1,720,973 research outputs found
ON THE TRANSIENCE OF PROCESSES DEFINED ON GALTON-WATSON TREES
No full textWe introduce a simple technique for proving the transience of certain processes defined on the random tree G generated by a supercritical branching process. We prove the transience for once-reinforced random walks on G , that is, a generalization of a result of Durrett, Kesten and Limic [Probab. Theory Related Fields 122 (2002) 567–592]. Moreover, we give a new proof for the transience of a family of biased random walks defined on G . Other proofs of this fact can be found in [Ann. Probab. 16 (1988) 1229–1241] and [Ann. Probab. 18 (1990) 931–958] as part of more general results. A similar technique is applied to a vertex-reinforced jump process. A by-product of our result is that this process is transient on the 3-ary tree. Davis and Volkov [Probab. Theory Related Fields 128 (2004) 42–62] proved that a vertex-reinforced jump process defined on the b-ary tree is transient if b ≥ 4 and recurrent if b = 1. The case b = 2 is still open
BOUNDS ON THE SPEED AND ON REGENERATION TIMES FOR CERTAIN PROCESSES ON REGULAR TREES
No full textWe develop a technique that provides a lower bound on the speed of tran- sient random walk in a random environment on regular trees. A refinement of this technique yields upper bounds on the first regeneration level and re- generation time. In particular, a lower and upper bound on the covariance in the annealed invariance principle follows. We emphasize the fact that our methods are general and also apply in the case of once-reinforced random walk. Durrett, Kesten and Limic [Probab. Theory Related Fields. 122 (2002) 567–592] prove an upper bound of the form b/(b + δ) for the speed on the b-ary tree, where δ is the reinforcement parameter. For δ > 1 we provide a lower bound of the form γ 2b/(b + δ), where γ is the survival probability of an associated branching process
The probability of nontrivial common knowledge
Full text linkWe study the probability that two or more agents can attain common knowledge of nontrivial events when the size of the state space grows large. We adopt the standard epistemic model where the knowledge of an agent is represented by a partition of the state space. Each agent is endowed with a partition generated by a random scheme consistent with his cognitive capacity. Assuming that agents' partitions are independently distributed, we prove that the asymptotic probability of nontrivial common knowledge undergoes a phase transition. Regardless of the number of agents, when their cognitive capacity is sufficiently large, the probability goes to one; and when it is small, it goes to zero. Our proofs rely on a graph-theoretic characterization of common knowledge that has independent interest
Phase Transitions For Dilute Particle Systems with Lennard-Jones Potential
No full textWe consider a classical dilute particle system in a large box with pair- interaction given by a Lennard-Jones-type potential. The inverse temperature is picked proportionally to the logarithm of the particle density. We identify the free energy per particle in terms of a variational formula and show that this formula exhibits a cascade of phase transitions as the temperature parameter ranges from zero to infinity. Loosely speaking, the particle system separates into spatially distant components in such a way that within each phase all components are of the same size, which is the larger the lower the temperature. The main tool in our proof is a new large deviation principle for sparse point configurations
On a preferential attachment and generalized Polya's urn model
Full text linkWe study a general preferential attachment and Pólya’s 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, that is, 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
A variational formula for the free energy of an interacting many-particle system
Full text linkWe consider N bosons in a box in R(d) with volume N/rho under the influence of a mutually repellent pair potential. The particle density p is an element of (0, infinity) is kept fixed. Our main result is the identification of the limiting free energy, f(beta, p), at positive temperature 1/beta, in terms of an explicit variational formula, for any fixed rho if beta is sufficiently small, and for any fixed beta if rho is sufficiently small.
The thermodynamic equilibrium is described by the symmetrized trace of e-beta H(N), where H(N) denotes the corresponding Hamilton operator. The well-known Feynman-Kac formula reformulates this trace in terms of N interacting Brownian bridges. Due to the symmetrization, the bridges are organized in an ensemble of cycles of various lengths. The novelty of our approach is a description in terms of a marked Poisson point process whose marks are the cycles. This allows for an asymptotic analysis of the system via a large-deviations analysis of the stationary empirical field. The resulting variational formula ranges over random shift-invariant marked point fields and optimizes the sum of the interaction and the relative entropy with respect to the reference process.
In our proof of the lower bound for the free energy, we drop all interaction involving "infinitely long" cycles, and their possible presence is signalled by a loss of mass of the "finitely long" cycles in the variational formula. In the proof of the upper bound, we only keep the mass on the "finitely long" cycles. We expect that the precise relationship between these two bounds lies at the heart of Bose-Einstein condensation and intend to analyze it further in future
Going Beyond Counting First Authors in Author Co-citation Analysis
Full text linkThe present study examines one of the fundamental aspects of author co-citation analysis (ACA) - the way co-citation
counts are defined. Co-citation counting provides the data on which all subsequent statistical analyses and mappings
are based, and we compare ACA results based on two different types of co-citation counting - the traditional type that
only counts the first one among a cited work's authors on the one hand and a non-traditional type that takes into
account the first 5 authors of a cited work on the other hand. Results indicate that the picture produced through this non-traditional author co-citation counting contains more coherent author groups and is therefore considerably clearer. However, this picture represents fewer specialties in the research field being studied than that produced through the traditional first-author co-citation counting when the same number of top-ranked authors is selected and analyzed. Reasons for these effects are discussed
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