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Edge-reinforced random walk, vertex-reinforced jump process and the supersymmetric hyperbolic sigma model

Christophe Sabot, Pierre Tarrès (2015)

Journal of the European Mathematical Society

Edge-reinforced random walk (ERRW), introduced by Coppersmith and Diaconis in 1986 [8], is a random process which takes values in the vertex set of a graph G and is more likely to cross edges it has visited before. We show that it can be represented in terms of a vertex-reinforced jump process (VRJP) with independent gamma conductances; the VRJP was conceived by Werner and first studied by Davis and Volkov [10, 11], and is a continuous-time process favouring sites with more local time. We calculate,...

Effect algebras and ring-like structures

Enrico G. Beltrametti, Maciej J. Maczyński (2003)

Discussiones Mathematicae - General Algebra and Applications

The dichotomic physical quantities, also called propositions, can be naturally associated to maps of the set of states into the real interval [0,1]. We show that the structure of effect algebra associated to such maps can be represented by quasiring structures, which are a generalization of Boolean rings, in such a way that the ring operation of addition can be non-associative and the ring multiplication non-distributive with respect to addition. By some natural assumption on the effect algebra,...

Effective Hamiltonians and Quantum States

Lawrence C. Evans (2000/2001)

Séminaire Équations aux dérivées partielles

We recount here some preliminary attempts to devise quantum analogues of certain aspects of Mather’s theory of minimizing measures [M1-2, M-F], augmented by the PDE theory from Fathi [F1,2] and from [E-G1]. This earlier work provides us with a Lipschitz continuous function u solving the eikonal equation aėȧnd a probability measure σ solving a related transport equation.We present some elementary formal identities relating certain quantum states ψ and u , σ . We show also how to build out of u , σ an approximate...

Eigenmodes of the damped wave equation and small hyperbolic subsets

Gabriel Rivière (2014)

Annales de l’institut Fourier

We study stationary solutions of the damped wave equation on a compact and smooth Riemannian manifold without boundary. In the high frequency limit, we prove that a sequence of β -damped stationary solutions cannot be completely concentrated in small neighborhoods of a small fixed hyperbolic subset made of β -damped trajectories of the geodesic flow.The article also includes an appendix (by S. Nonnenmacher and the author) where we establish the existence of an inverse logarithmic strip without eigenvalues...

Eigenvalue asymptotics for the Pauli operator in strong nonconstant magnetic fields

Georgi D. Raikov (1999)

Annales de l'institut Fourier

We consider the Pauli operator H ( μ ) : = ∑ j = 1 m σ j - i ∂ ∂ x j - μ A j 2 + V selfadjoint in L 2 ( ℝ m ; ℂ 2 ) , m = 2 , 3 . Here σ j , j = 1 , ... , m , are the Pauli matrices, A : = ( A 1 , ... , A m ) is the magnetic potential, μ > 0 is the coupling constant, and V is the electric potential which decays at infinity. We suppose that the magnetic field generated by A satisfies some regularity conditions; in particular, its norm is lower-bounded by a positive constant, and, in the case m = 3 , its direction is constant. We investigate the asymptotic behaviour as μ → ∞ of the number of the eigenvalues of H ( μ ) smaller than...

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