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Random walk through a fertile site.

Michel Bauer1,2, P L Krapivsky3, Kirone Mallick1

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The total number of random walkers grows exponentially in low dimensions but may remain finite in higher dimensions. Critical multiplication rates and anomalous moment growth are observed in higher dimensions.

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Area of Science:

  • Statistical physics
  • Complex systems

Background:

  • Random walks are fundamental models in physics and mathematics.
  • Understanding population dynamics in confined or expanding spaces is crucial.

Purpose of the Study:

  • To investigate the dynamics of random walks with multiplication on hypercubic lattices.
  • To analyze the conditions for exponential growth and critical phenomena.

Main Methods:

  • Mathematical modeling of random walks on lattices.
  • Analysis of population growth rates and moment calculations.
  • Dimensionality and critical exponent analysis.

Main Results:

  • Exponential growth of walker numbers in dimensions d≤2, with Malthusian rates dependent on dimensionality and multiplication rate.
  • Existence of a critical multiplication rate (μd) in dimensions d>2, determining exponential growth.
  • Broad distributions and anomalous moment growth in critical regimes; universal stationary distributions observed.
  • Sublinear growth for random walks with exclusion in d≤2 and linear growth in d>2 above the critical rate.

Conclusions:

  • Dimensionality and multiplication rates critically influence random walk population dynamics.
  • A universal critical regime exists for higher dimensions, with distinct growth patterns.
  • Interactions like exclusion significantly alter population dynamics, leading to different growth behaviors.