We study the mixing time of the single-site update Markov chain, known as the Glauber dynamics, for generating a random independent set of a tree. Our focus is obtaining optimal convergence results for arbitrary trees. We consider the more general problem of sampling from the Gibbs distribution in the hard-core model where independent sets are weighted by a parameter $\lambda \gt 0$; the special case $\lambda =1$ corresponds to the uniform distribution over all independent sets. Previous work of Martinelli, Sinclair and Weitz (2004) obtained optimal mixing time bounds for the complete $\Delta$-regular tree for all $\lambda$. However, Restrepo, Stefankovic, Vera, Vigoda, and Yang (2014) showed that for sufficiently large $\lambda$ there are bounded-degree trees where optimal mixing does not hold. Recent work of Eppstein and Frishberg (2022) proved a polynomial mixing time bound for the Glauber dynamics for arbitrary trees, and more generally for graphs of bounded tree-width.

We establish an optimal bound on the relaxation time (i.e., inverse spectral gap) of $O(n)$ for the Glauber dynamics for unweighted independent sets on arbitrary trees. We stress that our results hold for arbitrary trees and there is no dependence on the maximum degree $\Delta$. Interestingly, our results extend (far) beyond the uniqueness threshold which is on the order $\lambda =O(1/\Delta )$. Our proof approach is inspired by recent work on spectral independence. In fact, we prove that spectral independence holds with a constant independent of the maximum degree for any tree, but this does not imply mixing for general trees as the optimal mixing results of Chen, Liu, and Vigoda (2021) only apply for bounded-degree graphs. We instead utilize the combinatorial nature of independent sets to directly prove approximate tensorization of variance via a non-trivial inductive proof.

The fixed points of the generalized Ricci flow are the Bismut Ricci flat (BRF) metrics, i.e., a generalized metric (g, H) on a manifold M, where g is a Riemannian metric and H a closed 3-form, such that H is g-harmonic and $\operatorname{Rc}(g)=\tfrac{1}{4} H_g^2$. Given two standard Einstein homogeneous spaces $G_i/K$, where each Gi is a compact simple Lie group and K is a closed subgroup of them holding some extra assumption, we consider $M=G_1\times G_2/\Delta K$. Recently, Lauret and Will proved the existence of a BRF metric on any of these spaces. We proved that this metric is always asymptotically stable for the generalized Ricci flow on M among a subset of G-invariant metrics and, if $G_1=G_2$, then it is globally stable.

We answer the following question: if the occupied (or vacant) set of a planar Poisson Boolean percolation model contains a crossing of an $n\times n$ square, how wide is this crossing? The answer depends on whether we consider the critical, sub-, or super-critical regime, and is different for the occupied and vacant sets.

Pipelines are used in many sectors to transport materials such as fluid from one place to another. These pipelines require regular inspection and maintenance to ensure proper operations and to avoid accidents. Many in-pipe navigation robots have been developed to perform the inspection. Soft in-pipe navigation robot is a special class of in-pipe robot, where the structure is made entirely of soft materials. The soft in-pipe robots are cheaper, lightweight, robust, and more adaptable to the environment inside pipelines as compared to the traditional rigid in-pipe navigation robot. This paper reviews the design of different types of soft in-pipe navigation in terms of the material, structure, locomotion strategy, and actuation techniques. These four different aspects of the design help researchers to narrow down their research and explore different opportunities within each of the design aspects. This paper also offers suggestions on the direction of research to improve the current soft in-pipe navigation robot design.

Many populist leaders politicise disputes with external financial ‘elites’, but most are forced by economic pressures to fundamentally change their ‘people-versus-elite’ problem representations and ‘concede defeat’. Notable exceptions are Hungary’s Viktor Orbán, Argentina’s Cristina Kirchner, and Turkey’s Recep Tayyip Erdoğan, who sooner or later resisted strong push-back and defied the IMF or distressed-debt funds. These instances of prolonged populist defiance differ widely across commonly used structural and agential explanatory factors at international, domestic, and individual levels. To explain how Orbán, Kirchner, and Erdoğan managed to ‘beat the elite’, this paper clusters several root causes into a parsimonious framework of two intervening variables. The temporality of strong ‘elite’ push-back and the openness of advisory systems are theorised as shaping distinct cognitive mechanisms of representational continuity or change, through which ‘people-versus-elite’ input is either preserved until – or discarded before – feeding into decision outputs. As the two-by-two matrix of early/later external shocks and open/closed inner circles explains, Orbán did not move beyond marginal representational adjustments; Kirchner’s contingent representational fluidity benefited from opportune situational developments; and Erdoğan defied significant socio-economic cost with fundamental representational continuity. These insights highlight the potential of studying populism at the intersection of foreign policy analysis and international political economy.

In this article, I identify a conceptually distinct form of epistemic appropriation: the creation and proliferation of the epistemic fata morgana. An epistemic fata morgana is a hermeneutical resource that is hollowed out, stripped of its meaning and political power, and yet, posited as if it were still accessible. This resource is taken up by dominant knowers in a way that preserves only its perception, but not access to it. This process is illustrated by an examination of the resource “sexual harassment” within the university context. The epistemic fata morgana is an important addition to the field of epistemic injustice for it lends itself to highlighting the frustration that is felt by marginalized people, especially within institutional contexts.