Effective Fitness Landscapes for Evolutionary Systems

In evolution theory the concept of a fitness landscape has played an important role, evolution itself being portrayed as a hill-climbing process on a rugged landscape. In this article it is shown that in general, in the presence of other genetic operators such as mutation and recombination, hill-climbing is the exception rather than the rule. This descrepency can be traced to the different ways that the concept of fitness appears --- as a measure of the number of fit offspring, or as a measure of the probability to reach reproductive age. Effective fitness models the former not the latter and gives an intuitive way to understand population dynamics as flows on an effective fitness landscape when genetic operators other than selection play an important role. The efficacy of the concept is shown using several simple analytic examples and also some more complicated cases illustrated by simulations.Comment: 11 pages, 8 postscript figure

Environmentally Friendly Renormalization

We analyze the renormalization of systems whose effective degrees of freedom are described in terms of fluctuations which are ``environment'' dependent. Relevant environmental parameters considered are: temperature, system size, boundary conditions, and external fields. The points in the space of \lq\lq coupling constants'' at which such systems exhibit scale invariance coincide only with the fixed points of a global renormalization group which is necessarily environment dependent. Using such a renormalization group we give formal expressions to two loops for effective critical exponents for a generic crossover induced by a relevant mass scale $g$. These effective exponents are seen to obey scaling laws across the entire crossover, including hyperscaling, but in terms of an effective dimensionality, d\ef=4-\gl, which represents the effects of the leading irrelevant operator. We analyze the crossover of an $O(N)$ model on a $d$ dimensional layered geometry with periodic, antiperiodic and Dirichlet boundary conditions. Explicit results to two loops for effective exponents are obtained using a [2,1] Pad\'e resummed coupling, for: the ``Gaussian model'' ($N=-2$), spherical model ($N=\infty$), Ising Model ($N=1$), polymers ($N=0$), XY-model ($N=2$) and Heisenberg ($N=3$) models in four dimensions. We also give two loop Pad\'e resummed results for a three dimensional Ising ferromagnet in a transverse magnetic field and corresponding one loop results for the two dimensional model. One loop results are also presented for a three dimensional layered Ising model with Dirichlet and antiperiodic boundary conditions. Asymptotically the effective exponents are in excellent agreement with known results.Comment: 76 pages of Plain Tex, Postscript figures available upon request from [email protected], preprint numbers THU-93/14, DIAS-STP-93-1

Geometry the Renormalization Group and Gravity

We discuss the relationship between geometry, the renormalization group (RG) and gravity. We begin by reviewing our recent work on crossover problems in field theory. By crossover we mean the interpolation between different representations of the conformal group by the action of relevant operators. At the level of the RG this crossover is manifest in the flow between different fixed points induced by these operators. The description of such flows requires a RG which is capable of interpolating between qualitatively different degrees of freedom. Using the conceptual notion of course graining we construct some simple examples of such a group introducing the concept of a ``floating'' fixed point around which one constructs a perturbation theory. Our consideration of crossovers indicates that one should consider classes of field theories, described by a set of parameters, rather than focus on a particular one. The space of parameters has a natural metric structure. We examine the geometry of this space in some simple models and draw some analogies between this space, superspace and minisuperspace.Comment: 16 pages of LaTex, DIAS-STP-92-3

Schemata as Building Blocks: Does Size Matter?

We analyze the schema theorem and the building block hypothesis using a recently derived, exact schemata evolution equation. We derive a new schema theorem based on the concept of effective fitness showing that schemata of higher than average effective fitness receive an exponentially increasing number of trials over time. The building block hypothesis is a natural consequence in that the equation shows how fit schemata are constructed from fit sub-schemata. However, we show that generically there is no preference for short, low-order schemata. In the case where schema reconstruction is favoured over schema destruction large schemata tend to be favoured. As a corollary of the evolution equation we prove Geiringer's theorem. We give supporting numerical evidence for our claims in both non-epsitatic and epistatic landscapes.Comment: 17 pages, 10 postscript figure

Direct innervation of capillary endothelial cells in the lamina propria of the ferret stomach

Direct innervation of capillary endothelial cells in lamina propria of ferret stomac