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  • 1
    Electronic Resource
    Electronic Resource
    On critical phenomena in interacting growth systems. Part I: General (1994)
    Springer
    Journal of statistical physics 74 (1994), S. 91-109 
    Details
    ISSN: 1572-9613
    Keywords: Random process ; local interaction ; critical phenomena ; invariant distribution ; growth ; eroder ; convexity
    Source: Springer Online Journal Archives 1860-2000
    Topics: Physics
    Notes: Abstract Components which are placed in a finite or infinite space have integer numbers as possible states. They interact in a discrete time in a local deterministic way, in addition to which all the components' states are incremented at every time step by independent identically distributed random variables. We assume that the deterministic interaction function is translation-invariant and monotonic and that its values are between the minimum and the maximum of its arguments. Theorems 1 and 2 (based on propositions which we give in a separate Part II), give sufficient conditions for a system to have an invariant distribution or a bounded mean. Other statements, proved herein, provide background for them by giving conditions when a system has no invariant distribution or the mean of its components' states tends to infinity. All our main results use one and the same geometrical criterion.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF02186808
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  • 2
    Electronic Resource
    Electronic Resource
    On critical phenomena in interacting growth systems. Part II: Bounded growth (1994)
    Springer
    Journal of statistical physics 74 (1994), S. 111-130 
    Details
    ISSN: 1572-9613
    Keywords: Random process ; local interaction ; critical phenomena growth ; combinatorics ; contour method ; graph theory
    Source: Springer Online Journal Archives 1860-2000
    Topics: Physics
    Notes: Abstract This paper completes the classification of some infinite and finite growth systems which was started in Part I. Components whose states are integer numbers interact in a local deterministic way, in addition to which every component's state grows by a positive integerk with a probability ε k (1-ε) at every moment of the discrete time. Proposition 1 says that in the infinite system which starts from the state “all zeros”, percentages of elements whose states exceed a given valuek≥0 never exceed (Cε) k , whereC=const. Proposition 2 refers to finite systems. It states that the same inequalities hold during a time which depends exponentially on the system size.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF02186809
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  • 3
    Electronic Resource
    Electronic Resource
    Simple one-dimensional interaction systems with superexponential relaxation times (1995)
    Springer
    Journal of statistical physics 80 (1995), S. 545-563 
    Details
    ISSN: 1572-9613
    Keywords: Random processes ; one-dimensional local interaction ; relaxation time ; smoothing ; Cramér-Edgeworth expansion ; harnesses
    Source: Springer Online Journal Archives 1860-2000
    Topics: Physics
    Notes: Abstract Finite one-dimensional random processes with local interaction are presented which keep some information of a topological nature about their initial conditions during time, the logarithm of whose expectation grows asymptotically at least asM 3, whereM is the “size” of the setR M of states of one component. ActuallyR M is a circle of lengthM. At every moment of the discrete time every component turns into some kind of average of its neighbors, after which it makes a random step along this circle. All these steps are mutually independent and identically distributed. In the present version the absolute values of the steps never exceed a constant. The processes are uniform in space, time, and the set of states. This estimation contributes to our awareness of what kind of stable behavior one can expect from one-dimensional random processes with local interaction.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF02178547
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