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We show that the
$4$
-state anti-ferromagnetic Potts model with interaction parameter
$w\in (0,1)$
on the infinite
$(d+1)$
-regular tree has a unique Gibbs measure if
$w\geq 1-\dfrac{4}{d+1_{_{\;}}}$
for all
$d\geq 4$
. This is tight since it is known that there are multiple Gibbs measures when
$0\leq w\lt 1-\dfrac{4}{d+1}$
and
$d\geq 4$
. We moreover give a new proof of the uniqueness of the Gibbs measure for the
$3$
-state Potts model on the
$(d+1)$
-regular tree for
$w\geq 1-\dfrac{3}{d+1}$
when
$d\geq 3$
and for
$w\in (0,1)$
when
$d=2$
.
We investigate expansions for connectedness functions in the random connection model of continuum percolation in powers of the intensity. Precisely, we study the pair-connectedness and the direct-connectedness functions, related to each other via the Ornstein–Zernike equation. We exhibit the fact that the coefficients of the expansions consist of sums over connected and 2-connected graphs. In the physics literature, this is known to be the case more generally for percolation models based on Gibbs point processes and stands in analogy to the formalism developed for correlation functions in liquid-state statistical mechanics.
We find a representation of the direct-connectedness function and bounds on the intensity which allow us to pass to the thermodynamic limit. In some cases (e.g., in high dimensions), the results are valid in almost the entire subcritical regime. Moreover, we relate these expansions to the physics literature and we show how they coincide with the expression provided by the lace expansion.
We show that the diameter of a uniformly drawn spanning tree of a simple connected graph on n vertices with minimal degree linear in n is typically of order
$\sqrt{n}$
. A byproduct of our proof, which is of independent interest, is that on such graphs the Cheeger constant and the spectral gap are comparable.
In this text, we provide a fully rigorous and complete proof of E.H. Lieb’s statement that (topological) entropy of square ice (or six-vertex model, XXZ spin chain for anisotropy parameter $\Delta =1/2$) is equal to $\tfrac 32\log _{2} (4/3)$.
Signal-to-interference-plus-noise ratio (SINR) percolation is an infinite-range dependent variant of continuum percolation modeling connections in a telecommunication network. Unlike in earlier works, in the present paper the transmitted signal powers of the devices of the network are assumed random, independent and identically distributed, and possibly unbounded. Additionally, we assume that the devices form a stationary Cox point process, i.e., a Poisson point process with stationary random intensity measure, in two or more dimensions. We present the following main results. First, under suitable moment conditions on the signal powers and the intensity measure, there is percolation in the SINR graph given that the device density is high and interferences are sufficiently reduced, but not vanishing. Second, if the interference cancellation factor
$\gamma$
and the SINR threshold
$\tau$
satisfy
$\gamma \geq 1/(2\tau)$
, then there is no percolation for any intensity parameter. Third, in the case of a Poisson point process with constant powers, for any intensity parameter that is supercritical for the underlying Gilbert graph, the SINR graph also percolates with some small but positive interference cancellation factor.
We determine the asymptotics of the number of independent sets of size
$\lfloor \beta 2^{d-1} \rfloor$
in the discrete hypercube
$Q_d = \{0,1\}^d$
for any fixed
$\beta \in (0,1)$
as
$d \to \infty$
, extending a result of Galvin for
$\beta \in (1-1/\sqrt{2},1)$
. Moreover, we prove a multivariate local central limit theorem for structural features of independent sets in
$Q_d$
drawn according to the hard-core model at any fixed fugacity
$\lambda>0$
. In proving these results we develop several general tools for performing combinatorial enumeration using polymer models and the cluster expansion from statistical physics along with local central limit theorems.
We construct coloured lattice models whose partition functions represent symplectic and odd orthogonal Demazure characters and atoms. We show that our lattice models are not solvable, but we are able to show the existence of sufficiently many solutions of the Yang–Baxter equation that allow us to compute functional equations for the corresponding partition functions. From these functional equations, we determine that the partition function of our models are the Demazure atoms and characters for the symplectic and odd orthogonal Lie groups. We coin our lattice models as quasi-solvable. We use the natural bijection of admissible states in our models with Proctor patterns to give a right key algorithm for reverse King tableaux and Sundaram tableaux.
We expand the critical point for site percolation on the d-dimensional hypercubic lattice in terms of inverse powers of 2d, and we obtain the first three terms rigorously. This is achieved using the lace expansion.
We reduce the upper bound for the bond percolation threshold of the cubic lattice from 0.447 792 to 0.347 297. The bound is obtained by a growth process approach which views the open cluster of a bond percolation model as a dynamic process. A three-dimensional dynamic process on the cubic lattice is constructed and then projected onto a carefully chosen plane to obtain a two-dimensional dynamic process on a triangular lattice. We compare the bond percolation models on the cubic lattice and their projections, and demonstrate that the bond percolation threshold of the cubic lattice is no greater than that of the triangular lattice. Applying the approach to the body-centered cubic lattice yields an upper bound of 0.292 893 for its bond percolation threshold.
Distinguishing between continuous and first-order phase transitions is a major challenge in random discrete systems. We study the topic for events with recursive structure on Galton–Watson trees. For example, let
$\mathcal{T}_1$
be the event that a Galton–Watson tree is infinite and let
$\mathcal{T}_2$
be the event that it contains an infinite binary tree starting from its root. These events satisfy similar recursive properties:
$\mathcal{T}_1$
holds if and only if
$\mathcal{T}_1$
holds for at least one of the trees initiated by children of the root, and
$\mathcal{T}_2$
holds if and only if
$\mathcal{T}_2$
holds for at least two of these trees. The probability of
$\mathcal{T}_1$
has a continuous phase transition, increasing from 0 when the mean of the child distribution increases above 1. On the other hand, the probability of
$\mathcal{T}_2$
has a first-order phase transition, jumping discontinuously to a non-zero value at criticality. Given the recursive property satisfied by the event, we describe the critical child distributions where a continuous phase transition takes place. In many cases, we also characterise the event undergoing the phase transition.
We present a polynomial-time Markov chain Monte Carlo algorithm for estimating the partition function of the antiferromagnetic Ising model on any line graph. The analysis of the algorithm exploits the ‘winding’ technology devised by McQuillan [CoRR abs/1301.2880 (2013)] and developed by Huang, Lu and Zhang [Proc. 27th Symp. on Disc. Algorithms (SODA16), 514–527]. We show that exact computation of the partition function is #P-hard, even for line graphs, indicating that an approximation algorithm is the best that can be expected. We also show that Glauber dynamics for the Ising model is rapidly mixing on line graphs, an example being the kagome lattice.
The Parisi formula is a self-contained description of the infinite-volume limit of the free energy of mean-field spin glass models. We showthat this quantity can be recast as the solution of a Hamilton–Jacobi equation in the Wasserstein space of probability measures on the positive half-line.
We establish an upper bound for the ground state energy per unit volume of a dilute Bose gas in the thermodynamic limit, capturing the correct second-order term, as predicted by the Lee–Huang–Yang formula. This result was first established in [20] by H.-T. Yau and J. Yin. Our proof, which applies to repulsive and compactly supported
$V \in L^3 (\mathbb {R}^3)$
, gives better rates and, in our opinion, is substantially simpler.
We consider the problem of computing the partition function
$\sum _x e^{f(x)}$
, where
$f: \{-1, 1\}^n \longrightarrow {\mathbb R}$
is a quadratic or cubic polynomial on the Boolean cube
$\{-1, 1\}^n$
. In the case of a quadratic polynomial f, we show that the partition function can be approximated within relative error
$0 < \epsilon < 1$
in quasi-polynomial
$n^{O(\ln n - \ln \epsilon )}$
time if the Lipschitz constant of the non-linear part of f with respect to the
$\ell ^1$
metric on the Boolean cube does not exceed
$1-\delta $
, for any
$\delta>0$
, fixed in advance. For a cubic polynomial f, we get the same result under a somewhat stronger condition. We apply the method of polynomial interpolation, for which we prove that
$\sum _x e^{\tilde {f}(x)} \ne 0$
for complex-valued polynomials
$\tilde {f}$
in a neighborhood of a real-valued f satisfying the above mentioned conditions. The bounds are asymptotically optimal. Results on the zero-free region are interpreted as the absence of a phase transition in the Lee–Yang sense in the corresponding Ising model. The novel feature of the bounds is that they control the total interaction of each vertex but not every single interaction of sets of vertices.
We prove that the energy of any eigenvector of a sum of several independent large Wigner matrices is equally distributed among these matrices with very high precision. This shows a particularly strong microcanonical form of the equipartition principle for quantum systems whose components are modelled by Wigner matrices.
We provide the first generic exact simulation algorithm for multivariate diffusions. Current exact sampling algorithms for diffusions require the existence of a transformation which can be used to reduce the sampling problem to the case of a constant diffusion matrix and a drift which is the gradient of some function. Such a transformation, called the Lamperti transformation, can be applied in general only in one dimension. So, completely different ideas are required for the exact sampling of generic multivariate diffusions. The development of these ideas is the main contribution of this paper. Our strategy combines techniques borrowed from the theory of rough paths, on the one hand, and multilevel Monte Carlo on the other.
Fractal percolation exhibits a dramatic topological phase transition, changing abruptly from a dust-like set to a system-spanning cluster. The transition points are unknown and difficult to estimate. In many classical percolation models the percolation thresholds have been approximated well using additive geometric functionals, known as intrinsic volumes. Motivated by the question of whether a similar approach is possible for fractal models, we introduce corresponding geometric functionals for the fractal percolation process F. They arise as limits of expected functionals of finite approximations of F. We establish the existence of these limit functionals and obtain explicit formulas for them as well as for their finite approximations.
We provide a new proof of the existence of Gibbs point processes with infinite range interactions, based on the compactness of entropy levels. Our main existence theorem holds under two assumptions. The first one is the standard stability assumption, which means that the energy of any finite configuration is superlinear with respect to the number of points. The second assumption is the so-called intensity regularity, which controls the long range of the interaction via the intensity of the process. This assumption is new and introduced here since it is well adapted to the entropy approach. As a corollary of our main result we improve the existence results by Ruelle (1970) for pairwise interactions by relaxing the superstabilty assumption. Note that our setting is not reduced to pairwise interaction and can contain infinite-range multi-body counterparts.
In this paper we propose a polynomial-time deterministic algorithm for approximately counting the k-colourings of the random graph G(n, d/n), for constant d>0. In particular, our algorithm computes in polynomial time a
$(1\pm n^{-\Omega(1)})$
-approximation of the so-called ‘free energy’ of the k-colourings of G(n, d/n), for
$k\geq (1+\varepsilon) d$
with probability
$1-o(1)$
over the graph instances.
Our algorithm uses spatial correlation decay to compute numerically estimates of marginals of the Gibbs distribution. Spatial correlation decay has been used in different counting schemes for deterministic counting. So far algorithms have exploited a certain kind of set-to-point correlation decay, e.g. the so-called Gibbs uniqueness. Here we deviate from this setting and exploit a point-to-point correlation decay. The spatial mixing requirement is that for a pair of vertices the correlation between their corresponding configurations becomes weaker with their distance.
Furthermore, our approach generalizes in that it allows us to compute the Gibbs marginals for small sets of nearby vertices. Also, we establish a connection between the fluctuations of the number of colourings of G(n, d/n) and the fluctuations of the number of short cycles and edges in the graph.