Spanning forests and the q-state Potts model in the limit q \to 0

We study the q-state Potts model with nearest-neighbor coupling v=e^{\beta J}-1 in the limit q,v \to 0 with the ratio w = v/q held fixed. Combinatorially, this limit gives rise to the generating polynomial of spanning forests; physically, it provides information about the Potts-model phase diagram in the neighborhood of (q,v) = (0,0). We have studied this model on the square and triangular lattices, using a transfer-matrix approach at both real and complex values of w. For both lattices, we have computed the symbolic transfer matrices for cylindrical strips of widths 2 \le L \le 10, as well as the limiting curves of partition-function zeros in the complex w-plane. For real w, we find two distinct phases separated by a transition point w=w_0, where w_0 = -1/4 (resp. w_0 = -0.1753 \pm 0.0002) for the square (resp. triangular) lattice. For w > w_0 we find a non-critical disordered phase, while for w < w_0 our results are compatible with a massless Berker-Kadanoff phase with conformal charge c = -2 and leading thermal scaling dimension x_{T,1} = 2 (marginal operator). At w = w_0 we find a "first-order critical point": the first derivative of the free energy is discontinuous at w_0, while the correlation length diverges as w \downarrow w_0 (and is infinite at w = w_0). The critical behavior at w = w_0 seems to be the same for both lattices and it differs from that of the Berker-Kadanoff phase: our results suggest that the conformal charge is c = -1, the leading thermal scaling dimension is x_{T,1} = 0, and the critical exponents are \nu = 1/d = 1/2 and \alpha = 1.Comment: 131 pages (LaTeX2e). Includes tex file, three sty files, and 65 Postscript figures. Also included are Mathematica files forests_sq_2-9P.m and forests_tri_2-9P.m. Final journal versio

Rate and severity of radiological features of physical abuse in children during the first UK-wide COVID-19 enforced national lockdown

Rate and severity of radiological features of physical abuse in children during the first UK-wide COVID-19 enforced national lockdown. Objective: To assess the number, type and outcome of radiological investigations for children presenting to hospital with suspected physical abuse (SPA; including abusive head trauma) during the first national COVID-19 enforced lockdown compared with the prelockdown period. Design: Multicentre, retrospective, observational, interrupted time series analysis. Setting: Eight secondary/tertiary paediatric centres between January 2018 and July 2020 inclusive. Participants: 1587 hospital assessed children undergoing radiographic skeletal surveys (SkS) and head CT imaging performed for SPA/child protection concerns. Main outcome measures: Incidence and severity of fractures identified on SkS; head injury (composed of incidence rates and ratios of skull fracture, intracranial haemorrhage (ICH) and hypoxic ischaemic injury (HII)) on head CT imaging; and ratio of antemortem and postmortem SkS. Results: 1587 SkS were performed: 1282 (81%) antemortem, 762 (48%) male, and positive findings in 582 (37%). Median patient age was 6 months. There were 1.7 fractures/child prelockdown versus 1.1 fractures/child during lockdown. There was no difference between positive/negative SkS rates, the absolute ratio of antemortem/postmortem SkS or absolute numbers of head injury occurring between January 2018 and February 2020 and the lockdown period April–July 2020. Likewise, prelockdown incidence and rates of skull fracture 30/244 (12%), ICH 28/220 (13%) and HIE 10/205 (5%) were similar to lockdown, 142/1304 (11%), 171/1152 (15%) and 68/1089 (6%), respectively. Conclusion: The first UK COVID-19 lockdown did not lead to an increase in either the number of antemortem or postmortem radiological investigations performed for SPA, or the number or severity of fractures and intracranial injuries identified by these investigations