35 research outputs found
One-sided versus two-sided stochastic descriptions
It is well-known that discrete-time finite-state Markov Chains, which are
described by one-sided conditional probabilities which describe a dependence on
the past as only dependent on the present, can also be described as
one-dimensional Markov Fields, that is, nearest-neighbour Gibbs measures for
finite-spin models, which are described by two-sided conditional probabilities.
In such Markov Fields the time interpretation of past and future is being
replaced by the space interpretation of an interior volume, surrounded by an
exterior to the left and to the right.
If we relax the Markov requirement to weak dependence, that is, continuous
dependence, either on the past (generalising the Markov-Chain description) or
on the external configuration (generalising the Markov-Field description), it
turns out this equivalence breaks down, and neither class contains the other.
In one direction this result has been known for a few years, in the opposite
direction a counterexample was found recently. Our counterexample is based on
the phenomenon of entropic repulsion in long-range Ising (or "Dyson") models.Comment: 13 pages, Contribution for "Statistical Mechanics of Classical and
Disordered Systems
On the convergence of cluster expansions for polymer gases
We compare the different convergence criteria available for cluster
expansions of polymer gases subjected to hard-core exclusions, with emphasis on
polymers defined as finite subsets of a countable set (e.g. contour expansions
and more generally high- and low-temperature expansions). In order of
increasing strength, these criteria are: (i) Dobrushin criterion, obtained by a
simple inductive argument; (ii) Gruber-Kunz criterion obtained through the use
of Kirkwood-Salzburg equations, and (iii) a criterion obtained by two of us via
a direct combinatorial handling of the terms of the expansion. We show that for
subset polymers our sharper criterion can be proven both by a suitable
adaptation of Dobrushin inductive argument and by an alternative --in fact,
more elementary-- handling of the Kirkwood-Salzburg equations. In addition we
show that for general abstract polymers this alternative treatment leads to the
same convergence region as the inductive Dobrushin argument and, furthermore,
to a systematic way to improve bounds on correlations
The Analyticity of a Generalized Ruelle's Operator
In this work we propose a generalization of the concept of Ruelle operator
for one dimensional lattices used in thermodynamic formalism and ergodic
optimization, which we call generalized Ruelle operator, that generalizes both
the Ruelle operator proposed in [BCLMS] and the Perron Frobenius operator
defined in [Bowen]. We suppose the alphabet is given by a compact metric space,
and consider a general a-priori measure to define the operator. We also
consider the case where the set of symbols that can follow a given symbol of
the alphabet depends on such symbol, which is an extension of the original
concept of transition matrices from the theory of subshifts of finite type. We
prove the analyticity of the Ruelle operator and present some examples
Cluster and virial expansions for the multi-species tonks gas
We consider a mixture of non-overlapping rods of different lengths ℓk moving in R or Z. Our main result are necessary and sufficient convergence criteria for the expansion of the pressure in terms of the activities zk and the densities ρk. This provides an explicit example against which to test known cluster expansion criteria, and illustrates that for non-negative interactions, the virial expansion can converge in a domain much larger than the activity expansion. In addition, we give explicit formulas that generalize the well-known relation between non-overlapping rods and labelled rooted trees. We also prove that for certain choices of the activities, the system can undergo a condensation transition akin to that of the zero-range process. The key tool is a fixed point equation for the pressure
Duality Theorems in Ergodic Transport
We analyze several problems of Optimal Transport Theory in the setting of
Ergodic Theory. In a certain class of problems we consider questions in Ergodic
Transport which are generalizations of the ones in Ergodic Optimization.
Another class of problems is the following: suppose is the shift
acting on Bernoulli space , and, consider a fixed
continuous cost function . Denote by the set
of all Borel probabilities on , such that, both its and
marginal are -invariant probabilities. We are interested in the
optimal plan which minimizes among the probabilities on
.
We show, among other things, the analogous Kantorovich Duality Theorem. We
also analyze uniqueness of the optimal plan under generic assumptions on .
We investigate the existence of a dual pair of Lipschitz functions which
realizes the present dual Kantorovich problem under the assumption that the
cost is Lipschitz continuous. For continuous costs the corresponding
results in the Classical Transport Theory and in Ergodic Transport Theory can
be, eventually, different.
We also consider the problem of approximating the optimal plan by
convex combinations of plans such that the support projects in periodic orbits
Convergence of density expansions of correlation functions and the Ornstein-Zernike equation
We prove absolute convergence of the multi-body correlation functions as a power series in the density uniformly in their arguments. This is done by working in the context of the cluster expansion in the canonical ensemble and by expressing the correlation functions as the derivative of the logarithm of an appropriately extended partition function. In the thermodynamic limit, due to combinatorial cancellations, we show that the coeffi- cients of the above series are expressed by sums over some class of two-connected graphs. Furthermore, we prove the convergence of the density expansion of the “direct correlation function” which is based on a completely different approach and it is valid only for some inte- gral norm. Precisely, this integral norm is suitable to derive the Ornstein-Zernike equation. As a further outcome, we obtain a rigorous quantification of the error in the Percus-Yevick approximation
An improvement of the Lovász local lemma via cluster expansion
An old result by Shearer relates the Lov´asz Local Lemma with the independent set
polynomial on graphs, and consequently, as observed by Scott and Sokal, with the partition
function of the hard core lattice gas on graphs. We use this connection and a recent result on
the analyticity of the logarithm of the partition function of the abstract polymer gas to get
an improved version of the Lov´asz Local Lemma. As applications we obtain tighter bounds
on conditions for the existence of latin transversal matrices and the satisfiability of k-SAT
forms
Sufficient Conditions for Uniform Bounds in Abstract Polymer Systems and Explorative Partition Schemes
We present several new sufficient conditions for uniform boundedness of the reduced correlations and free energy of an abstract polymer system in a complex multidisc around zero fugacity. They resolve a discrepancy between two incomparable and previously known extensions of Dobrushin’s classic condition. All conditions arise from an extension of the tree-operator approach introduced by Fernández and Procacci combined with a novel family of partition schemes of the spanning subgraph complex of a cluster. The key technique is the increased transfer of structural information from the partition scheme to a tree-operator on an enhanced space