183 research outputs found
Sandpile models : The infinite volume model, Zhang's model and limiting shapes
No full textMeester, R.W.J. [Promotor]Redig, F. [Copromotor
RANDOM WALKS IN DYNAMIC RANDOM ENVIRONMENTS: A TRANSFERENCE PRINCIPLE
No full textWe study a general class of random walks driven by a uniquely ergodic Markovian environment. Under a coupling condition on the environment we obtain strong ergodicity properties for the environment as seen from the position of the walker, that is, the environment process. We can transfer the rate of mixing in time of the environment to the rate of mixing of the environment process with a loss of at most polynomial order. Therefore the method is applicable to environments with sufficiently fast polynomial mixing. We obtain unique ergodicity of the environment process. Moreover, the unique invariant measure of the environment process depends continuously on the jump rates of the walker. As a consequence we obtain the law of large numbers and a central limit theorem with nondegenerate variance for the position of the walk.NWO [600.065.100.07N14
Hydrodynamics for the partial exclusion process in random environment
No full textIn this paper, we introduce a random environment for the exclusion process in Zd obtained by assigning a maximal occupancy to each site. This maximal occupancy is allowed to randomly vary among sites, and partial exclusion occurs. Under the assumption of ergodicity under translation and uniform ellipticity of the environment, we derive a quenched hydrodynamic limit in path space by strengthening the mild solution approach initiated in Nagy (2002) and Faggionato (2007). To this purpose, we prove, employing the technology developed for the random conductance model, a homogenization result in the form of an arbitrary starting point quenched invariance principle for a single particle in the same environment, which is a result of independent interest. The self-duality property of the partial exclusion process allows us to transfer this homogenization result to the particle system and, then, apply the tightness criterion in Redig et al. (2020)
Asymmetric Stochastic Transport Models with Uq(su(1,1)) Symmetry
No full textBy using the algebraic construction outlined in Carinci et al. (arXiv:?1407.?3367, 2014), we introduce several Markov processes related to the Uq(su(1,1)) quantum Lie algebra. These processes serve as asymmetric transport models and their algebraic structure easily allows to deduce duality properties of the systems. The results include: (a) the asymmetric version of the Inclusion Process, which is self-dual; (b) the diffusion limit of this process, which is a natural asymmetric analogue of the and which turns out to have the Symmetric Inclusion Process as a dual process; (c) the asymmetric analogue of the KMP Process, which also turns out to have a symmetric dual process. We give applications of the various duality relations by computing exponential moments of the current.Delft Institute of Applied MathematicsElectrical Engineering, Mathematics and Computer Scienc
Spatial fluctuation theorem
No full textFor non-equilibrium systems of interacting particles and for interacting diffusions in d-dimensions, a novel fluctuation relation is derived. The theorem
establishes a quantitative relation between the probabilities of observing two current values in different spatial directions. The result is a consequence of
spatial symmetries of the microscopic dynamics, generalizing in this way the Gallavotti–Cohen fluctuation theorem related to the time-reversal symmetry.
This new perspective opens up the possibility of direct experimental measurements of fluctuation relations of vectorial observables
Orthogonal Dualities of Markov Processes and Unitary Symmetries
No full textWe study self-duality for interacting particle systems, where the particles move as continuous time random walkers having either exclusion interaction or inclusion interaction. We show that orthogonal self-dualities arise from unitary symmetries of the Markov generator. For these symmetries we provide two equivalent expressions that are related by the Baker-Campbell-Hausdorff formula. The first expression is the exponential of an anti Hermitian operator and thus is unitary by inspection; the second expression is factorized into three terms and is proved to be unitary by using generating functions. The factorized form is also obtained by using an independent approach based on scalar products, which is a new method of independent interest that we introduce to derive (bi)orthogonal duality functions from non-orthogonal duality functions.AnalysisApplied Probabilit
Correlation Inequalities for Interacting Particle Systems with Duality
No full textContains fulltext :
83659.pdf (Publisher’s version ) (Closed access
A generalized asymmetric exclusion process with Uq(sl2) stochastic duality
No full textWe study a new process, which we call ASEP(q, j ), where particles move asymmetrically on a one-dimensional integer lattice with a bias determined by q ? (0, 1) and where at most 2 j ? N particles per site are allowed. The process is constructed from a (2 j + 1)-dimensional representation of a quantum Hamiltonian with Uq (sl2) invariance by applying a suitable ground-state transformation. After showing basic properties of the process ASEP(q, j ), we prove self-duality with several selfduality functions constructed from the symmetries of the quantum Hamiltonian. By making use of the self-duality property we compute the first q-exponential moment of the current for step initial conditions (both a shock or a rarefaction fan) as well as when the process is started from a homogeneous product measure.Delft Institute of Applied MathematicsElectrical Engineering, Mathematics and Computer Scienc
Dualities in population genetics: A fresh look with new dualities
No full textWe apply our general method of duality, introduced in [15], to models
of population dynamics. The classical dualities between forward
and ancestral processes can be viewed as a change of representation
in the classical creation and annihilation operators, both for diffusions
dual to coalescents of Kingman’s type, as well as for models with finite
population size.
Next, using SU(1, 1) raising and lowering operators, we find new
dualities between the Wright-Fisher diffusion with d types and the
Moran model, both in presence and absence of mutations. These new
dualities relates two forward evolutions. From our general scheme we
also identify self-duality of the Moran model
Non-criticality criteria for Abelian sandpile models with sources and sinks
No full textWe prove that the Abelian sandpile model on a random binary and binomial tree, as introduced in Redig, Ruszel, and Saada [J. Stat. Phys. 147, 653-677 (2012)], is not critical for all branching probabilities p < 1; by estimating the tail of the annealed survival time of a random walk on the binary tree with randomly placed traps, we obtain some more information about the exponential tail of the avalanche radius. Next we study the sandpile model on Zd with some additional dissipative sites: we provide examples and sufficient conditions for non-criticality; we also make a connection with the parabolic Anderson model. Finally we initiate the study of the sandpile model with both sources and sinks and give a sufficient condition for non-criticality in the presence of a finite number of sources, using a connection with the homogeneous pinning model.Applied Probabilit
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