality of the dynamics, the macroscopic behavior of all models in this class is governed by the classical heat equation, with a diffusion coefficient depending explicitly on the underlying microscopic interactions.
math.PR↗
Ultra-Discretization of Yang-Baxter Maps, Probability Distributions and Independence Preserving Property
We study the relationship between Yang-Baxter maps and the independence preserving (IP) property, motivated by their role in integrable systems, from the perspective of ultra-discretization. Yang-Baxter maps satisfy the set-theoretic Yang-Baxter equation, while the IP property ensures independence of transformed random variables. The relationship between these two seemingly unrelated properties has recently started to be studied by Sasada and Uozumi (2024). Ultra-discretization is a concept primarily used in the context of integrable systems and is an area of active research, serving as a method for exploring the connections between different integrable systems. However, there are few studies on how the stationary distribution for integrable systems changes through ultra-discretization. In this paper, we introduce the concept of ultra-discretization for probability distributions, and prove that the properties of being a Yang-Baxter map and having the IP property are both preserved under ultra-discretization. Applying this to quadrirational Yang-Baxter maps, we confirm that their ultra-discrete versions retain these properties, yielding new examples of piecewise linear maps having the IP property. We also explore implications of our results for stationary distributions of integrable systems and pose several open questions.
nlin.SI↗
Unveiling Optimal Diffusion for Infection Control in Brownian Particle Systems
Understanding the spread of infectious diseases requires integrating movement, physical constraints, and spatial configurations into epidemiological models. In this study, we investigate how particle diffusivity, hardcore interactions, and non-equilibrium initial conditions influence infection dynamics within a system of Brownian particles. Using numerical simulations and theoretical analysis, we reveal a nontrivial relationship between diffusivity and the speed of infection spread. Specifically, when particles are initially positioned at uniform distances greater than the infection radius -- a non-equilibrium configuration -- there exists an optimal diffusion coefficient that minimizes the infection propagation speed. This counterintuitive result arises from the competition between diffusive timescales and the rate of infection transmission. The presence of an optimal diffusivity is observed both in systems with and without hardcore interactions, provided that the infection radius exceeds the mean lattice spacing. Our findings provide a theoretical framework for understanding and controlling the spread of infections in confined and diffusive environments, with potential implications for designing movement-based strategies for infection control.
cond-mat.other↗
Hydrodynamic limit for some gradient and attractive spin models
We study the hydrodynamic limit for three gradient spin models: generalized Kipnis-Marchioro-Presutti (KMP), its discrete version and a family of harmonic models, under symmetric and nearest-neighbor interactions. These three models share some universal properties: occupation variables are unbounded, all these processes are of gradient type, their invariant measures are product with spatially homogeneous weights, and, notably, they are all attractive, meaning that the process preserves the partial order of measures along the dynamics. In view of hydrodynamics of large-scale interacting systems, dealing with processes taking values in unbounded configuration spaces is known to be a challenging problem. In the present paper, we show the hydrodynamic limit for all three models listed above in a comprehensive way, and show as a main result, that, under the diffusive time scaling, the hydrodynamic equation is given by the heat equation with model-dependent diffusion coefficient. Our novelty is showing the attractiveness for each model, which is crucial for the proof of hydrodynamics.
math.PR↗
Scaling limits of solitons in the box-ball system
We study the space-time scaling limits of solitons in the box-ball system with random initial distribution. In particular, we show that any recentered tagged soliton converges to a Brownian motion in the diffusive space-time scale, and also prove the large deviation principle for the tagged soliton under certain shift-ergodic invariant distributions, including Bernoulli product measures and two-sided Markov distributions. Furthermore, in the diffusive space-time scaling, we show that two tagged solitons converge to the same Brownian motion even if they are macroscopically far apart.
math.PR↗
On Interactions for Large Scale Interacting Systems
Statistical mechanics explains the properties of macroscopic phenomena based on the movements of microscopic particles such as atoms and molecules. Movements of microscopic particles can be represented by large-scale interacting systems. In this article, we study a combinatorial object which we call interactions, given as a symmetric directed graph representing the possible transition of states on adjacent sites of large-scale interacting systems. Such interactions underlie various standard processes such as the exclusion processes, generalized exclusion processes, multi-species exclusion processes, lattice-gas processes with energy, and the multi-lane particle processes. We introduce the notion of equivalences of interactions using their space of conserved quantities. This allows for the classification of interactions reflecting corresponding macroscopic properties. In particular, we prove that when the set of local states consists of two, three or four elements, then the number of equivalence classes of separable interactions are respectively one, two and five. We also define the wedge sums and box products of interactions, which give systematic methods for constructing new interactions from existing ones. Furthermore, we prove that the irreducibly quantified condition for interactions, which has implicitly played an important role in the theory of hydrodynamic limits, is preserved by wedge sums and box products. Our results provide a systematic method to construct and classify interactions, offering abundant examples suitable for considering hydrodynamic limits.
math.PR↗
Topological Structures of Large Scale Interacting Systems via Uniform Functions and Forms
In this article, we investigate the topological structure of large scale interacting systems on infinite graphs, by constructing a suitable cohomology which we call the uniform cohomology. The central idea for the construction is the introduction of a class of functions called uniform functions. Uniform cohomology provides a new perspective for the identification of macroscopic observables from the microscopic system. As a straightforward application of our theory when the underlying graph has a free action of a group, we prove a certain decomposition theorem for shift-invariant closed uniform forms. This result is a uniform version in a very general setting of the decomposition result for shift-invariant closed $L^2$-forms originally proposed by Varadhan, which has repeatedly played a key role in the proof of the hydrodynamic limits of nongradient large scale interacting systems. In a subsequent article, we use this result as a key to prove Varadhan's decomposition theorem for a general class of large scale interacting systems.
math.PR↗
On Uniform Functions on Configuration Spaces of Large Scale Interacting Systems
Stochastic large scale interacting systems can be studied via the observables, i.e. functions on the underlying configuration space. In our previous article, we introduced the concept of uniform functions, which are suitable class of functions on configuration spaces underlying stochastic systems on infinite graphs. An important consequence is the successful characterization of conserved quantities without introducing the notion of stationary distributions. In this article, we further develop the theory of uniform functions and construct the theory independent of any choice of a base state. Furthermore, we generalize the notion of interactions given in our previous article to accommodate the case where there are multiple possible state transitions on adjacent vertices. We then prove that if the interaction is exchangeable, then any uniform function which gives a global conserved quantity can be expressed as a sum of local conserved quantities of the interaction. Contrary to our previous article, we do not need to assume that the interaction is irreducibly quantified. This shows that our theory of uniform functions on configuration spaces over infinite graphs with transition structure given by an exchangeable interaction is a natural framework to study general stochastic large scale interacting systems. While some of the ideas in this article are based on our previous article, the article is logically independent and self-contained.
math.PR↗
Varadhan's Decomposition of Shift-Invariant Closed $L^2$-forms for Large Scale Interacting Systems on the Euclidean Lattice
We rigorously formulate and prove for a relatively general class of interactions Varadhan's Decomposition of shift-invariant closed $L^2$-forms for a large scale interacting system on the Euclidean lattice with finite range. Such decomposition of closed forms has played an essential role in proving the diffusive scaling limit of nongradient systems. A general expression in terms of conserved quantities was sought from observations for specific models, but a precise formulation or rigorous proof up until now had been elusive. Our result is based on a general decomposition theorem of shift-invariant closed uniform forms studied in our previous article (arXiv:2009.04699). In the present article, we show that the same universal structure also appears for $L^2$-forms. The essential assumptions are: (i) the set of states on each vertex is a finite set, (ii) the measure on the configuration space is the product measure, and (iii) there is a certain uniform spectral gap estimate for the mean field version of the interaction. As a special case, our result gives Varadhan's decomposition for the case of the multi-species exclusion process, which is a new result that could not be proved by existing methods. Our result also gives complete proofs of Varadhan's decompositions for finite range interactions, whose detailed proofs have been missing in the literature - presumably because there exists an obstruction to the standard method of proof.
math.PR↗
The incompressible Navier-Stokes limit from the lattice BGK Boltzmann equation
In this paper, we prove that a local weak solution to the $d$-dimensional incompressible Navier-Stokes equations ($d \geq 2$) can be constructed by taking the hydrodynamic limit of a velocity-discretized Boltzmann equation with a simplified BGK collision operator. Moreover, in the case when the dimension is $d=2,3$, we characterize the combinations of finitely many particle velocities and probabilities that lead to the incompressible Navier-Stokes equations in the hydrodynamic limit. Numerical computations conducted in 2D provide information about the rate with which this hydrodynamic limit is achieved when the Knudsen number tends to zero.
math.AP↗
Characterization of Gradient Condition for Asymmetric Partial Exclusion Processes and Their Scaling Limits
We consider partial exclusion processes~(PEPs) on the one-dimensional square lattice, that is, a system of interacting particles where each particle random walks according to a jump rate satisfying an exclusion rule that allows up to a certain number of particles can exist on each site. Particularly, we assume that the jump rate is given as a product of two functions depending on occupation variables on the original and target sites. Our interest is to study the limiting behavior, especially to derive some macroscopic PDEs by means of (fluctuating) hydrodynamics, of fluctuation fields associated with PEPs, starting from an invariant measure. The so-called gradient condition, meaning that the symmetric part of the instantaneous current is written in a gradient form, and that the invariant measures are given as a product measure is technically crucial. Our first main result is to clarify the relationship between these two conditions, and we show that the gradient condition and the existence of product invariant measures are mutually equivalent, provided the jump rate is given in the above simple form, as it is imposed in most of the literature, and the dynamics is asymmetric. Moreover, when the width of the lattice tends to zero and the process is accelerated in diffusive time-scaling, we show that the family of fluctuation fields converges to the stationary energy solution of the stochastic Burgers equation (SBE), under the setting that the jump rate to the right neighboring site is a bit larger than the one to the left side, of which discrepancy is given as square root of the width of the underlying lattice. This fills the gap at the level of universality of SBE since it has been proved for the exclusion process (a special case of PEP) and for the zero-range process.
math.PR↗
An elementary and direct computation of cohomology with and without a group action
Recently, we introduced a configuration space with interaction structure and a uniform local cohomology on it with co-authors in arXiv:2009.04699. The notion is used to understand a common structure of infinite product spaces appeared in the proof of Varadhan's non-gradient method. For this, the cohomology of the configuration space with a group action is the main target to study, but the cohomology is easily obtained from that of the configuration space without a group action by applying a well-known property on the group cohomology. In fact, the analysis of the cohomology of the configuration space without a group action is the essential part of arXiv:2009.04699. In this article, we give an elementary and direct proof to obtain the cohomology of a space with a group action from that without a group action under a certain condition including the setting of the configuration space with interaction structure. In particular, no knowledge of group cohomology is required.
math.AT↗
Yang-Baxter maps and independence preserving property
We study a surprising relationship between two properties for bijective functions $F : \mathcal{X} \times \mathcal{X} \to \mathcal{X} \times \mathcal{X}$ for a set $\mathcal{X}$ which are introduced from very different backgrounds. One of the property is that $F$ is a Yang-Baxter map, namely it satisfies the "set-theoretical" Yang-Baxter equation, and the other property is the independence preserving property (IP property for short), which means that there exist independent (non-constant) $\mathcal{X}$-valued random variables $X,Y$ such that $U,V$ are also independent with $(U,V)=F(X,Y)$. Recently in the study of invariant measures for a discrete integrable system, a class of functions having these two properties were found. Motivated by this, we analyze a relationship between the Yang-Baxter maps and the IP property, which has never been studied as far as we are aware, focusing on the case $\mathcal{X}=\mathbb{R}_+$. Our first main result is that all quadrirational Yang-Baxter maps $F : \mathbb{R}_+ \times \mathbb{R}_+ \to \mathbb{R}_+ \times \mathbb{R}_+$ in the most interesting subclass have the independence preserving property. In particular, we find new classes of bijections having the IP property. Our second main result is that these newly introduce bijections are fundamental in the class of (known) bijections with the IP property, in the sense that most of known bijections having the IP property are derived from these maps by taking special parameters or performing some limiting procedure. This reveals that the IP property, which has been investigated for specific functions individually, can be understood in a unified manner.
nlin.SI↗
Varadhan's decomposition of shift-invariant closed uniform forms for large scale interacting systems on general crystal lattices
We prove a uniform version of Varadhan decomposition for shift-invariant closed uniform forms associated to large scale interacting systems on general crystal lattices. In particular, this result includes the case of translation invariant processes on Euclidean lattices $\mathbf{Z}^d$ with finite range. Our result generalizes the result of arXiv:2009.04699 which was valid for systems on transferable graphs. In subsequent research, we will use the result of this article to prove Varadhan's decomposition of closed $L^2$-forms for large scale interacting systems on general crystal lattices.
math.PR↗
Bi-infinite solutions for KdV- and Toda-type discrete integrable systems based on path encodings
We define bi-infinite versions of four well-studied discrete integrable models, namely the ultra-discrete KdV equation, the discrete KdV equation, the ultra-discrete Toda equation, and the discrete Toda equation. For each equation, we show that there exists a unique solution to the initial value problem when the given data lies within a certain class, which includes the support of many shift ergodic measures. Our unified approach, which is also applicable to other integrable systems defined locally via lattice maps, involves the introduction of a path encoding (that is, a certain antiderivative) of the model configuration, for which we are able to describe the dynamics more generally than in previous work on finite size systems, periodic systems and semi-infinite systems. In particular, in each case we show that the behaviour of the system is characterized by a generalization of the classical 'Pitman's transformation' of reflection in the past maximum, which is well-known to probabilists. The picture presented here also provides a means to identify a natural 'carrier process' for configurations within the given class, and is convenient for checking that the systems we discuss are all-time reversible. Finally, we investigate links between the different systems, such as showing that bi-infinite all-time solutions for the ultra-discrete KdV (resp. Toda) equation may appear as ultra-discretizations of corresponding solutions for the discrete KdV (resp. Toda) equation.
nlin.SI↗