When a biological population expands into new territory, genetic drift develops an enormous influence on evolution at the propagating front. In such range expansion processes, fluctuations in allele frequencies occur through stochastic spatial wandering of both genetic lineages and the boundaries between genetically segregated sectors. Laboratory experiments on microbial range expansions have shown that this stochastic wandering, transverse to the front, is superdiffusive due to the front's growing roughness, implying much faster loss of genetic diversity than predicted by simple flat front diffusive models. We study the evolutionary consequences of this superdiffusive wandering using two complementary numerical models of range expansions: the stepping stone model, and a new interpretation of the model of directed paths in random media, in the context of a roughening population front. Through these approaches we compute statistics for the times since common ancestry for pairs of individuals with a given spatial separation at the front, and we explore how environmental heterogeneities can locally suppress these superdiffusive fluctuations.arXiv:1812.09279v1 [physics.bio-ph] 21 Dec 2018 1, the spatial fluctuations for each

The probability distribution for the free energy of directed polymers in random media (DPRM) with uncorrelated noise in d = 1 + 1 dimensions satisfies the Tracy-Widom distribution. We inquire if and how this universal distribution is modified in the presence of spatially correlated noise. The width of the distribution scales as the DPRM length to an exponent β, in good (but not full) agreement with previous renormalization group and numerical results. The scaled probability is well described by the Tracy-Widom form for uncorrelated noise, but becomes symmetric with increasing correlation exponent. We thus find a class of distributions that continuously interpolates between Tracy-Widom and Gaussian forms. DOI: 10.1103/PhysRevE.94.010101The Tracy-Widom (TW) distribution was originally introduced in connection with the probability for the largest eigenvalue of a random matrix [1]. It has since acquired iconic status [2] due to applications ranging from bioinformatic sequence alignments [3] to aircraft fault detection [4]. Like the Gumbel and Gaussian distributions, TW is universal in being independent of various underlying (microscopic) details. However, whereas it is known how the addition of fat-tailed random variables modifies a Gaussian to a Lévy distribution, corresponding limitations for TW are not known. We take up this question in the context of directed polymers in random media (DPRM) [5][6][7], one of the more highly studied systems in the TW class [8,9].The DPRM problem considers configurations of a directed path (no overhangs) traversing a random energy landscape. Unlike the traveling salesman problem (which allows overhangs and loops), the optimization problem can be solved in polynomial time with a transfer matrix formalism [5][6][7]. The optimal energy path (or the free energy at finite temperature) exhibits sample to sample fluctuations, which scale with the path length t, as t β . In 1+1 dimensions, and for uncorrelated random energies, the scaled probability of these fluctuations satisfies the TW distribution [8,9]. It is known, however, that the exponent β is modified if the random energies have long-range (power-law) correlations [10]. We examine energy fluctuations in such correlated energy landscapes, and inquire if and how the TW form changes along with the exponent β.As one of the simplest random processes described by the Kardar-Parisi-Zhang (KPZ) equation [10,11], DPRM has been extensively studied over the past three decades [12][13][14], with renewed recent interest [15][16][17] due to its connection to TW. It is closely related to the Eden [18][19][20], the restricted solid-onsolid (RSOS) [21,22], and ballistic deposition (BD) models [19,23]. (Extensive reviews from both statistical physics [6,7] and mathematical [9] perspectives provide an excellent background on the subject.) In the continuum limit, the partition function W (x,t) of a polymer of length t terminating at a point x ∈ R d satisfies the stochastic heat equationwhere ν is related to the polymer line tension, and η(x,t) is th...

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We analyze the statistics of the shortest and fastest paths on the road network between randomly sampled end points. We find that, to a good approximation, the optimal paths can be described as directed polymers in a disordered medium, which belong to the Kardar-Parisi-Zhang universality class of interface roughening. Comparing the scaling behavior of our data with simulations of directed polymers and previous theoretical results, we are able to point out the few characteristics of the road network that are relevant to the large-scale statistics of optimal paths. Indeed, we show that the local structure is akin to a disordered environment with a power-law distribution which become less important at large scales where long-ranged correlations in the network control the scaling behavior of the optimal paths. DOI: 10.1103/PhysRevE.96.050301 Complex networks of nodes and links can be used to model a wide array of systems. Examples range from biological networks such as those formed by neurons and synapses in the brain or chemical reactions inside a cell, to social or transportation networks and the World Wide Web. Their topology in the abstract space of edges and vertices has been much studied, allowing one to identify widespread properties such as "small-world" effects, scale-free connectivity, and a high degree of clustering, which can be captured by simple physical models [1][2][3][4][5]. Comparatively, less is understood about the spatial organization of complex networks embedded in a Euclidean space, a very active subject of research (see Ref.[6] for a review). The effect of geometry becomes especially relevant when the network is strongly constrained by the environment or when the "cost" to maintain edges increases significantly with their length (e.g., rivers [7], railways [8], or vascular networks [9]). The spatial structure of streets is another example that has been particularly studied to gain insight into the structure of cities and their development [10][11][12].Much information about the geometry of a network can be obtained by studying the shortest paths between the nodes of the network. In many cases, it is also a problem of practical importance to characterize the paths that optimize a given cost function. For example, in transportation networks, one would like to understand the properties of the paths that minimize the travel time, the distance, or the monetary cost to travel between two points. An obvious application is in the development of efficient global positioning system (GPS) routing algorithms which could use prior information on optimal paths to perform better [13]. The shortest paths between two generating nodes on the power grid are also important to predict the overloading of electric lines [14]. Understanding the properties of these optimal paths appears challenging since they are expected to depend strongly on the geometry of the network which can be shaped by various factors, from natural obstacles to historical development or differences in policy.The theory of directed polymers...

Reflectionless CMV matrices are studied using scattering theory. By changing a single Verblunsky coefficient, a full-line CMV matrix can be decoupled and written as the sum of two half-line operators. Explicit formulas for the scattering matrix associated to the coupled and decoupled operators are derived. In particular, it is shown that a CMV matrix is reflectionless iff the scattering matrix is off-diagonal which in turns provides a short proof of an important result of [1]. These developments parallel those recently obtained for Jacobi matrices [10].