Rigorous Probabilistic Analysis of Equilibrium Crystal Shapes

The rigorous microscopic theory of equilibrium crystal shapes has made enormous progress during the last decade. We review here the main results which have been obtained, both in two and higher dimensions. In particular, we describe how the phenomenological Wulff and Winterbottom constructions can be derived from the microscopic description provided by the equilibrium statistical mechanics of lattice gases. We focus on the main conceptual issues and describe the central ideas of the existing approaches.Comment: To appear in the March 2000 special issue of Journal of Mathematical Physics on Probabilistic Methods in Statistical Physic

Inflammatory skin diseases and the risk of chronic kidney disease: population-based case-control and cohort analyses.

BACKGROUND: Emerging evidence suggests an association between common inflammatory skin diseases and chronic kidney disease (CKD). OBJECTIVES: To explore the association between CKD stages 3-5 (CKD3-5) and atopic eczema, psoriasis, rosacea and hidradenitis suppurativa. METHODS: We undertook two complementary analyses; a prevalent case-control study and a cohort study using routinely collected primary care data [UK Clinical Practice Research Datalink (CPRD)]. We matched individuals with CKD3-5 in CPRD in March 2018 with up to five individuals without CKD for general practitioner practice, age and sex. We compared the prevalence of CKD3-5 among individuals with and without each inflammatory skin disease. We included individuals in CPRD with diabetes mellitus (2004-2018) in a cohort analysis to compare the incidence of CKD3-5 among people with and without atopic eczema and psoriasis. RESULTS: Our study included 56 602 cases with CKD3-5 and 268 305 controls. Cases were more likely than controls to have a history of atopic eczema [odds ratio (OR) 1·14, 99% confidence interval (CI) 1·11-1·17], psoriasis (OR 1·13, 99% CI 1·08-1·19) or hidradenitis suppurativa (OR 1·49, 99% CI 1·19-1·85), but were slightly less likely to have been diagnosed with rosacea (OR 0·92, 99% CI 0·87-0·97), after adjusting for age, sex, practice (matching factors), index of multiple deprivation, diabetes, smoking, harmful alcohol use and obesity. Results remained similar after adjusting for hypertension and cardiovascular disease. In the cohort with diabetes (N = 335 827), there was no evidence that CKD3-5 incidence was associated with atopic eczema or psoriasis. CONCLUSIONS: Atopic eczema, psoriasis and hidradenitis suppurativa are weakly associated with CKD3-5. Future research is needed to elucidate potential mechanisms and the clinical significance of our findings

The Alexander-Orbach conjecture holds in high dimensions

We examine the incipient infinite cluster (IIC) of critical percolation in regimes where mean-field behavior has been established, namely when the dimension d is large enough or when d>6 and the lattice is sufficiently spread out. We find that random walk on the IIC exhibits anomalous diffusion with the spectral dimension d_s=4/3, that is, p_t(x,x)= t^{-2/3+o(1)}. This establishes a conjecture of Alexander and Orbach. En route we calculate the one-arm exponent with respect to the intrinsic distance.Comment: 25 pages, 2 figures. To appear in Inventiones Mathematica

Cooperative Behavior of Kinetically Constrained Lattice Gas Models of Glassy Dynamics

Kinetically constrained lattice models of glasses introduced by Kob and Andersen (KA) are analyzed. It is proved that only two behaviors are possible on hypercubic lattices: either ergodicity at all densities or trivial non-ergodicity, depending on the constraint parameter and the dimensionality. But in the ergodic cases, the dynamics is shown to be intrinsically cooperative at high densities giving rise to glassy dynamics as observed in simulations. The cooperativity is characterized by two length scales whose behavior controls finite-size effects: these are essential for interpreting simulations. In contrast to hypercubic lattices, on Bethe lattices KA models undergo a dynamical (jamming) phase transition at a critical density: this is characterized by diverging time and length scales and a discontinuous jump in the long-time limit of the density autocorrelation function. By analyzing generalized Bethe lattices (with loops) that interpolate between hypercubic lattices and standard Bethe lattices, the crossover between the dynamical transition that exists on these lattices and its absence in the hypercubic lattice limit is explored. Contact with earlier results are made via analysis of the related Fredrickson-Andersen models, followed by brief discussions of universality, of other approaches to glass transitions, and of some issues relevant for experiments.Comment: 59 page

Phase Transitions on Nonamenable Graphs

We survey known results about phase transitions in various models of statistical physics when the underlying space is a nonamenable graph. Most attention is devoted to transitive graphs and trees

Metastable lifetimes in a kinetic Ising model: Dependence on field and system size

The lifetimes of metastable states in kinetic Ising ferromagnets are studied by droplet theory and Monte Carlo simulation, in order to determine their dependences on applied field and system size. For a wide range of fields, the dominant field dependence is universal for local dynamics and has the form of an exponential in the inverse field, modified by universal and nonuniversal power-law prefactors. Quantitative droplet-theory predictions are numerically verified, and small deviations are shown to depend nonuniversally on the details of the dynamics. We identify four distinct field intervals in which the field dependence and statistical properties of the lifetimes are different. The field marking the crossover between the weak-field regime, in which the decay is dominated by a single droplet, and the intermediate-field regime, in which it is dominated by a finite droplet density, vanishes logarithmically with system size. As a consequence the slow decay characteristic of the former regime may be observable in systems that are macroscopic as far as their equilibrium properties are concerned.Comment: 18 pages single spaced. RevTex Version 3. FSU-SCRI-94-1

Glauber Dynamics for the mean-field Potts Model

We study Glauber dynamics for the mean-field (Curie-Weiss) Potts model with $q\geq 3$ states and show that it undergoes a critical slowdown at an inverse-temperature $\beta_s(q)$ strictly lower than the critical $\beta_c(q)$ for uniqueness of the thermodynamic limit. The dynamical critical $\beta_s(q)$ is the spinodal point marking the onset of metastability. We prove that when $\beta<\beta_s(q)$ the mixing time is asymptotically $C(\beta, q) n \log n$ and the dynamics exhibits the cutoff phenomena, a sharp transition in mixing, with a window of order $n$. At $\beta=\beta_s(q)$ the dynamics no longer exhibits cutoff and its mixing obeys a power-law of order $n^{4/3}$. For $\beta>\beta_s(q)$ the mixing time is exponentially large in $n$. Furthermore, as $\beta \uparrow \beta_s$ with $n$, the mixing time interpolates smoothly from subcritical to critical behavior, with the latter reached at a scaling window of $O(n^{-2/3})$ around $\beta_s$. These results form the first complete analysis of mixing around the critical dynamical temperature --- including the critical power law --- for a model with a first order phase transition.Comment: 45 pages, 5 figure

Regularity Properties and Pathologies of Position-Space Renormalization-Group Transformations

We reconsider the conceptual foundations of the renormalization-group (RG) formalism, and prove some rigorous theorems on the regularity properties and possible pathologies of the RG map. Regarding regularity, we show that the RG map, defined on a suitable space of interactions (= formal Hamiltonians), is always single-valued and Lipschitz continuous on its domain of definition. This rules out a recently proposed scenario for the RG description of first-order phase transitions. On the pathological side, we make rigorous some arguments of Griffiths, Pearce and Israel, and prove in several cases that the renormalized measure is not a Gibbs measure for any reasonable interaction. This means that the RG map is ill-defined, and that the conventional RG description of first-order phase transitions is not universally valid. For decimation or Kadanoff transformations applied to the Ising model in dimension $d \ge 3$, these pathologies occur in a full neighborhood $\{ \beta > \beta_0 ,\, |h| < \epsilon(\beta) \}$ of the low-temperature part of the first-order phase-transition surface. For block-averaging transformations applied to the Ising model in dimension $d \ge 2$, the pathologies occur at low temperatures for arbitrary magnetic-field strength. Pathologies may also occur in the critical region for Ising models in dimension $d \ge 4$. We discuss in detail the distinction between Gibbsian and non-Gibbsian measures, and give a rather complete catalogue of the known examples. Finally, we discuss the heuristic and numerical evidence on RG pathologies in the light of our rigorous theorems.Comment: 273 pages including 14 figures, Postscript, See also ftp.scri.fsu.edu:hep-lat/papers/9210/9210032.ps.

Atopic eczema in adulthood and risk of depression and anxiety: a population-based cohort study

BACKGROUND: Atopic eczema is a common and debilitating condition associated with depression and anxiety, but the nature of this association remains unclear. // OBJECTIVE: To explore the temporal relationship between atopic eczema and new depression/anxiety. // METHODS: A matched cohort study using routinely-collected data from the UK Clinical Practice Research Datalink, linked to hospital admissions data. We identified adults with atopic eczema (1998-2016) using a validated algorithm, and up to five individuals without atopic eczema matched on date of diagnosis, age, sex and general practice. We estimated the hazard ratio (HR) for new depression/anxiety using stratified Cox regression to account for age, sex, calendar period, Index of Multiple Deprivation, glucocorticoid treatment, obesity, smoking and harmful alcohol use. // RESULTS: We identified 526,808 adults with atopic eczema who were matched to 2,569,030 without. Atopic eczema was associated with increased incidence of new depression (HR 1.14; 99% confidence interval [CI] 1.12-1.16), and anxiety (HR 1.17; 99% CI 1.14-1.19). We observed a stronger effect of atopic eczema on depression with increasing atopic eczema severity (HR [99% CI] compared to no atopic eczema: mild 1.10 [1.08-1.13]; moderate 1.19 [1.15-1.23]; severe 1.26 [1.17-1.37]). A dose-response association, however, was less apparent for new anxiety diagnosis (HR [99% CI] compared to no atopic eczema: mild 1.14 [1.11-1.18]; moderate 1.21 [1.17-1.26]; severe 1.15; [1.05-1.25]). // CONCLUSIONS: Adults with atopic eczema are more likely to develop new depression and anxiety. For depression, we observed a dose-response relationship with atopic eczema severity