Giant leaps and long excursions: fluctuation mechanisms in systems with long-range memory

Many random processes lack memory – that is, the probability distribution for change over any time interval depends on the present state of the process but remains independent of how it behaved previously. Such simple “Markovian” processes stand in contrast to more complicated yet realistic processes which depend on memory, meaning that the probabilities of various outcomes at any moment depend on what has happened in the past. Many examples arise in fields ranging from biology to telecommunications to finance. Importantly such non-Markovian processes exhibit unusual fluctuation behaviour and long-term outcomes. In the classic example of the Pólya Urn, for example, the large-time limit of the process is indeterminate, varying from one realisation to the next.
In a new paper, LML External Fellow Rosemary Harris and Robert Jack of the University of Cambridge consider non-Markovian systems where the dynamics depend explicitly on a time-averaged current, whose value at time t is denoted by q_t. This simple type of memory occurs in many contexts including models of growing clusters. Using methods of large deviation theory, they show how long-range memory can lead to anomalous fluctuations of q_t. In particular, in the parameter regimes considered in the paper, q_t exhibits super-diffusive behaviour.
The authors illustrate the underlying mechanisms by analysing current fluctuations in two specific models, in which large and rare fluctuations at early times bias the subsequent evolution and trigger anomalous fluctuations which persist for large times. In the limit of long process times the two mechanisms they consider are: (i) a diverging initial current flow in a finite time interval, which results in anomalously large deviations and (ii) a large-but-finite initial current which occurs over a diverging time interval. The latter case even leads to a breakdown of the central limit theorem for the average current q_t, in other words it does not approach a normal distribution_. The authors believe these two cases, which they refer to as an “initial giant leap” or “long initial excursion,” should be useful in classifying and predicting fluctuation behaviour in a wide-range of non-Markovian models.
The paper is available as a pre-print at https://arxiv.org/pdf/2003.03587.pdf