Random Walk Probabilistic Model

A Random Walk Probabilistic Model is a Probabilistic Model that models the random process of a random variable taking an independent random step in any given direction in each time period.

• AKA: Random Walk Model, Random Walk Process, Random Walker Model.
• Context:
  • It can be visualized in a Random Graph Walk.
  • It can be artificially simulated by Random Walk Algorithm.
  • It can range from being an One-Dimensional Random Walk Model to being a N-Dimensional Random Walk Model.
  • It can range from being a Lattice Random Walk Model, to being a Gaussian Random Walk Model, to being a Graph-based Random Walk Model.
  • It can range from being a Self-Interacting Random Walk Model, Correlated Random Walk, to being a Long-range Correlated Random Walk Model.
  • It can range from being a Biased Random Walk Model to being a Maximal Entropy Random Walk Model.
• Example(s):
  • a Simple Random Walk (SRW) Model,
  • a Self-Interacting Random Walk Model such as:
    • Loop-Erased Random Walk Model,
    • Self-Avoiding Walk Model,
  • …
• Counter-Example(s):
  • Branching Random Walk,
  • Random Surfing Model,
  • Wiener Process,
  • Bernoulli Process,
  • Poisson Process,
  • Markov Process,
  • Levy Process.
• See: Brownian Motion, Markov Chain, Riemannian Manifold, Anomalous Diffusion, Statistical Physics, Levy Flight Model, Random-Walk-based Natural Language Processing Algorithm, Random-Walk-based Image Processing Algorithm, Random Walk with Restart Algorithm.

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References

2021a

• (Wikipedia, 2021) ⇒ https://en.wikipedia.org/wiki/Random_walk Retrieved:2021-8-12.
  • In mathematics, a random walk is a mathematical object, known as a stochastic or random process, that describes a path that consists of a succession of random steps on some mathematical space such as the integers.

    An elementary example of a random walk is the random walk on the integer number line, [math]\displaystyle{ \mathbb Z }[/math] , which starts at 0 and at each step moves +1 or −1 with equal probability. Other examples include the path traced by a molecule as it travels in a liquid or a gas (see Brownian motion), the search path of a foraging animal, the price of a fluctuating stock and the financial status of a gambler: all can be approximated by random walk models, even though they may not be truly random in reality.

    As illustrated by those examples, random walks have applications to engineering and many scientific fields including ecology, psychology, computer science, physics, chemistry, biology, economics, and sociology. Random walks explain the observed behaviors of many processes in these fields, and thus serve as a fundamental model for the recorded stochastic activity. As a more mathematical application, the value of can be approximated by the use of a random walk in an agent-based modeling environment. The term random walk was first introduced by Karl Pearson in 1905.

    Various types of random walks are of interest, which can differ in several ways. The term itself most often refers to a special category of Markov chains, but many time-dependent processes are referred to as random walks, with a modifier indicating their specific properties. Random walks (Markov or not) can also take place on a variety of spaces: commonly studied ones include graphs, others on the integers or the real line, in the plane or higher-dimensional vector spaces, on curved surfaces or higher-dimensional Riemannian manifolds, and also on groups finite, finitely generated or Lie. The time parameter can also be manipulated. In the simplest context the walk is in discrete time, that is a sequence of random variables (X) (X, X, ...) indexed by the natural numbers. However, it is also possible to define random walks which take their steps at random times, and in that case, the position has to be defined for all times t ∈ [0,+∞). Specific cases or limits of random walks include the Lévy flight and diffusion models such as Brownian motion.

    Random walks are a fundamental topic in discussions of Markov processes. Their mathematical study has been extensive. Several properties, including dispersal distributions, first-passage or hitting times, encounter rates, recurrence or transience, have been introduced to quantify their behavior.

2021b

• (Biskup, 2021) ⇒ https://www.math.ucla.edu/~biskup/PDFs/PCMI/PCMI-notes-1 Retrieved:2021-8-12.
  • QUOTE: Suppose that $X_1, X_2, \ldots$ is a sequence of $\R^d$ -valued independent and identically distributed random variables. A random walk started at $z \in \R^d$ is the sequence $\left(S_n\right)_{n\geq 0}$ where $S_0 = z$ and
    $S_n = S_{n−1} + X_n,\quad n \geq 1$.
    The quantities $\left(X_n\right)$ are referred to as steps of the random walk.

    Our interpretation of the above formula is as follows: The variable $S_n$ marks the position of the walk at time $n$. At each time the walk chooses a step at random — with the same step distribution at each time — and adds the result to its current position.

2021c

• (Kardar, 2021) ⇒ https://www.mit.edu/~kardar/teaching/projects/chemotaxis(AndreaSchmidt)/random.htm Retrieved:2021-8-12.
  • QUOTE: A random walk is the process by which randomly-moving objects wander away from where they started. The video below shows 7 black dots that start in one place randomly walking away. We will come back to this video when we know a little more about random walks.
    

2008

• (Britannica, 2008) ⇒ Britannica, The Editors of Encyclopaedia. "Random walk". Encyclopedia Britannica, 24 Jun. 2008, https://www.britannica.com/science/random-walk. Accessed 12 August 2021.
  • QUOTE: Random walk, in probability theory, a process for determining the probable location of a point subject to random motions, given the probabilities (the same at each step) of moving some distance in some direction. Random walks are an example of Markov processes, in which future behaviour is independent of past history. A typical example is the drunkard’s walk, in which a point beginning at the origin of the Euclidean plane moves a distance of one unit for each unit of time, the direction of motion, however, being random at each step. The problem is to find, after some fixed time, the probability distribution function of the distance of the point from the origin. Many economists believe that stock market fluctuations, at least over the short run, are random walks.

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