Codina Cotar

Should females prefer to mate with low-quality males?

We analyse a model of mate choice when males differ in reproductive quality and provide care for their offspring. Females choose males on the basis of the success they will obtain from breeding with them and a male chooses his care time on the basis of his quality so as to maximise his long-term rate of reproductive success. We use this model to establish whether high-quality males should devote a longer period of care to their broods than low-quality males and whether females obtain greater reproductive success from mating with higher quality males. We give sufficient conditions for optimal care times to decrease with increasing male quality. When care times decrease, this does not necessarily mean that high-quality males are less valuable to the female because quality may more than compensate for the lack of care. We give a necessary and sufficient condition for high-quality males to be less valuable mates, and hence for females to prefer low-quality males. Females can prefer low-quality males if offspring produced and cared for by high-quality males do well even if care is short, and do not significantly benefit from additional care, while offspring produced and cared for by low-quality males do well only if they receive a long period of care.

On a Preferential Attachment and Generalized Pólya's Urn Model

We study a general preferential attachment and Polyas urn model. At each step a new vertex is introduced, which can be connected to at most one existing vertex. If it is disconnected, it becomes a pioneer vertex. Given that it is not disconnected, it joins an existing pioneer vertex with probability proportional to a function of the degree of that vertex. This function is allowed to be vertex-dependent, and is called the reinforcement function. We prove that there can be at most three phases in this model, depending on the behavior of the reinforcement function. Consider the set whose elements are the vertices with cardinality tending a.s. to infinity. We prove that this set either is empty, or it has exactly one element, or it contains all the pioneer vertices. Moreover, we describe the phase transition in the case where the reinforcement function is the same for all vertices. Our results are general, and in particular we are not assuming monotonicity of the reinforcement function. Finally, consider the regime where exactly one vertex has a degree diverging to infinity. We give a lower bound for the probability that a given vertex ends up being the leading one, i.e. its degree diverges to infinity. Our proofs rely on a generalization of the Rubin construction given for edge-reinforced random walks, and on a Brownian motion embedding.

Infinite-body optimal transport with Coulomb cost

We introduce and analyze symmetric infinite-body optimal transport (OT) problems with cost function of pair potential form. We show that for a natural class of such costs, the optimizer is given by the independent product measure all of whose factors are given by the one-body marginal. This is in striking contrast to standard finite-body OT problems, in which the optimizers are typically highly correlated, as well as to infinite-body OT problems with Gangbo–Swiech cost. Moreover, by adapting a construction from the study of exchangeable processes in probability theory, we prove that the corresponding

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N

Attraction time for strongly reinforced walks

N-body OT problem is well approximated by the infinite-body problem. To our class belongs the Coulomb cost which arises in many-electron quantum mechanics. The optimal cost of the Coulombic N-body OT problem as a function of the one-body marginal density is known in the physics and quantum chemistry literature under the name SCE functional, and arises naturally as the semiclassical limit of the celebrated Hohenberg-Kohn functional. Our results imply that in the inhomogeneous high-density limit (i.e.

Edge- and vertex-reinforced random walks with super-linear reinforcement on infinite graphs

N\rightarrow \infty