The shape of a memorised random walk

Abstract

We view random walks as the paths of foraging animals, perhaps searching for food or avoiding predators while forming a mental map of their surroundings. The formation of such maps requires them to memorise the locations they have visited. We model memory using a kernel, proportional to the number of locations recalled as a function of the time since they were first observed. We give exact analytic expressions relating the elongation of the memorised walk to the structure of the memory kernel, and confirm these by simulation. We find that more slowly decaying memories lead to less elongated mental maps.

Size and shape of tracked Brownian bridges

“…This observation arises from Campbell's theorem [30,31] on the characteristic function of a time-function summed over a point process (see also appendix A in [26]) and our numerical evidence from the simulation section 5. The moments of Poisson sums/integrals with stochastic summands/integrands were investigated in [32].…”

“…Hence, bridges tracked at lower intensity have larger asphericity. We also note from figure 4 (LHS) that as λ → ∞ the asphericity tends to 2 3 which is identical to the asphericity of a standard, untethered random walk tracked using an exponential strategy [26] (note that in [26], μ was viewed as the memory kernel of a forager). This is a result of the fact that in the earliest part of its motion a bridge behaves like an untethered random walk, and for large λ only the earliest parts of the walk are tracked.…”

“…To characterise the shapes of tracked Brownian bridges we use gyration tensors, which were originally used by Rudnick and Gaspari [21] to define a measure of asphericity of random walks. In our previous paper [26] we worked with egocentric asphericity which differs from the standard definition of asphericity in that the moments which form the elements of the gyration tensor are taken about the current location of the walker, rather than the centre of mass of the walk. In the current paper moments are taken about the tether point (0, 0) of the bridge, thought of as the home location of the walker.…”

“…If the strategy usually generates point sets with approximately equal eigenvalues, then the tether point asphericity of the collection of random sets L c will be close to zero, meaning that they are typically not elongated. Because the elements of our gyration tensor are mean square deviations about the tether point and not the centre of mass of the tracked points, a set of points which would be perfectly spherical by the centre of mass definition are not spherical by ours (see [26] for some extreme examples). Despite this, our definition remains a measure of typical elongation (see figure 6) and due to its connection to the original definition we keep the name.…”

“…The function μ may be given other interpretations than tracking. For example it could be used to model scent decay rates [24], time variations in the density of communication events from mobile phones [25], or memory (as in our previous work [26]). Here we motivate our study with a possible application to the spread of disease.…”