Spontaneous symmetry breaking in amnestically induced persistence

Abstract

We investigate a recently proposed non-Markovian random walk model characterized by loss of memories of the recent past and amnestically induced persistence. We report numerical and analytical results showing the complete phase diagram, consisting of 4 phases, for this system: (i) classical nonpersistence, (ii) classical persistence (iii) log-periodic nonpersistence and (iv) logperiodic persistence driven by negative feedback. The first two phases possess continuous scale invariance symmetry, however log-periodicity breaks this symmetry. Instead, log-periodic motion satisfies discrete scale invariance symmetry, with complex rather than real fractal dimensions. We find for log-periodic persistence evidence not only of statistical but also of geometric self-similarity.Nonpersistent random walkers with negative feedback tend not to repeat past behavior [1], but what happens when they forget their recent past [2]? Remarkably, they become persistent for sufficiently large memory losses. This recently reported phenomenon of amnestically induced persistence [2, 3] allows log-periodic [4] superdiffusion [5,6,7] driven by negative feedback. Its practical importance stems from the conceptual advance of quantitatively relating, on a causal level, two otherwise apparently unconnected phenomena: repetitive or persistent behavior on the one hand, and recent memory loss on the other, e.g., in Alzheimer's disease [2]. Precisely how does persistence depend on recent memory loss? Here, we answer this question and report numerical and analytical results showing the complete phase diagram for the problem, comprising 4 phases: (i) classical nonpersistence (ii) classical persistence, (iii) log-periodic nonpersistence and (iv) log-periodic persistence driven by negative feedback. The former two phases possess continuous scale invariance symmetry, which breaks down in the other two.Random walkers without memory have a mean square displacement x 2 that scales with time t according to x 2 ∼ t 2H , with Hurst exponent H = 1/2 as demanded by the Central Limit Theorem, assuming finite moments. Hurst exponents H > 1/2 indicate persistence and can arise due to long-range memory. Most random walks with and without memory display continuous scale invariance symmetry, i.e., continuous scale transformations by a "zoom" factor λ leave the Hurst exponent unchanged: t 2H → λ 2H t 2H as t → λt. Schütz and Trimper [1] pioneered a novel approach for studying walks with long-range memory [6,7,8,9], which we have adapted [2] for studying memory loss. Consider a random walker that starts at the origin at time t 0 = 0, with memory of the initial f t time steps of its complete history (0 ≤ f ≤ 1). At each time step the random walker moves either one step to the right or left. Let v t = ±1 represent the "velocity" at time t, such that the position followsAt time t, we randomly choose a previous time 1 ≤ t ′ < f t with equal a priori probabilities. The walker then chooses the current step direction v t based on the value of v t ′ , using the following rule. W...

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