We study random integer-valued Lipschitz functions on regular trees. It was shown by Peled, Samotij, and Yehudayoff [22] that such functions are localized; however, finer questions about the structure of Gibbs measures remain unanswered. Our main result is that the weak limit of a uniformly chosen 1-Lipschitz function with 0 boundary condition on a $d$-ary tree of height $n$ exists as $n \to \infty$ if $2 \le d \le 7$, but not if $d \ge 8$, thereby partially answering a question posed by Peled, Samotij and Yehudayoff. For large $d$, the value at the root alternates between being almost entirely concentrated on 0 for even $n$ and being roughly uniform on $\{-1,0,1\}$ for odd $n$, leading to different limits as $n$ approaches infinity along evens or odds. For $d \ge 8$, the essence of this phenomenon is preserved, which obstructs the convergence. For $d \le 7$, this phenomenon ceases to exist, and the law of the value at the root loses its connection with the parity of $n$. Along the way, we also obtain an alternative proof of localization. The key idea is a fixed point convergence result for a related operator on $\ell ^\infty$ and a procedure to show that the iterations get into a ‘basin of attraction’ of the fixed point. We also prove some accompanying analogous ‘even-odd phenomenon’ type results about $M$-Lipschitz functions on general non-amenable graphs with high enough expansion (this includes for example the large $d$ case for regular trees). We also prove a convergence result for 1-Lipschitz functions with $\{0,1\}$ boundary condition. This last result relies on an absolute value FKG for uniform 1-Lipschitz functions when shifted by $1/2$.

This article examines the ways in which explanation has been achieved in scientific work on language change over the last two hundred years. Explanations have come in many forms and at many levels and are greatly influenced by what are taken as the leading questions, which themselves have varied significantly since the early nineteenth century.

We model voting behaviour in the multi-group setting of a two-tier voting system using sequences of de Finetti measures. Our model is defined by using the de Finetti representation of a probability measure (i.e. as a mixture of conditionally independent probability measures) describing voting behaviour. The de Finetti measure describes the interaction between voters and possible outside influences on them. We assume that for each population size there is a (potentially) different de Finetti measure, and as the population grows, the sequence of de Finetti measures converges weakly to the Dirac measure at the origin, representing a tendency toward weakening social cohesion as the population grows large. The resulting model covers a wide variety of behaviours, ranging from independent voting in the limit under fast convergence, a critical convergence speed with its own pattern of behaviour, to a subcritical convergence speed which yields a model in line with empirical evidence of real-world voting data, contrary to previous probabilistic models used in the study of voting. These models can be used, e.g., to study the problem of optimal voting weights in two-tier voting systems.

A 4-component clay-polymer-salt-water system was studied by neutron scattering. The clay-salt-water system consisted of n-butylammonium vermiculite, n-butylammonium chloride and heavy water, and the volume fraction of clay in the system was held constant, at r = 0.01. Three polymers in the molecular weight range 10,000 to 30,000 were studied, poly(vinyl methyl ether) (PVME), poly (ethylene oxide) (PEO) and poly(acrylic acid) (PAA), at a polymer volume fraction of v = 0.01. The addition of PAA suppressed the clay swelling, irrespective of the salt concentration, c. The addition of the neutral polymers had no effect on the phase transition temperature, Tc, between the gel and tactoid phases of the system, its value remaining at 14 °C for c = 0.1 M and 30 °C for c = 0.01 M. At c = 0.01 M, the neutral polymers also had a negligible effect on the lattice constant d along the swelling axis of the clay colloid, but at c = 0.1 M, the d-value was significantly lower than in the system without added polymer. For a PVME sample of molecular weight 18,000, both d and Tc were measured as a function of ν, for volume fractions between 0 and 0.04. The addition of polymer, up to v = 0.04, had no effect on Tc. However, even for v values as low as 0.001, the vermiculite layers in the gel phase were more parallel and more regularly spaced than in the system without added polymer. In the gel phase, d decreased exponentially as a function of v, from 12 nm at v = 0 to 8 nm at v = 0.04. In the tactoid phase, at T < 14 °C, the d-value in the crystalline regions was equal to 1.94 nm at v = 0 and v = 0.04, showing that the spacing between the vermiculite layers is not affected by the added polymer when they are collapsed by an increase in temperature. The addition of a PVME sample of molecular weight 110,000, at v = 0.001, had no noticeable effect on either d or Tc.

The swelling of n-butylammonium vermiculite in water was investigated as a function of the sol concentration (r), the salt concentration (c) and the temperature (T).

The interlayer spacing in the gel phase was investigated as a function of r and c by neutron diffraction and by laboratory experiments which measured how many times its own volume a crystal would absorb. The salt concentration was found to be the stronger variable with the interlayer spacing decreasing proportional to c0.5, which is consistent with previous results and with the Coulombic attraction theory. The sol concentration was found to affect the swelling for two reasons, the salt fractionation effect and the trapped salt effect. Both of these cause the salt concentration in the supernatant fluid to be greater than that originally added to the crystals and so reduce the swelling.

A new method was used for extracting the solution from inside the gels by collapsing the gels by the addition of potassium hydrogen carbonate. The Volhardt titration was carried out on the extracted and supernatant solutions from about 250 gels. The ratio of the external to the internal chloride concentration was found to be approximately constant across the range of salt concentrations. Its average value was equal to 2.6, again in agreement with Coulombic attraction theory and showing the surface potential to be constant at about 70 mV.

The (r, c, T) boundary of the two phase colloid region was investigated by three methods. A plot of log c against Tc was linear within experimental accuracy, with a gradient of 0.077 K−1 or 13 K per log unit. This shows that the surface potential varies by only 1 mV per decade in the salt concentration. The system is therefore governed by the Dirichlet boundary condition and not by the Nernst equation.

We consider a model of binary opinion dynamics where one opinion is inherently “superior” than the other, and social agents exhibit a “bias” toward the superior alternative. Specifically, it is assumed that an agent updates its choice to the superior alternative with probability α > 0 irrespective of its current opinion and opinions of other agents. With probability $1-\alpha$, it adopts majority opinion among two randomly sampled neighbors and itself. We are interested in the time it takes for the network to converge to a consensus on the superior alternative. In a complete graph of size n, we show that irrespective of the initial configuration of the network, the average time to reach consensus scales as $\Theta(n\,\log n)$ when the bias parameter α is sufficiently high, that is, $\alpha \gt \alpha_c$ where αc is a threshold parameter that is uniquely characterized. When the bias is low, that is, when $\alpha \in (0,\alpha_c]$, we show that the same rate of convergence can only be achieved if the initial proportion of agents with the superior opinion is above certain threshold $p_c(\alpha)$. If this is not the case, then we show that the network takes $\Omega(\exp(\Theta(n)))$ time on average to reach consensus.

The Ginzburg–Landau functional is a phase transition model which is suitable for classification type problems. We study the asymptotics of a sequence of Ginzburg–Landau functionals with anisotropic interaction potentials on point clouds Ψn where n denotes the number data points. In particular, we show the limiting problem, in the sense of Γ-convergence, is related to the total variation norm restricted to functions taking binary values, which can be understood as a surface energy. We generalize the result known for isotropic interaction potentials to the anisotropic case and add a result concerning the rate of convergence.

The consistency of a non-local anisotropic Ginzburg–Landau type functional for data classification and clustering is studied. The Ginzburg–Landau objective functional combines a double well potential, that favours indicator valued functions, and the p-Laplacian, that enforces regularity. Under appropriate scaling between the two terms, minimisers exhibit a phase transition on the order of ɛ = ɛn, where n is the number of data points. We study the large data asymptotics, i.e. as n → ∝, in the regime where ɛn → 0. The mathematical tool used to address this question is Γ-convergence. It is proved that the discrete model converges to a weighted anisotropic perimeter.