Phase Transitions in the Time Synchronization Model

Малышев, В. А.; Manita, Anatoly

doi:10.1137/s0040585x97981524

Cited by 14 publications

(13 citation statements)

References 3 publications

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“…Indeed, any mean-field model can be reduced to the corresponding model satisfying (9), by time and space rescaling. So, in all the proofs we do assume (9), in which case…”

Section: Theorem 3 Assume (1) Thenmentioning

confidence: 99%

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Large-scale behavior of a particle system with mean-field interaction: Traveling wave solutions

Stolyar¹

2020

Preprint

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We use probabilistic methods to study properties of mean-field models, arising as large-scale limits of certain particle systems with mean-field interaction. The underlying particle system is such that n particles move forward on the real line. Specifically, each particle "jumps forward" at some time points, with the instantaneous rate of jumps given by a decreasing function of the particle's location quantile within the overall distribution of particle locations. A mean-field model describes the evolution of the particles' distribution, when n is large. It is essentially a solution to an integro-differential equation within a certain class. Our main results concern the existence and uniqueness of -and attraction tomean-field models which are traveling waves, under general conditions on the jump-rate function and the jump-size distribution.

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“…Indeed, any mean-field model can be reduced to the corresponding model satisfying (9), by time and space rescaling. So, in all the proofs we do assume (9), in which case…”

Section: Theorem 3 Assume (1) Thenmentioning

confidence: 99%

“…Subsequently to [5,6], there has been a line of work generally focused on modeling synchronization between elements (particles) of a large system, with mean-field interaction of the elements (particles), cf. [9][10][11][12][13][14].…”

Section: Other Related Workmentioning

confidence: 99%

Large-scale behavior of a particle system with mean-field interaction: Traveling wave solutions

Stolyar¹

2020

Preprint

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“…There are already many particular examples justifying this belief. Thus the existence of the three time stages in the long time behavior was already proved for system with two types of deterministic particles and pairwise stochastic synchronizations [6], for discrete time random walks with a 3-particle anysotropic interaction [5], for continuous time random walks with symmetric k-particle synchronizations [7].…”

Section: Introductionmentioning

confidence: 87%

Time scales in large systems of Brownian particles with stochastic synchronization

Manita

2010

Preprint

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We consider a system x(t) = (x 1 (t), . . . , x N (t)) consisting of N Brownian particles with synchronizing interaction between them occurring at random time momentsUnder assumption that the free Brownian motions and the sequence {τ n } ∞ n=1 are independent we study asymptotic properties of the system when both the dimension N and the time t go to infinity. We find three time scales t = t(N ) of qualitatively different behavior of the system.

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“…where averaging E S is taken over distribution of the map-valued random variable S introduced in (40). Hence by (42) we get…”

Section: Recurrent Equationsmentioning

confidence: 99%

“…The problem is to find different time scales (t = t N → ∞ as N → ∞) on which the synchronization system x(t N ) demonstrates completely different qualitative behaviour. The complete description of times scales was obtained for several models [40,[46][47][48]51]. For example, in [47] it was shown that the "BM N -model" passes three different phases before it reaches the final synchronization.…”

Section: Introductionmentioning

confidence: 99%

Intrinsic scales for high-dimensional LEVY-driven models with non-Markovian synchronizing updates

Manita

2014

Preprint

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We propose stochastic N -component synchronization models (x 1 (t), . . . , x N (t)), x j ∈ R d , t ∈ R + , whose dynamics is described by Lévy processes and synchronizing jumps. We prove that symmetric models reach synchronization in a stochastic sense: differences between components d (N ) kj (t) = x k (t)−x j (t) have limits in distribution as t → ∞. We give conditions of existence of natural (intrinsic) space scales for large synchronized systems, i.e., we are looking for such sequences {b N } that distribution of d (N ) kj (∞)/b N converges to some limit as N → ∞. It appears that such sequence exists if the Lévy process enters a domain of attraction of some stable law. For Markovian synchronization models based on α-stable Lévy processes this results holds for any finite N in the precise form with b N = (N − 1) 1/α . For non-Markovian models similar results hold only in the asymptotic sense. The class of limiting laws includes the Linnik distributions. We also discuss generalizations of these theorems to the case of non-uniform matrix-based intrinsic scales. The central point of our proofs is a representation of characteristic functions of d (N ) kj (t) via probability distribution of a superposition of N independent renewal processes.

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Phase Transitions in the Time Synchronization Model

Cited by 14 publications

References 3 publications

Large-scale behavior of a particle system with mean-field interaction: Traveling wave solutions

Large-scale behavior of a particle system with mean-field interaction: Traveling wave solutions

Time scales in large systems of Brownian particles with stochastic synchronization

Intrinsic scales for high-dimensional LEVY-driven models with non-Markovian synchronizing updates

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