We study the first-passage dynamics of a non-Markovian stochastic process with time-averaged feedback, which we model as a one-dimensional Ornstein–Uhlenbeck process wherein the particle drift is modified by the empirical mean of its trajectory. This process maps onto a class of self-interacting diffusions. Using weak-noise large deviation theory, we calculate the leading order asymptotics of the time-dependent distribution of the particle position, derive the most probable paths that reach the specified position at a given time and quantify their likelihood via the action functional. We compute the feedback-modified Kramers rate and its inverse, which approximates the mean first-passage time, and show that the feedback accelerates dynamics by storing finite-time fluctuations, thereby lowering the effective energy barrier and shifting the optimal first-passage time from infinite to finite. Although we identify alternative mechanisms, such as slingshot and ballistic trajectories, we find that they remain sub-optimal and hence do not accelerate the dynamics. These results show how memory feedback reshapes rare event statistics, thereby offering a mechanism to potentially control first-passage dynamics.

We investigate fluctuations in the average speed or current of a self-interacting diffusion (SID) on a ring, mimicking the non-Markovian behaviour of an agent influenced by its own path. We derive the SID’s phase diagram, showing a delocalisation-localisation phase transition from self-repelling to self-attracting. Current fluctuations are analysed using: (i) an adiabatic approximation, where the system reaches its stationary distribution before developing current fluctuations, and (ii) an original extension of level 2.5 large deviations for Markov processes combined with perturbation theory. Both methods provide lower bounds to current fluctuations, with the former tighter than the latter in all localised regimes, and both equally tight in the self-repelling region. Both methods accurately estimate the asymptotic variance and suggest a phase transition at the onset of the localised regime.

Although higher-order interactions are known to affect the typical state of dynamical processes giving rise to new collective behavior, how they drive the emergence of rare events and fluctuations is still an open problem. We investigate how fluctuations of a dynamical quantity of a random walk exploring a higher-order network arise over time. In the quenched case, where the hypergraph structure is fixed, through large deviation theory we show that the appearance of rare events is hampered in nodes with many higher-order interactions, and promoted elsewhere. Dynamical fluctuations are further boosted in an annealed scenario, where both the diffusion process and higher-order interactions evolve in time. Here, extreme fluctuations generated by optimal higher-order configurations can be predicted in the limit of a saddle-point approximation. Our study lays the groundwork for a wide and general theory of fluctuations and rare events in higher-order networks.

In this work, we introduce a generalization of the Landauer bound for erasure processes that stems from absolutely irreversible dynamics. Assuming that the erasure process is carried out in an absolutely irreversible way so that the probability of observing some trajectories is zero in the forward process but finite in the reverse process, we derive a generalized form of the bound for the average erasure work, which is valid also for imperfect erasure and asymmetric bits. The generalized bound obtained is tighter than or, at worst, as tight as existing ones. Our theoretical predictions are supported by numerical experiments and the comparison with data from previous works.

We analytically derive universal bounds that describe the tradeoff between thermodynamic cost and precision in a sequence of events related to some internal changes of an otherwise hidden physical system. The precision is quantified by the fluctuations in either the number of events counted over time or the waiting times between successive events. Our results are valid for the same broad class of nonequilibrium driven systems considered by the thermodynamic uncertainty relation, but they extend to both time-symmetric and asymmetric observables. We show how optimal precision saturating the bounds can be achieved. For waiting-time fluctuations of asymmetric observables, a phase transition in the optimal configuration arises, where higher precision can be achieved by combining several signals.

We study the appearance of first-order dynamical phase transitions (DPTs) as intermittent coexisting phases in the fluctuations of random walks on graphs. We show that the diverging timescale leading to critical behavior is the waiting time to jump from one phase to another. This timescale is crucial for observing the system's relaxation to stationarity and demonstrate ergodicity of the system at criticality. We illustrate these results through three analytical examples which provide insights into random walks exploring random graphs.

We study the performance of a stochastic algorithm based on the power method that adaptively learns the large deviation functions characterizing the fluctuations of additive functionals of Markov processes, used in physics to model nonequilibrium systems. This algorithm was introduced in the context of risk-sensitive control of Markov chains and was recently adapted to diffusions evolving continuously in time. Here we provide an in-depth study of the convergence of this algorithm close to dynamical phase transitions, exploring the speed of convergence as a function of the learning rate and the effect of including transfer learning. We use as a test example the mean degree of a random walk on an Erdős-Rényi random graph, which shows a transition between high-degree trajectories of the random walk evolving in the bulk of the graph and low-degree trajectories evolving in dangling edges of the graph. The results show that the adaptive power method is efficient close to dynamical phase transitions, while having many advantages in terms of performance and complexity compared to other algorithms used to compute large deviation functions.

The integral fluctuation theorem (IFT) for entropy production is among the few equalities that are known to be valid for physical systems arbitrarily driven far from equilibrium. Microscopically, it can be understood as an inherent symmetry for the fluctuating entropy production rate implying the second law of thermodynamics. Here, we examine an IFT statistical estimator based on regular sampling and discuss its limitations for nonequilibrium systems, when sampling rare events becomes pivotal. Furthermore, via a large deviation study, we discuss a method to carefully setup an experiment in the parameter region where the IFT estimator safely converges and also show how to improve the convergence region for Markov chains with finite correlation time. We corroborate our arguments with two illustrative examples.

We consider random walks evolving on two models of connected and undirected graphs and study the exact large deviations of a local dynamical observable. We prove, in the thermodynamic limit, that this observable undergoes a first-order dynamical phase transition (DPT). This is interpreted as a coexistence of paths in the fluctuations that visit the highly connected bulk of the graph (delocalization) and paths that visit the boundary (localization). The methods we used also allow us to characterize analytically the scaling function that describes the finite-size crossover between the localized and delocalized regimes. Remarkably, we also show that the DPT is robust with respect to a change in the graph topology, which only plays a role in the crossover regime. All results support the view that a first-order DPT may also appear in random walks on infinite-size random graphs.

We consider discrete-time Markov chains and study large deviations of the pair empirical occupation measure, which is useful to compute fluctuations of pure-additive and jump-type observables. We provide an exact expression for the finite-time moment generating function, which is split in cycles and paths contributions, and scaled cumulant generating function of the pair empirical occupation measure via a graph-combinatorial approach. The expression obtained allows us to give a physical interpretation of interaction and entropic terms, and of the Lagrange multipliers, and may serve as a starting point for sub-leading asymptotics. We illustrate the use of the method for a simple two-state Markov chain.