1,464 research outputs found
Gibbs Ensembles of Nonintersecting Paths
We consider a family of determinantal random point processes on the
two-dimensional lattice and prove that members of our family can be interpreted
as a kind of Gibbs ensembles of nonintersecting paths. Examples include
probability measures on lozenge and domino tilings of the plane, some of which
are non-translation-invariant.
The correlation kernels of our processes can be viewed as extensions of the
discrete sine kernel, and we show that the Gibbs property is a consequence of
simple linear relations satisfied by these kernels. The processes depend on
infinitely many parameters, which are closely related to parametrization of
totally positive Toeplitz matrices.Comment: 6 figure
On a conjecture of Widom
We prove a conjecture of H.Widom stated in [W] (math/0108008) about the
reality of eigenvalues of certain infinite matrices arising in asymptotic
analysis of large Toeplitz determinants. As a byproduct we obtain a new proof
of A.Okounkov's formula for the (determinantal) correlation functions of the
Schur measures on partitions.Comment: 9 page
Universal exit probabilities in the TASEP
We study the joint exit probabilities of particles in the totally asymmetric
simple exclusion process (TASEP) from space-time sets of given form. We extend
previous results on the space-time correlation functions of the TASEP, which
correspond to exits from the sets bounded by straight vertical or horizontal
lines. In particular, our approach allows us to remove ordering of time moments
used in previous studies so that only a natural space-like ordering of particle
coordinates remains. We consider sequences of general staircase-like boundaries
going from the northeast to southwest in the space-time plane. The exit
probabilities from the given sets are derived in the form of Fredholm
determinant defined on the boundaries of the sets. In the scaling limit, the
staircase-like boundaries are treated as approximations of continuous
differentiable curves. The exit probabilities with respect to points of these
curves belonging to arbitrary space-like path are shown to converge to the
universal Airy process.Comment: 46 pages, 7 figure
Research of nanocomposite structure of boron nitride at proton radiation
Using roentgen diffraction and electron microscopy, the influence of nanosecond irradiation by ion beams of high energy on forming of self-organized nanoblocks in near surface's layers of boron nitride (BN) has been studied. It was shown that low temperature transitions from hexagonal to wrutz boron nitrides is associated with changes of shape and sizes of self-organized particles consisting the nanoblocks. We have calculated the parameters of nanoblocks using the meanings of interplane distances and properties of subreflexes orders. The collective shifting deformations of layers in nanoblocks provides phase transition under the screen and forming the set of nanotubes with escaping of five order axes of symmetry. It has been realized that pentagons and stars arranged in points of entrance of five order axis of symmetry are associated with peculiarity of self-organization of the spiral-cyclic structures
Airy processes and variational problems
We review the Airy processes; their formulation and how they are conjectured
to govern the large time, large distance spatial fluctuations of one
dimensional random growth models. We also describe formulas which express the
probabilities that they lie below a given curve as Fredholm determinants of
certain boundary value operators, and the several applications of these
formulas to variational problems involving Airy processes that arise in
physical problems, as well as to their local behaviour.Comment: Minor corrections. 41 pages, 4 figures. To appear as chapter in "PASI
Proceedings: Topics in percolative and disordered systems
Non-colliding Brownian Motions and the extended tacnode process
We consider non-colliding Brownian motions with two starting points and two
endpoints. The points are chosen so that the two groups of Brownian motions
just touch each other, a situation that is referred to as a tacnode. The
extended kernel for the determinantal point process at the tacnode point is
computed using new methods and given in a different form from that obtained for
a single time in previous work by Delvaux, Kuijlaars and Zhang. The form of the
extended kernel is also different from that obtained for the extended tacnode
kernel in another model by Adler, Ferrari and van Moerbeke. We also obtain the
correlation kernel for a finite number of non-colliding Brownian motions
starting at two points and ending at arbitrary points.Comment: 38 pages. In the revised version a few arguments have been expanded
and many typos correcte
From interacting particle systems to random matrices
In this contribution we consider stochastic growth models in the
Kardar-Parisi-Zhang universality class in 1+1 dimension. We discuss the large
time distribution and processes and their dependence on the class on initial
condition. This means that the scaling exponents do not uniquely determine the
large time surface statistics, but one has to further divide into subclasses.
Some of the fluctuation laws were first discovered in random matrix models.
Moreover, the limit process for curved limit shape turned out to show up in a
dynamical version of hermitian random matrices, but this analogy does not
extend to the case of symmetric matrices. Therefore the connections between
growth models and random matrices is only partial.Comment: 18 pages, 8 figures; Contribution to StatPhys24 special issue; minor
corrections in scaling of section 2.
Form factor approach to dynamical correlation functions in critical models
We develop a form factor approach to the study of dynamical correlation
functions of quantum integrable models in the critical regime. As an example,
we consider the quantum non-linear Schr\"odinger model. We derive
long-distance/long-time asymptotic behavior of various two-point functions of
this model. We also compute edge exponents and amplitudes characterizing the
power-law behavior of dynamical response functions on the particle/hole
excitation thresholds. These last results confirm predictions based on the
non-linear Luttinger liquid method. Our results rely on a first principles
derivation, based on the microscopic analysis of the model, without invoking,
at any stage, some correspondence with a continuous field theory. Furthermore,
our approach only makes use of certain general properties of the model, so that
it should be applicable, with possibly minor modifications, to a wide class of
(not necessarily integrable) gapless one dimensional Hamiltonians.Comment: 33 page
Advice coins for classical and quantum computation
We study the power of classical and quantum algorithms equipped with nonuniform advice, in the form of a coin whose bias encodes useful information. This question takes on particular importance in the quantum case, due to a surprising result that we prove: a quantum finite automaton with just two states can be sensitive to arbitrarily small changes in a coin’s bias. This contrasts with classical probabilistic finite automata, whose sensitivity to changes in a coin’s bias is bounded by a classic 1970 result of Hellman and Cover.
Despite this finding, we are able to bound the power of advice coins for space-bounded classical and quantum computation. We define the classes BPPSPACE/coin and BQPSPACE/coin, of languages decidable by classical and quantum polynomial-space machines with advice coins. Our main theorem is that both classes coincide with PSPACE/poly. Proving this result turns out to require substantial machinery. We use an algorithm due to Neff for finding roots of polynomials in NC; a result from algebraic geometry that lower-bounds the separation of a polynomial’s roots; and a result on fixed-points of superoperators due to Aaronson and Watrous, originally proved in the context of quantum computing with closed timelike curves
How long does it take to pull an ideal polymer into a small hole?
We present scaling estimates for characteristic times and
of pulling ideal linear and randomly branched polymers of
monomers into a small hole by a force . We show that the absorbtion process
develops as sequential straightening of folds of the initial polymer
configuration. By estimating the typical size of the fold involved into the
motion, we arrive at the following predictions: and , and we also confirm them by
the molecular dynamics experiment.Comment: 4 pages, 3 figure
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