We present rigorous mathematical analyses of a number of wellknown mathematical models for genetic mutations. In these models, the genome is represented by a vertex of the n-dimensional binary hypercube, for some n, a mutation involves the flipping of a single bit, and each vertex is assigned a real number, called its fitness, according to some rules. Our main concern is with the issue of existence of (selectively) accessible paths; that is, monotonic paths in the hypercube along which fitness is always increasing. Our main results resolve open questions about three such models, which in the biophysics literature are known as house of cards (HoC), constrained house of cards (CHoC) and rough Mount Fuji (RMF). We prove that the probability of there being at least one accessible path from the all-zeroes node v 0 to the all-ones node v 1 tends respectively to 0, 1 and 1, as n tends to infinity. A crucial idea is the introduction of a generalization of the CHoC model, in which the fitness of v 0 is set to some α = αn ∈ [0, 1]. We prove that there is a very sharp threshold at αn = ln n n for the existence of accessible paths from v 0 to v 1 . As a corollary we prove significant concentration, for α below the threshold, of the number of accessible paths about the expected value (the precise statement is technical; see Corollary 1.4). In the case of RMF, we prove that the probability of accessible paths from v 0 to v 1 existing tends to 1 provided the drift parameter θ = θn satisfies nθn → ∞, and for any fitness distribution which is continuous on its support and whose support is connected.

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The Paradoxissirnus Siltstone, representins the Middle Cambrian zone with Tomagnostusjssus and PQcha,onostus alaius in the Oland area, was deposited in a bay under the influence of rhythmic or pendulating curtent action from NE. Longer periods of mud deposition and a n abundant animal life alternated with short periods of silt influx causing a devastation of the mud-burrowing zoocoenoses over large areas. Occasionally the currents turned, transporting some silt in the opposite direction. Among the characteristic animals of the muddy bottoms was the large trilobite Paradoxides parndoxissimur which is found in abundance in the siltstone but only as fossil exuviae; in the shale, representing the environment in which it lived, shelly and trace fossils have been obliterated by diagenetic processes. The silt coiitent in the Paradoxissirnus Beds shows lateral variations within the a l a n d area, and the mainly shaly portions, mostly in the lower part of the sequence, contain better preserved dead trilobites and esuviae in situ.Primary sedimentary structures in connection with the current activity (ripples of different types, traction m a r h , and priels) are briefly discussed, and the trace fossils are dealt with in more detail. Practical problems in the treatment of trace fossils and ichnocoenoses are discussed as well as some ichnosystematic aspects of the Cambrian "genera" Halopoa and Scotolithus.

We consider first-passage percolation on the class of "high-dimensional" graphs that can be written as an iterated Cartesian product G G . . . G of some base graph G as the number of factors tends to infinity. We propose a natural asymptotic lower bound on the first-passage time between (v, v, . . . , v) and (w, w, . . . , w) as n, the number of factors, tends to infinity, which we call the critical time t * G (v, w). Our main result characterizes when this lower bound is sharp as n → ∞. As a corollary, we are able to determine the limit of the socalled diagonal time-constant in Z n as n → ∞ for a large class of distributions of passage times.2010 Mathematics Subject Classification. Primary 60C05; secondary 60K35, 82B43.

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