159,446 research outputs found
Thermodynamic formalism for dissipative quantum walks
We consider the dynamical properties of dissipative continuous-time quantum
walks on directed graphs. Using a large-deviation approach we construct a
thermodynamic formalism allowing us to define a dynamical order parameter, and
to identify transitions between dynamical regimes. For a particular class of
dissipative quantum walks we propose a quantum generalization of the the
classical PageRank vector, used to rank the importance of nodes in a directed
graph. We also provide an example where one can characterize the dynamical
transition from an effective classical random walk to a dissipative quantum
walk as a thermodynamic crossover between distinct dynamical regimes.Comment: 8 page
Asymptotic Behaviour and Cyclic Properties of Weighted Shifts on Directed Trees
In this paper we investigate a new class of operators called weighted shifts
on directed trees introduced recently in [Z. J. Jablonski, I. B. Jung and J.
Stochel, A Non-hyponormal Operator Generating Stieltjes Moment Sequences, J.
Funct. Anal. 262 (2012), no. 9, 3946--3980.]. This class is a natural
generalization of the so called weighted bilateral, unilateral and backward
shift operators. In the first part of the paper we calculate the asymptotic
limit and the isometric asymptote of a contractive weighted shift on a directed
tree and that of the adjoint. Then we use the asymptotic behaviour and
similarity properties to deal with cyclicity. We also show that a weighted
backward shift operator is cyclic if and only if there is at most one zero
weight.Comment: 22 page
Distributed convergence to Nash equilibria in two-network zero-sum games
This paper considers a class of strategic scenarios in which two networks of
agents have opposing objectives with regards to the optimization of a common
objective function. In the resulting zero-sum game, individual agents
collaborate with neighbors in their respective network and have only partial
knowledge of the state of the agents in the other network. For the case when
the interaction topology of each network is undirected, we synthesize a
distributed saddle-point strategy and establish its convergence to the Nash
equilibrium for the class of strictly concave-convex and locally Lipschitz
objective functions. We also show that this dynamics does not converge in
general if the topologies are directed. This justifies the introduction, in the
directed case, of a generalization of this distributed dynamics which we show
converges to the Nash equilibrium for the class of strictly concave-convex
differentiable functions with locally Lipschitz gradients. The technical
approach combines tools from algebraic graph theory, nonsmooth analysis,
set-valued dynamical systems, and game theory
The abelian sandpile and related models
The Abelian sandpile model is the simplest analytically tractable model of
self-organized criticality. This paper presents a brief review of known results
about the model. The abelian group structure allows an exact calculation of
many of its properties. In particular, one can calculate all the critical
exponents for the directed model in all dimensions. For the undirected case,
the model is related to q= 0 Potts model. This enables exact calculation of
some exponents in two dimensions, and there are some conjectures about others.
We also discuss a generalization of the model to a network of communicating
reactive processors. This includes sandpile models with stochastic toppling
rules as a special case. We also consider a non-abelian stochastic variant,
which lies in a different universality class, related to directed percolation.Comment: Typos and minor errors fixed and some references adde
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