Distribution of sizes of erased loops for loop-erased random walks

We study the distribution of sizes of erased loops for loop-erased random walks on regular and fractal lattices. We show that for arbitrary graphs the probability $P(l)$ of generating a loop of perimeter $l$ is expressible in terms of the probability $P_{st}(l)$ of forming a loop of perimeter $l$ when a bond is added to a random spanning tree on the same graph by the simple relation $P(l)=P_{st}(l)/l$. On $d$-dimensional hypercubical lattices, $P(l)$ varies as $l^{-\sigma}$ for large $l$, where $\sigma=1+2/z$ for $1<d<4$, where z is the fractal dimension of the loop-erased walks on the graph. On recursively constructed fractals with $\tilde{d} < 2$ this relation is modified to $\sigma=1+2\bar{d}/{(\tilde{d}z)}$, where $\bar{d}$ is the hausdorff and $\tilde{d}$ is the spectral dimension of the fractal.Comment: 4 pages, RevTex, 3 figure

Boundary conditions and defect lines in the Abelian sandpile model

We add a defect line of dissipation, or crack, to the Abelian sandpile model. We find that the defect line renormalizes to separate the two-dimensional plane into two half planes with open boundary conditions. We also show that varying the amount of dissipation at a boundary of the Abelian sandpile model does not affect the universality class of the boundary condition. We demonstrate that a universal coefficient associated with height probabilities near the defect can be used to classify boundary conditions.Comment: 8 pages, 1 figure; suggestions from referees incorporated; to be published in Phys. Rev.

Self-supervised Contrastive Video-Speech Representation Learning for Ultrasound

In medical imaging, manual annotations can be expensive to acquire and sometimes infeasible to access, making conventional deep learning-based models difficult to scale. As a result, it would be beneficial if useful representations could be derived from raw data without the need for manual annotations. In this paper, we propose to address the problem of self-supervised representation learning with multi-modal ultrasound video-speech raw data. For this case, we assume that there is a high correlation between the ultrasound video and the corresponding narrative speech audio of the sonographer. In order to learn meaningful representations, the model needs to identify such correlation and at the same time understand the underlying anatomical features. We designed a framework to model the correspondence between video and audio without any kind of human annotations. Within this framework, we introduce cross-modal contrastive learning and an affinity-aware self-paced learning scheme to enhance correlation modelling. Experimental evaluations on multi-modal fetal ultrasound video and audio show that the proposed approach is able to learn strong representations and transfers well to downstream tasks of standard plane detection and eye-gaze prediction.Comment: MICCAI 2020 (early acceptance

Lack of consensus in social systems

We propose an exactly solvable model for the dynamics of voters in a two-party system. The opinion formation process is modeled on a random network of agents. The dynamical nature of interpersonal relations is also reflected in the model, as the connections in the network evolve with the dynamics of the voters. In the infinite time limit, an exact solution predicts the emergence of consensus, for arbitrary initial conditions. However, before consensus is reached, two different metastable states can persist for exponentially long times. One state reflects a perfect balancing of opinions, the other reflects a completely static situation. An estimate of the associated lifetimes suggests that lack of consensus is typical for large systems.Comment: 4 pages, 6 figures, submitted to Phys. Rev. Let

Elementary derivation of Spitzer's asymptotic law for Brownian windings and some of its physical applications

A simple derivation of Spitzer'z asymptotic law for Brownian windings [Trans.Am.Math.Soc.87,187 (1958)]is presented along with its generalizations >.These include the cases of planar Brownian walks interacting with a single puncture and Brownian walks on a single truncated cone with variable conical angle interacting with the truncated conical tip.Such situations are typical in the theories of quantum Hall effect and 2+1 quantum gravity, respectively .They also have some applications in polymer physic

The Sydney Multisite Intervention of LaughterBosses and ElderClowns (SMILE) study: Cluster randomised trial of humour therapy in nursing homes

Objectives: To determine whether humour therapy reduces depression (primary outcome), agitation and behavioural disturbances and improves social engagement and quality-of-life in nursing home residents. Design: The Sydney Multisite Intervention of LaughterBosses and ElderClowns study was a singleblind cluster randomised controlled trial of humour therapy. Setting: 35 Sydney nursing homes. Participants: All eligible residents within geographically defined areas within each nursing home were invited to participate. Intervention: Professional 'ElderClowns' provided 9-12 weekly humour therapy sessions, augmented by resident engagement by trained staff 'LaughterBosses'. Controls received usual care. Measurements: Depression scores on the Cornell Scale for Depression in Dementia, agitation scores on the Cohen-Mansfield Agitation Inventory, behavioural disturbance scores on the Neuropsychiatric Inventory, social engagement scores on the withdrawal subscale of Multidimensional Observation Scale for Elderly Subjects, and self-rated and proxy-rated quality-of-life scores on a health-related quality-of-life tool for dementia, the DEMQOL. All outcomes were measured at the participant level by researchers blind to group assignment. Randomisation: Sites were stratified by size and level of care then assigned to group using a random number generator. Results: Seventeen nursing homes (189 residents) received the intervention and 18 homes (209 residents) received usual care. Groups did not differ significantly over time on the primary outcome of depression, or on behavioural disturbances other than agitation, social engagement and quality of life. The secondary outcome of agitation was significantly reduced in the intervention group compared with controls over 26 weeks (time by group interaction adjusted for covariates: p=0.011). The mean difference in change from baseline to 26 weeks in Blom-transformed agitation scores after adjustment for covariates was 0.17 (95% CI 0.004 to 0.34, p=0.045). Conclusions: Humour therapy did not significantly reduce depression but significantly reduced agitation

Cluster size distributions in particle systems with asymmetric dynamics

We present exact and asymptotic results for clusters in the one-dimensional totally asymmetric exclusion process (TASEP) with two different dynamics. The expected length of the largest cluster is shown to diverge logarithmically with increasing system size for ordinary TASEP dynamics and as a logarithm divided by a double logarithm for generalized dynamics, where the hopping probability of a particle depends on the size of the cluster it belongs to. The connection with the asymptotic theory of extreme order statistics is discussed in detail. We also consider a related model of interface growth, where the deposited particles are allowed to relax to the local gravitational minimum.Comment: 12 pages, 3 figures, RevTe

Conservative interacting particles system with anomalous rate of ergodicity

We analyze certain conservative interacting particle system and establish ergodicity of the system for a family of invariant measures. Furthermore, we show that convergence rate to equilibrium is exponential. This result is of interest because it presents counterexample to the standard assumption of physicists that conservative system implies polynomial rate of convergence.Comment: 16 pages; In the previous version there was a mistake in the proof of uniqueness of weak Leray solution. Uniqueness had been claimed in a space of solutions which was too large (see remark 2.6 for more details). Now the mistake is corrected by introducing a new class of moderate solutions (see definition 2.10) where we have both existence and uniquenes

Robust and Generalisable Segmentation of Subtle Epilepsy-causing Lesions: a Graph Convolutional Approach

Focal cortical dysplasia (FCD) is a leading cause of drug-resistant focal epilepsy, which can be cured by surgery. These lesions are extremely subtle and often missed even by expert neuroradiologists. "Ground truth" manual lesion masks are therefore expensive, limited and have large inter-rater variability. Existing FCD detection methods are limited by high numbers of false positive predictions, primarily due to vertex- or patch-based approaches that lack whole-brain context. Here, we propose to approach the problem as semantic segmentation using graph convolutional networks (GCN), which allows our model to learn spatial relationships between brain regions. To address the specific challenges of FCD identification, our proposed model includes an auxiliary loss to predict distance from the lesion to reduce false positives and a weak supervision classification loss to facilitate learning from uncertain lesion masks. On a multi-centre dataset of 1015 participants with surface-based features and manual lesion masks from structural MRI data, the proposed GCN achieved an AUC of 0.74, a significant improvement against a previously used vertex-wise multi-layer perceptron (MLP) classifier (AUC 0.64). With sensitivity thresholded at 67%, the GCN had a specificity of 71% in comparison to 49% when using the MLP. This improvement in specificity is vital for clinical integration of lesion-detection tools into the radiological workflow, through increasing clinical confidence in the use of AI radiological adjuncts and reducing the number of areas requiring expert review.Comment: accepted at MICCAI 202