Exact Large Deviation Research Articles

Comment on "Exact large deviation statistics and trajectory phase transition of a deterministic boundary driven cellular automaton".

Buča etal. [Phys. Rev. E 100, 020103(R) (2019)2470-004510.1103/PhysRevE.100.020103] study the dynamical large deviations of a boundary-driven cellular automaton. They take a double limit in which first time and then space is made infinite, and interpret the resulting large-deviation singularity as evidence of a first-order phase transition and the accompanying coexistence of two distinct dynamical phases. This view is characteristic of an approach to dynamical large deviations in which time is interpreted as if it were a spatial coordinate of a thermodynamic system [Jack, Eur. Phys. J. B 93, 74 (2020)1434-602810.1140/epjb/e2020-100605-3]. Here, I argue that the large-deviation function produced in this double limit is not consistent with the basic features of the model of Buča etal. I show that a modified limiting procedure results in a nonsingular large-deviation function consistent with those features, and that neither supports the idea of coexisting dynamical phases.

Reply to "Comment on 'Exact large deviation statistics and trajectory phase transition of a deterministic boundary driven cellular automaton'".

We reply to Whitelam's Comment [Phys. Rev. E 108, 036105 (2023)2470-004510.1103/PhysRevE.108.036105] on our paper [Phys. Rev. E 100, 020103(R) (2019)2470-004510.1103/PhysRevE.100.020103] where we compute the exact large deviation (LD) statistics of a wide class of observables in the rule 54 cellular automaton. Using some heuristic arguments, Whitelam states that despite the fact that the LD functions we compute display singular behavior, this is not indicative of a LD phase transition or of dynamical phase coexistence. Here, we refute this observation and confirm that the (standard) interpretation of our exact results stands.

Delocalization-localization dynamical phase transition of random walks on graphs.

We consider random walks evolving on two models of connected and undirected graphs and study the exact large deviations of a local dynamical observable. We prove, in the thermodynamic limit, that this observable undergoes a first-order dynamical phase transition (DPT). This is interpreted as a "coexistence" of paths in the fluctuations that visit the highly connected bulk of the graph (delocalization) and paths that visit the boundary (localization). The methods we used also allow us to characterize analytically the scaling function that describes the finite-size crossover between the localized and delocalized regimes. Remarkably, we also show that the DPT is robust with respect to a change in the graph topology, which only plays a role in the crossover regime. All results support the view that a first-order DPT may also appear in random walks on infinite-size random graphs.

Inverse Scattering Method Solves the Problem of Full Statistics of Nonstationary Heat Transfer in the Kipnis-Marchioro-Presutti Model

We determine the full statistics of nonstationary heat transfer in the Kipnis-Marchioro-Presutti lattice gas model at long times by uncovering and exploiting complete integrability of the underlying equations of the macroscopic fluctuation theory. These equations are closely related to the derivative nonlinear Schrödinger equation (DNLS), and we solve them by the Zakharov-Shabat inverse scattering method (ISM) adapted by D. J. Kaup and A. C. Newell, J. Math. Phys. 19, 798 (1978)JMAPAQ0022-248810.1063/1.523737 for the DNLS. We obtain explicit results for the exact large deviation function of the transferred heat for an initially localized heat pulse, where we uncover a nontrivial symmetry relation.

Exact solution of the “Rule 150” reversible cellular automaton

We study the dynamics and statistics of the Rule 150 reversible cellular automaton (RCA). This is a one-dimensional lattice system of binary variables with synchronous (Floquet) dynamics that corresponds to a bulk deterministic and reversible discretized version of the kinetically constrained "exclusive one-spin facilitated" (XOR) Fredrickson-Andersen (FA) model, where the local dynamics is restricted: A site flips if and only if its adjacent sites are in different states from each other. Similar to other RCA that have been recently studied, such as Rule 54 and Rule 201, the Rule 150 RCA is integrable, however, in contrast is noninteracting: The emergent quasiparticles, which are identified by the domain walls, behave as free fermions. This property allows us to solve the model by means of matrix product ansatz. In particular, we find the exact equilibrium and nonequilibrium stationary states for systems with closed (periodic) and open (stochastic) boundaries, respectively, resolve the full spectrum of the time evolution operator and, therefore, gain access to the relaxation dynamics, and obtain the exact large deviation statistics of dynamical observables in the long-time limit.

On the Existence Conditions for Exact Large Deviation Principles

On the Existence Conditions for Exact Large Deviation Principles

Rule 54: exactly solvable model of nonequilibrium statistical mechanics

We review recent results on an exactly solvable model of nonequilibrium statistical mechanics, specifically the classical rule 54 reversible cellular automaton and some of its quantum extensions. We discuss the exact microscopic description of nonequilibrium dynamics as well as the equilibrium and nonequilibrium stationary states. This allows us to obtain a rigorous handle on the corresponding emergent hydrodynamic description, which is treated as well. Specifically, we focus on two different paradigms of rule 54 dynamics. Firstly, we consider a finite chain driven by stochastic boundaries, where we provide exact matrix product descriptions of the nonequilibrium steady state, most relevant decay modes, as well as the eigenvector of the tilted Markov chain yielding exact large deviations for a broad class of local and extensive observables. Secondly, we treat the explicit dynamics of macro-states on an infinite lattice and discuss exact closed form results for dynamical structure factor, multi-time-correlation functions and inhomogeneous quenches. Remarkably, these results prove that the model, despite its simplicity, behaves like a regular fluid with coexistence of ballistic (sound) and diffusive (heat) transport. Finally, we briefly discuss quantum interpretation of rule 54 dynamics and explicit results on dynamical spreading of local operators and operator entanglement.

Large Deviations in the Symmetric Simple Exclusion Process with Slow Boundaries

We obtain the exact large deviation functions of the density profile and of the current, in the non-equilibrium steady state of a one dimensional symmetric simple exclusion process coupled to boundary reservoirs with slow rates. Compared to earlier results, where rates at the boundaries are comparable to the bulk ones, we show how macroscopic fluctuations are modified when the boundary rates are slower by an order of inverse of the system length.

On Exact Large Deviation Principles for Compound Renewal Processes

We consider two large deviation principles (LDPs): the “ordinary” LDP (when the “strong” Cramer condition is met) and the “extended” LDP when only the standard Cramer condition is met and the devia...

On exact large deviation principles for compound renewal processes

Рассматриваются два принципа больших уклонений (п.б.у.) - "обычный" (при выполнении "усиленного" условия Крамера) и "расширенный", когда выполнено лишь стандартное условие Крамера, а функционал уклонений может быть конечным и для разрывных траекторий. Стандартная формулировка этих принципов содержит две асимптотические оценки (сверху и снизу) для логарифмов вероятностей того, что нормированная траектория процесса принадлежит заданному множеству $B$. Найдены условия на множество $B$, при которых эти оценки совпадают и принципы больших уклонений принимают форму точных асимптотических равенств. Такие п.б.у. названы точными. Установлено, что оценивающий отрезок обычного п.б.у. вложен в оценивающий отрезок расширенного п.б.у. и что, стало быть, выполнение точного расширенного п.б.у. влечет за собой выполнение точного обычного п.б.у. Полученные результаты в полной мере справедливы и актуальны для случайных блужданий (частного случая обобщенных процессов восстановления).

Current fluctuations in a stochastic system of classical particles with next-nearest-neighbor interactions

A totally asymmetric exclusion process consisting of classical particles with next-nearest-neighbor interactions has been considered on a 1D discrete lattice with a ring geometry. Using large deviation techniques, we have investigated fluctuations of particle current in two and three-particle sectors. In a two-particle sector, we have obtained exact large deviation function for the probability distribution of particle current. In this sector, we have also calculated the the propensity of each configuration to exhibit atypical particle current. Analytical results for both the scaled cumulant generating function of particle current and average particle current are compared with those obtained from cloning Monte Carlo simulations. Numerical results in three-particle sector have also been presented.

Exact large deviation statistics and trajectory phase transition of a deterministic boundary driven cellular automaton.

We study the statistical properties of the long-time dynamics of the rule 54 reversible cellular automaton (CA), driven stochastically at its boundaries. This CA can be considered as a discrete-time and deterministic version of the Fredrickson-Andersen kinetically constrained model (KCM). By means of a matrix product ansatz, we compute the exact large deviation cumulant generating functions for a wide range of time-extensive observables of the dynamics, together with their associated rate functions and conditioned long-time distributions over configurations. We show that for all instances of boundary driving the CA dynamics occurs at the point of phase coexistence between competing active and inactive dynamical phases, similar to what happens in more standard KCMs. We also find the exact finite size scaling behavior of these trajectory transitions, and provide the explicit "Doob-transformed" dynamics that optimally realizes rare dynamical events.

Exact large deviation function of spin current for the one dimensional XX spin chain with domain wall initial condition

We investigate the fluctuations of the spin current for the one-dimensional XX spin chain starting from the domain wall initial condition. The generating function of the current is shown to be written as a determinant with the Bessel kernel. An exact analytical expression for the large deviation function is obtained by applying the Coulomb gas method. Our results are also compared with DMRG calculations.

Large deviations of surface height in the 1 + 1-dimensional Kardar–Parisi–Zhang equation: exact long-time results for λH

We study atypically large fluctuations of height H in the 1 + 1-dimensional Kardar–Parisi–Zhang (KPZ) equation at long times t, when starting from a ‘droplet’ initial condition. We derive exact large deviation function of height for , where λ is the nonlinearity coefficient of the KPZ equation. This large deviation function describes a crossover from the Tracy–Widom distribution tail at small , which scales as , to a different tail at large , which scales as . The latter tail exists at all times t > 0. It was previously obtained in the framework of the optimal fluctuation method. It was also obtained at short times from exact representation of the complete height statistics. The crossover between the two tails, at long times, occurs at as previously conjectured. Our analytical findings are supported by numerical evaluations using exact representation of the complete height statistics.

Universal large deviations for the tagged particle in single-file motion.

We consider a gas of point particles moving in a one-dimensional channel with a hard-core interparticle interaction that prevents particle crossings--this is called single-file motion. Starting from equilibrium initial conditions we observe the motion of a tagged particle. It is well known that if the individual particle dynamics is diffusive, then the tagged particle motion is subdiffusive, while for ballistic particle dynamics, the tagged particle motion is diffusive. Here we compute the exact large deviation function for the tagged particle displacement and show that this is universal, independent of the individual dynamics.

Non-Equilibrium Steady States in Conformal Field Theory

We present a construction of non-equilibrium steady states in one-dimensional quantum critical systems carrying energy and charge fluxes. This construction is based on a scattering approach within a real-time hamiltonian reservoir formulation. Using conformal field theory techniques, we prove convergence towards steady states at large time. We discuss in which circumstances these states describe the universal non-equilibrium regime at low temperatures. We compute the exact large deviation functions for both energy and charge transfers, which encode for the quantum and statistical fluctuations of these transfers at large time. They are universal, depending only on fundamental constants (\({\hbar, k_B}\)), on the central charge and on the external parameters such as the temperatures or the chemical potentials, and they satisfy fluctuation relations. A key point consists in relating the derivatives of these functions to the linear response functions but at complex shifted external parameters.

Large Deviations and Ratio Limit Theorems for Pattern-Avoiding Permutations

For a fixed permutation τ, let$\mathcal{S}_N(\tau)$be the set of permutations onNelements that avoid the pattern τ. Madras and Liu (2010) conjectured that$\lim_{N\rightarrow\infty}\frac{|\mathcal{S}_{N+1}(\tau)|}{ |\mathcal{S}_N(\tau)|}$exists; if it does, it must equal the Stanley–Wilf limit. We prove the conjecture for every permutation τ of length 5 or less, as well as for some longer cases (including 704 of the 720 permutations of length 6). We also consider permutations drawn at random from$\mathcal{S}_N(\tau)$, and we investigate properties of their graphs (viewing permutations as functions on {1,. . .,N}) scaled down to the unit square [0,1]2. We prove exact large deviation results for these graphs when τ has length 3; it follows, for example, that it is exponentially unlikely for a random 312-avoiding permutation to have points above the diagonal strip |y−x| < ε, but not unlikely to have points below the strip. For general τ, we show that some neighbourhood of the upper left corner of [0,1]2is exponentially unlikely to contain a point of the graph if and only if τ starts with its largest element. For patterns such as τ=4231 we establish that this neighbourhood can be extended along the sides of [0,1]2to come arbitrarily close to the corner points (0,0) and (1,1), as simulations had suggested.

Heat and work fluctuations for a harmonic oscillator

The formalism of Kundu et al. [J. Stat. Mech. P03007 (2011)], for computing the large deviations of heat flow in harmonic systems, is applied to the case of single Brownian particle in a harmonic trap and coupled to two heat baths at different temperatures. The large-τ form of the moment generating function <e(-λQ)>≈g(λ)exp[τμ(λ)], of the total heat flow Q from one of the baths to the particle in a given time interval τ, is studied and exact explicit expressions are obtained for both μ(λ) and g(λ). For a special case of the single particle problem that corresponds to the work done by an external stochastic force on a harmonic oscillator coupled to a thermal bath, the large-τ form of the moment generating function is analyzed to obtain the exact large deviation function as well as the complete asymptotic forms of the probability density function of the work.

Applications of Stein’s method for concentration inequalities

Stein’s method for concentration inequalities was introduced to prove concentration of measure in problems involving complex dependencies such as random permutations and Gibbs measures. In this paper, we provide some extensions of the theory and three applications: (1) We obtain a concentration inequality for the magnetization in the Curie–Weiss model at critical temperature (where it obeys a nonstandard normalization and super-Gaussian concentration). (2) We derive exact large deviation asymptotics for the number of triangles in the Erdős–Rényi random graph G(n, p) when p ≥ 0.31. Similar results are derived also for general subgraph counts. (3) We obtain some interesting concentration inequalities for the Ising model on lattices that hold at all temperatures.

Stochastic hybrid simulation

This is a summary of the author’s PhD thesis supervised by Bart Scheers and Antoine Van de Capelle and defended on 20 November 2009 at the Royal Military Academy, Brussels. The thesis (https://lirias.kuleuven.be/handle/123456789/246952) is written in English and is available from the author upon request. This work deals with an extension to the hybrid simulation paradigm, i.e. the combination of event-driven simulation and analytical modelling, applied to packet telecommunication networks. In order to speed up the simulation only a small part of all packets, the foreground traffic, is processed in an event-driven way. On each arrival of a foreground packet, the waiting time of the packet is sampled from the virtual waiting time distribution function of the combined foreground and background traffic. This distribution function is stochastically modelled by the exact large deviations asymptotic of the virtual waiting time in a many sources regime. This novel methodology is not only valid for wired point-to-point queueing networks having a fixed transmission capacity, but it can also be applied to queueing networks for which the transmission capacity varies with the traffic load of all the elements in the network. The results obtained by the stochastic hybrid simulator are compared to full-blown event-driven simulations. An important reduction in simulation run-time is gained without sacrificing accuracy.