In this work we focus on fluctuations of time-integrated observables for a particle diffusing in a one-dimensional periodic potential in the weak-noise asymptotics. Our interest goes to rare trajectories presenting an atypical value of the observable, that we study through a biased dynamics in a largedeviation framework. We determine explicitly the effective probability-conserving dynamics which makes rare trajectories of the original dynamics become typical trajectories of the effective one. Our approach makes use of a weak-noise path-integral description in which the action is minimised by the rare trajectories of interest. For 'current-type' additive observables, we find the emergence of a propagative trajectory minimising the action for large enough deviations, revealing the existence of a dynamical phase transition at a fluctuating level. In addition, we provide a new method to determine the scaled cumulant generating function of the observable without having to optimise the action. It allows one to show that the weak-noise and the large-time limits commute in this problem. Finally, we show how the biased dynamics can be mapped in practice to an effective driven dynamics, which takes the form of a driven Langevin dynamics in an effective potential. The non-trivial shape of this effective potential is key to understand the link between the dynamical phase transition in the large deviations of current and the standard depinning transition of a particle in a tilted potential. ‡ During the preparation of this manuscript, we became aware that works parallel to ours were completed using similar approaches [22,23].

Highlights d Core vpda neuron morphology is established during embryogenesis d The primary branch grows deterministically but secondary branches are stochastic d Tree architecture can increase or decrease the local probability of branch survival d Contact-induced retraction selects secondary branches perpendicular to the primary

Dynamical phase transitions (DPTs) in the space of trajectories are one of the most intriguing phenomena of nonequilibrium physics, but their nature in realistic high-dimensional systems remains puzzling. Here we observe for the first time a DPT in the current vector statistics of an archetypal two-dimensional (2d) driven diffusive system, and characterize its properties using macroscopic fluctuation theory. The complex interplay among the external field, anisotropy and vector currents in 2d leads to a rich phase diagram, with different symmetry-broken fluctuation phases separated by lines of 1 st -and 2 nd -order DPTs. Remarkably, different types of 1d order in the form of jammed density waves emerge to hinder transport for low-current fluctuations, revealing a connection between rare events and self-organized structures which enhance their probability.Introduction-The theory of critical phenomena is a cornerstone of modern theoretical physics [1, 2]. Indeed, phase transitions of all sorts appear ubiquitously in most domains of physics, from cosmological scales to the quantum world of elementary particles. In a typical 2 nd -order phase transition order emerges continuously at some critical point, as captured by an order parameter, signaling the spontaneous breaking of a symmetry and an associated non-analyticity of the relevant thermodynamic potential. Conversely, 1 st -order transitions are characterized by an abrupt jump in the order parameter and a coexistente of different phases [1, 2]. In recent years these ideas have been extended to the realm of fluctuations, where dynamical phase transitions (i.e. in the space of trajectories) have been identified in different systems, both classical [3][4][5][6][7][8][9][10][11][12][13][14][15][16][17] and quantum [18][19][20][21]. Important examples include glass formers [22][23][24][25][26][27][28][29], micromasers and superconducting transistors [30,31], or applications such as DPT-based quantum thermal switches [32][33][34].DPTs appear when conditioning a system to have a fixed value of some time-integrated observable, as e.g. the current or the activity. The different dynamical phases correspond to different types of trajectories adopted by the system to sustain atypical values of this observable. Interestingly, some dynamical phases may display emergent order and collective rearrangements in their trajectories, including symmetry-breaking phenomena [5,[9][10][11], while the large deviation functions (LDFs) [35] controlling the statistics of these fluctuations exhibit nonanalyticities and Lee-Yang singularities [36][37][38][39][40][41][42][43] at the DPT reminiscent of standard critical behavior. This is a finding of crucial importance in nonequilibrium physics, as these LDFs play a role akin to the equilibrium thermodynamic potentials for nonequilibrium systems, where no bottom-up approach exists yet connecting microscopic dynamics with macroscopic properties [3, 4,44]. Moreover, the emergence of coherent structures associated * phurtado@onsager.ugr.es to rare f...