Universality Classes of Hitting Probabilities of Jump Processes Nicolas Levernier, 1 Olivier B´ enichou, 2 and Raphaël Voituriez 2,3 1 Aix Marseille Universit´ e, CNRS, IUSTI, 13453 Marseille, France 2 Laboratoire de Physique Th´ eorique de la Mati` ere Condens´ ee, UMR 7600 CNRS/UPMC, 4 Place Jussieu, 75255 Paris Cedex, France 3 Laboratoire Jean Perrin, UMR 8237 CNRS/UPMC, 4 Place Jussieu, 75255 Paris Cedex, France (Received 17 November 2020; revised 26 January 2021; accepted 11 February 2021; published 10 March 2021) Quantifying the efficiency of random target search strategies is a key question of random walk theory, with applications in various fields. If many results do exist for recurrent processes, for which the probability of eventually finding a target in infinite space—so called hitting probability—is one, much less is known in the opposite case of transient processes, for which the hitting probability is strictly less than one. Here, we determine the universality classes of the large distance behavior of the hitting probability for general d- dimensional transient jump processes, which we show are parametrized by a transience exponent that is explicitly given. DOI: 10.1103/PhysRevLett.126.100602 Quantifying the statistics of encounter events of a random walker with a target has become a central question of random walk theory [1,2], with applications ranging from chemical reaction kinetics to animal foraging behav- iors [3–5]. In the simplest setting of a single target in an unbounded d-dimensional space, two radically different scenarios naturally emerge: either the target is eventually found with probability Π ¼ 1 (compact or recurrent case), or with probability Π < 1 (noncompact or transient case). This defines the hitting probability Π [1,6]. In the recurrent case, a classical observable is the survival probability SðtÞ , i.e., the probability that the target has not been found until time t. This quantity typically decreases at long timescales as t −θ , where θ is the persistence exponent, which has been the focus of numer- ous studies [7]. In the transient case, the survival proba- bility admits a nonzero large time limit, which is readily expressed in terms of the hitting probability S→ t→∞ 1 − Π. This makes the hitting probability a key quantifier of the search process in the transient case. The hitting probability is expected to decrease with the distance R from the starting position of the random walk and the target of radius a according to Π ∼ Cða=RÞ ψ . The transience exponent ψ recently introduced in [8] parallels the persistence exponent of recurrent processes and is an intrinsic characteristic of transient processes that has, however, been largely unex- plored so far; in turn, the prefactor C depends on the process and is required for the full determination of the asymptotics of the hitting probability. For jump processes [9,10], target capture events must be defined. Two conventions have been adopted [8,11,12]: the target can be found either when a jump ends inside the target—arrival convention, or when a jump crosses the target—crossing convention (see Fig. 1). Importantly, for d ¼ 1, it has been shown that arrival and crossing conventions can lead to strikingly different first-passage properties [11,13] in the case of Levy processes. These can be defined as the continuous time limit of Levy flights, which are discrete time jump processes whose jump-length distribution has a power law tail pðlÞ ∝ 1=l 1þα with index α ∈ ½0; 2. For d ≥ 2, exact results for the hitting proba- bility are sparse and available only for the arrival con- vention, either for examples of jump distributions with finite variance [14–16], or for the specific case of Levy processes [17]. However, the hitting probability with the crossing convention—which is clearly larger than with the arrival convention and nontrivial for d ≥ 2 only—has not been studied, despite its relevance to various examples of target search problems. In particular, the hitting probability of Levy flights with the crossing convention gives access to the hitting probability of Levy walks, which play an important role in the context of animal behavior [18]. In this Letter, we consider jump processes with a general distribution of jump length, determine the large R behavior of the hitting probability for both conventions, and reveal its universality classes that we show are parametrized by the transience exponent, which is explicitly given. Our results are summarized in Table I. More precisely, in the case of processes whose jump distribution has a finite variance we FIG. 1. Hitting probability of jump processes with crossing convention (plain line trajectory) or arrival convention (dashed line trajectory). The target is detected when the path turns red. PHYSICAL REVIEW LETTERS 126, 100602 (2021) 0031-9007=21=126(10)=100602(5) 100602-1 © 2021 American Physical Society