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I'd like to propose a novel type of random walk problem. Suppose we are looking at a 1D discrete time random walk. Suppose we restrict the movement of the walker in the following way: If the walker has just taken two steps to right in the previous two steps of time, then on their next step the walker is not allowed step to the right. If the walker "attempts" to step to right, then time is still incremented, but their position does not change. The walker will not be able to move until they attempt a step to the left. Therefore, more than one increment of time can pass before the movement to the left occurs.

If we impose a symmetric "dynamical constraint" on the left, then the walker will not have a drift, but will be moving at a slower rate. NOTE: this slower rate has no association with the walker's location in space.

I'm wondering if this kind of problem has been studied?

The reason I ask is that I am studying glassy systems in mathematical physics, and in this field we have "dynamically constrained" or "kinetically constrained" models. The idea is that the system is not prevented from reaching any point in its state space, but it is prevented from reaching certain points by certain routes. As a result, the system may need, for example, five steps to get to a state that "should" be only one step away. Again, the system will "move" at a slower rate. If these dynamic constraints become stronger as temperature goes down, then at some point the system may freeze, i.e. the glassy transition.

I believe the basic idea of a dynamically constrained model is the same as the random walk problem described above. The idea is that the system is slowed by the constraint, but not in a way that has any association with the location of the system in real space.

I'm not looking for a solution to this random walk problem, I'm looking for any terminology that can help me search for work that other people might have done.

By the way, I do not like the terms "dynamically constrained" or "kinetically constrained". I'd love a different terminology. I wonder if I can call this, "frustrated dynamics"

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  • $\begingroup$ Is this in the last two actual time steps, they went right, then they can't go right again, but then in the following time step they can go right? $\endgroup$
    – Ian
    Commented Jan 2, 2023 at 18:39
  • $\begingroup$ @Ian I will clarify my question. Please check back in one minute. Thanks. $\endgroup$
    – Chris
    Commented Jan 2, 2023 at 18:40
  • $\begingroup$ @Ian I clarified my question. However, I think both ways are interesting. Maybe the walker is free to step however they like on the following step, or maybe they are forever frozen until they attempt a step to the left. I will have to think more carefully about which scenario most closely resembles the dynamically constrained models in physics. $\endgroup$
    – Chris
    Commented Jan 2, 2023 at 18:45
  • $\begingroup$ After a local time rescaling (removing the holding times), the effect is different; one is deterministic movement left for one step, and the other is 75% chance to move left for one step. So they have different net drift. The time rescaling is different as well (you slow time more if you have to wait for a leftward movement than if you just have to wait one step). $\endgroup$
    – Ian
    Commented Jan 2, 2023 at 18:50
  • $\begingroup$ @Ian This is all true. But if we are trying to connect this to glassy systems and dynamically constrained models, then we cannot remove the holding times. The holding times are what slow down a glassy system. $\endgroup$
    – Chris
    Commented Jan 2, 2023 at 18:53

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