133 research outputs found
Random permutations of a regular lattice
Spatial random permutations were originally studied due to their connections
to Bose-Einstein condensation, but they possess many interesting properties of
their own. For random permutations of a regular lattice with periodic boundary
conditions, we prove existence of the infinite volume limit under fairly weak
assumptions. When the dimension of the lattice is two, we give numerical
evidence of a Kosterlitz-Thouless transition, and of long cycles having an
almost sure fractal dimension in the scaling limit. Finally we comment on
possible connections to Schramm-L\"owner curves.Comment: 23 pages, 8 figure
Precise coupling terms in adiabatic quantum evolution
It is known that for multi-level time-dependent quantum systems one can
construct superadiabatic representations in which the coupling between
separated levels is exponentially small in the adiabatic limit. For a family of
two-state systems with real-symmetric Hamiltonian we construct such a
superadiabatic representation and explicitly determine the asymptotic behavior
of the exponentially small coupling term. First order perturbation theory in
the superadiabatic representation then allows us to describe the
time-development of exponentially small adiabatic transitions. The latter
result rigorously confirms the predictions of Sir Michael Berry for our family
of Hamiltonians and slightly generalizes a recent mathematical result of George
Hagedorn and Alain Joye.Comment: 24 page
Gibbs measures with double stochastic integrals on a path space
We investigate Gibbs measures relative to Brownian motion in the case when
the interaction energy is given by a double stochastic integral. In the case
when the double stochastic integral is originating from the Pauli-Fierz model
in nonrelativistic quantum electrodynamics, we prove the existence of its
infinite volume limit.Comment: 17 page
Breaking the chain
We consider the motion of a Brownian particle in , moving between
a particle fixed at the origin and another moving deterministically away at
slow speed . The middle particle interacts with its neighbours via
a potential of finite range , with a unique minimum at , where
. We say that the chain of particles breaks on the left- or right-hand
side when the middle particle is greater than a distance from its left or
right neighbour, respectively. We study the asymptotic location of the first
break of the chain in the limit of small noise, in the case where and is the noise intensity.Comment: 13 pages, 2 figures. v2: Corrected a mistake in proof of second part
of main theore
Stable states of perturbed Markov chains
Given an infinitesimal perturbation of a discrete-time finite Markov chain,
we seek the states that are stable despite the perturbation, \textit{i.e.} the
states whose weights in the stationary distributions can be bounded away from
as the noise fades away. Chemists, economists, and computer scientists have
been studying irreducible perturbations built with exponential maps. Under
these assumptions, Young proved the existence of and computed the stable states
in cubic time. We fully drop these assumptions, generalize Young's technique,
and show that stability is decidable as long as is. Furthermore, if
the perturbation maps (and their multiplications) satisfy or , we prove the existence of and compute the stable states and the
metastable dynamics at all time scales where some states vanish. Conversely, if
the big- assumption does not hold, we build a perturbation with these maps
and no stable state. Our algorithm also runs in cubic time despite the general
assumptions and the additional work. Proving the correctness of the algorithm
relies on new or rephrased results in Markov chain theory, and on algebraic
abstractions thereof
Phase transition for loop representations of Quantum spin systems on trees
We consider a model of random loops on Galton-Watson trees with an offspring
distribution with high expectation. We give the configurations a weighting of
. For many these models are equivalent to
certain quantum spin systems for various choices of the system parameters. We
find conditions on the offspring distribution that guarantee the occurrence of
a phase transition from finite to infinite loops for the Galton-Watson tree.Comment: 16 pages, 1 figur
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