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Slowdown estimates and central limit theorem for random walks in random environment

Alain-Sol Sznitman (2000)

Journal of the European Mathematical Society

This work is concerned with asymptotic properties of multi-dimensional random walks in random environment. Under Kalikow’s condition, we show a central limit theorem for random walks in random environment on ℤ d , when d > 2 . We also derive tail estimates on the probability of slowdowns. These latter estimates are of special interest due to the natural interplay between slowdowns and the presence of traps in the medium. The tail behavior of the renewal time constructed in [25] plays an important role in...

Smallness problem for quantum affine algebras and quiver varieties

David Hernandez (2008)

Annales scientifiques de l'École Normale Supérieure

The geometric small property (Borho-MacPherson [2]) of projective morphisms implies a description of their singularities in terms of intersection homology. In this paper we solve the smallness problem raised by Nakajima [37, 35] for certain resolutions of quiver varieties [37] (analogs of the Springer resolution): for Kirillov-Reshetikhin modules of simply-laced quantum affine algebras, we characterize explicitly the Drinfeld polynomials corresponding to the small resolutions. We use an elimination...

Soft local times and decoupling of random interlacements

Serguei Popov, Augusto Teixeira (2015)

Journal of the European Mathematical Society

In this paper we establish a decoupling feature of the random interlacement process ℐ u ⊂ ℤ d at level u , d ≥ 3 . Roughly speaking, we show that observations of ℐ u restricted to two disjoint subsets A 1 and A 2 of ℤ d are approximately independent, once we add a sprinkling to the process ℐ u by slightly increasing the parameter u . Our results differ from previous ones in that we allow the mutual distance between the sets A 1 and A 2 to be much smaller than their diameters. We then provide an important application of this...

Solutions to the XXX type Bethe ansatz equations and flag varieties

E. Mukhin, A. Varchenko (2003)

Open Mathematics

We consider a version of the A N Bethe equation of XXX type and introduce a reporduction procedure constructing new solutions of this equation from a given one. The set of all solutions obtained from a given one is called a population. We show that a population is isomorphic to the sl N+1 flag variety and that the populations are in one-to-one correspondence with intersection points of suitable Schubert cycles in a Grassmanian variety. We also obtain similar results for the root systems B N and...

Spectral gap for an unrestricted Kawasaki type dynamics

Gustavo Posta (2010)

ESAIM: Probability and Statistics

We give an accurate asymptotic estimate for the gap of the generator of a particular interacting particle system. The model we consider may be informally described as follows. A certain number of charged particles moves on the segment [1,L] according to a Markovian law. One unitary charge, positive or negative, jumps from a site k to another site k'=k+1 or k'=k-1 at a rate which depends on the charge at site k and at site k'. The total charge of the system is preserved by the dynamics, in...

Spectral statistics for random Schrödinger operators in the localized regime

François Germinet, Frédéric Klopp (2014)

Journal of the European Mathematical Society

We study various statistics related to the eigenvalues and eigenfunctions of random Hamiltonians in the localized regime. Consider a random Hamiltonian at an energy E in the localized phase. Assume the density of states function is not too flat near E . Restrict it to some large cube Λ . Consider now I Λ , a small energy interval centered at E that asymptotically contains infintely many eigenvalues when the volume of the cube Λ grows to infinity. We prove that, with probability one in the large volume...

Spectral theory of corrugated surfaces

Vojkan Jakšić (2001)

Journées équations aux dérivées partielles

We discuss spectral and scattering theory of the discrete laplacian limited to a half-space. The interesting properties of such operators stem from the imposed boundary condition and are related to certain phenomena in surface physics.

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