Limits of permutation sequences

A permutation sequence is said to be convergent if the density of occurrences of every fixed permutation in the elements of the sequence converges. We prove that such a convergent sequence has a natural limit object, namely a Lebesgue measurable function $Z:[0,1]^2 \to [0,1]$ with the additional properties that, for every fixed $x \in [0,1]$, the restriction $Z(x,\cdot)$ is a cumulative distribution function and, for every $y \in [0,1]$, the restriction $Z(\cdot,y)$ satisfies a "mass" condition. This limit process is well-behaved: every function in the class of limit objects is a limit of some permutation sequence, and two of these functions are limits of the same sequence if and only if they are equal almost everywhere. An ingredient in the proofs is a new model of random permutations, which generalizes previous models and might be interesting for its own sake.Comment: accepted for publication in the Journal of Combinatorial Theory, Series B. arXiv admin note: text overlap with arXiv:1106.166

Surface tension in the dilute Ising model. The Wulff construction

We study the surface tension and the phenomenon of phase coexistence for the Ising model on \mathbbm{Z}^d ($d \geqslant 2$) with ferromagnetic but random couplings. We prove the convergence in probability (with respect to random couplings) of surface tension and analyze its large deviations : upper deviations occur at volume order while lower deviations occur at surface order. We study the asymptotics of surface tension at low temperatures and relate the quenched value $\tau^q$ of surface tension to maximal flows (first passage times if $d = 2$). For a broad class of distributions of the couplings we show that the inequality $\tau^a \leqslant \tau^q$ -- where $\tau^a$ is the surface tension under the averaged Gibbs measure -- is strict at low temperatures. We also describe the phenomenon of phase coexistence in the dilute Ising model and discuss some of the consequences of the media randomness. All of our results hold as well for the dilute Potts and random cluster models

Fourier transform of self-affine measures

Suppose $F$ is a self-affine set on $\mathbb{R}^d$, $d\geq 2$, which is not a singleton, associated to affine contractions $f_j = A_j + b_j$, $A_j \in \mathrm{GL}(d,\mathbb{R})$, $b_j \in \mathbb{R}^d$, $j \in \mathcal{A}$, for some finite $\mathcal{A}$. We prove that if the group $\Gamma$ generated by the matrices $A_j$, $j \in \mathcal{A}$, forms a proximal and totally irreducible subgroup of $\mathrm{GL}(d,\mathbb{R})$, then any self-affine measure $\mu = \sum p_j f_j \mu$, $\sum p_j = 1$, $0 < p_j < 1$, $j \in \mathcal{A}$, on $F$ is a Rajchman measure: the Fourier transform $\widehat{\mu}(\xi) \to 0$ as $|\xi| \to \infty$. As an application this shows that self-affine sets with proximal and totally irreducible linear parts are sets of rectangular multiplicity for multiple trigonometric series. Moreover, if the Zariski closure of $\Gamma$ is connected real split Lie group in the Zariski topology, then $\widehat{\mu}(\xi)$ has a power decay at infinity. Hence $\mu$ is $L^p$ improving for all $1 < p < \infty$ and $F$ has positive Fourier dimension. In dimension $d = 2,3$ the irreducibility of $\Gamma$ and non-compactness of the image of $\Gamma$ in $\mathrm{PGL}(d,\mathbb{R})$ is enough for power decay of $\widehat{\mu}$. The proof is based on quantitative renewal theorems for random walks on the sphere $\mathbb{S}^{d-1}$.Comment: v2: 27 pages, updated references. Accepted to Advances in Mat

Non-homogeneous polygonal Markov fields in the plane: graphical representations and geometry of higher order correlations

We consider polygonal Markov fields originally introduced by Arak and Surgailis (1989). Our attention is focused on fields with nodes of order two, which can be regarded as continuum ensembles of non-intersecting contours in the plane, sharing a number of features with the two-dimensional Ising model. We introduce non-homogeneous version of polygonal fields in anisotropic enviroment. For these fields we provide a class of new graphical constructions and random dynamics. These include a generalised dynamic representation, generalised and defective disagreement loop dynamics as well as a generalised contour birth and death dynamics. Next, we use these constructions as tools to obtain new exact results on the geometry of higher order correlations of polygonal Markov fields in their consistent regime.Comment: 54 page

A Note on Kuhn's Theorem with Ambiguity Averse Players

Kuhn's Theorem shows that extensive games with perfect recall can equivalently be analyzed using mixed or behavioral strategies, as long as players are expected utility maximizers. This note constructs an example that illustrate the limits of Kuhn's Theorem in an environment with ambiguity averse players who use maxmin decision rule and full Bayesian updating.Comment: 7 figure