Morphisms and currents in infinite nonlinear resistive networks

Soardi, Paolo M.

doi:10.1007/bf01049393

Cited by 8 publications

(4 citation statements)

References 29 publications

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“…For results on the L 2 -capacity of general networks (and more generally, L pcapacities, where the expression f (e)R e in the above definition of V (f ) is replaced by its (p−1) power), as part of Discrete Nonlinear Potential Theory, cf., e.g., [28], [36], [37] and the references therein.…”

Section: 2mentioning

confidence: 99%

“…For results on the L 2 -capacity of general networks (and more generally, L p -capacities, where the expression f (e)R e in the above definition of V ( f ) is replaced by its ( p − 1) power), as part of the Discrete Nonlinear Potential Theory, cf., e.g., [28,36,37] and the references therein. For our proofs, we will use the well-known fact that the L 2 -capacity of the tree T is precisely the effective conductance between the root ρ and the leaves ∂ T , denoted by C eff (ρ ↔ ∂ T ).…”

Section: -Capacitymentioning

confidence: 99%

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Mixing Time of Critical Ising Model on Trees is Polynomial in the Height

Ding

Lubetzky

Peres

2010

Commun. Math. Phys.

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Abstract:In the heat-bath Glauber dynamics for the Ising model on the lattice, physicists believe that the spectral gap of the continuous-time chain exhibits the following behavior. For some critical inverse-temperature β c , the inverse-gap is O(1) for β < β c , polynomial in the surface area for β = β c and exponential in it for β > β c . This has been proved for Z 2 except at criticality. So far, the only underlying geometry where the critical behavior has been confirmed is the complete graph. Recently, the dynamics for the Ising model on a regular tree, also known as the Bethe lattice, has been intensively studied. The facts that the inverse-gap is bounded for β < β c and exponential for β > β c were established, where β c is the critical spin-glass parameter, and the tree-height h plays the role of the surface area.In this work, we complete the picture for the inverse-gap of the Ising model on the b-ary tree, by showing that it is indeed polynomial in h at criticality. The degree of our polynomial bound does not depend on b, and furthermore, this result holds under any boundary condition. We also obtain analogous bounds for the mixing-time of the chain. In addition, we study the near critical behavior, and show that for β > β c , the inverse-gap and mixing-time are both exp[ ((β − β c )h)].

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Section: 2mentioning

confidence: 99%

Section: -Capacitymentioning

confidence: 99%

Mixing Time of Critical Ising Model on Trees is Polynomial in the Height

Ding

Lubetzky

Peres

2010

Commun. Math. Phys.

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“…Such a flow µ can be identified with a positive finite measure on ∂T where µ(e) is the measure of the set of paths in ∂T that traverse e. The total mass of this measure is the outflow from the root, |µ| := y : o→y µ(oy). (2.3) These capacities have been studied on more general networks as part of discrete nonlinear potential theory [see, e.g., Murakami and Yamasaki (1992), Soardi (1993Soardi ( , 1994 and the references therein]. However, all the properties of cap p that we will use follow readily from the definition.…”

mentioning

confidence: 99%

The critical Ising model on trees, concave recursions and nonlinear capacity

Pemantle¹,

Peres²

2010

Ann. Probab.

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We consider the Ising model on a general tree under various boundary conditions: all plus, free and spin-glass. In each case, we determine when the root is influenced by the boundary values in the limit as the boundary recedes to infinity. We obtain exact capacity criteria that govern behavior at critical temperatures. For plus boundary conditions, an L 3 capacity arises. In particular, on a spherically symmetric tree that has n α b n vertices at level n (up to bounded factors), we prove that there is a unique Gibbs measure for the ferromagnetic Ising model at the relevant critical temperature if and only if α ≤ 1/2. Our proofs are based on a new link between nonlinear recursions on trees and L p capacities.

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“…Zukowski and Wyatt (1984)), but a mechanical analogue in which flows are replaced by forces is given by Cohen and Horowitz (1991). Keady (1993), (1995) list over a dozen references and Soardi (1993) advances the ideas in infinite networks.…”

Section: Introductionmentioning

confidence: 99%

Braess's paradox in a queueing network with state-dependent routing

Calvert

Solomon

Ziedins

1997

Journal of Applied Probability

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We consider initially two parallel routes, each of two queues in tandem, with arriving customers choosing the route giving them the shortest expected time in the system, given the queue lengths at the customer's time of arrival. All interarrival and service times are exponential.We then augment this network to obtain a Wheatstone bridge, in which customers may cross from one route to the other between queues, again choosing the route giving the shortest expected time in the system, given the queue lengths ahead of them.We find that Braess's paradox can occur: namely in equilibrium the expected transit time in the augmented network, for some service rates, can be greater than in the initial network.

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Morphisms and currents in infinite nonlinear resistive networks

Cited by 8 publications

References 29 publications

Mixing Time of Critical Ising Model on Trees is Polynomial in the Height

Mixing Time of Critical Ising Model on Trees is Polynomial in the Height

The critical Ising model on trees, concave recursions and nonlinear capacity

Braess's paradox in a queueing network with state-dependent routing

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