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A random walk model with a mixed memory profile: Exponential and rectangular profile

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  • de Lacerda, K.J.C.C.
  • da Silva, L.R.
  • Viswanathan, G.M.
  • Cressoni, J.C.
  • da Silva, M.A.A.

Abstract

The theory of Markovian random walks is consolidated and very well understood, however the theory of non-Markovian random walks presents many challenges due to its remarkably rich phenomenology. An important open problem in this context is to study how the diffusive properties of random walk processes change when memory-induced correlations are introduced. In this work we propose a model of a random walk that evolves in time according to past memories selected from rectangular (flat) and exponentially decaying memory profiles. In this mixed memory profile model, the walker remembers either the last B steps with equal a priori probability or the steps A prior to B with exponentially decaying probability, for a total number of steps equal to A+B. The diffusive behavior of the walk is numerically examined through the Hurst exponent (H). Even in the lack of exact solutions, we are still able to show that the model can be mapped onto a RW model with rectangular memory profile.

Suggested Citation

  • de Lacerda, K.J.C.C. & da Silva, L.R. & Viswanathan, G.M. & Cressoni, J.C. & da Silva, M.A.A., 2022. "A random walk model with a mixed memory profile: Exponential and rectangular profile," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 597(C).
    • Handle: RePEc:eee:phsmap:v:597:y:2022:i:c:s0378437122002497
    DOI: 10.1016/j.physa.2022.127301
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    References listed on IDEAS

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    1. Scalas, Enrico, 2006. "The application of continuous-time random walks in finance and economics," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 362(2), pages 225-239.
    2. Moura, Thiago R.S. & Viswanathan, G.M. & da Silva, M.A.A. & Cressoni, J.C. & da Silva, L.R., 2016. "Transient superdiffusion in random walks with a q-exponentially decaying memory profile," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 453(C), pages 259-263.
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