Chordal SLE<inf>6</inf> explorations of a quantum disk

We consider a particular type of $\sqrt{8/3}$-Liouville quantum gravity surface called a doubly marked quantum disk (equivalently, a Brownian disk) decorated by an independent chordal SLE$_6$ curve $\eta$ between its marked boundary points. We obtain descriptions of the law of the quantum surfaces parameterized by the complementary connected components of $\eta([0,t])$ for each time $t \geq 0$ as well as the law of the left/right $\sqrt{8/3}$-quantum boundary length process for $\eta$

Scaling limit of the uniform infinite half-plane quadrangulation in the Gromov-Hausdorff-Prokhorov-uniform topology

We prove that the uniform infinite half-plane quadrangulation (UIHPQ), with either general or simple boundary, equipped with its graph distance, its natural area measure, and the curve which traces its boundary, converges in the scaling limit to the Brownian half-plane. The topology of convergence is given by the so-called Gromov-Hausdorff-Prokhorov-uniform (GHPU) metric on curve-decorated metric measure spaces, which is a generalization of the Gromov-Hausdorff metric whereby two such spaces $(X_1, d_1 , \mu_1,\eta_1)$ and $(X_2, d_2 , \mu_2,\eta_2)$ are close if they can be isometrically embedded into a common metric space in such a way that the spaces $X_1$ and $X_2$ are close in the Hausdorff distance, the measures $\mu_1$ and $\mu_2$ are close in the Prokhorov distance, and the curves $\eta_1$ and $\eta_2$ are close in the uniform distance.E.G. was supported by the U.S. Department of Defense via an NDSEG fellowship

Bifurcations and Chaos in Time Delayed Piecewise Linear Dynamical Systems

We reinvestigate the dynamical behavior of a first order scalar nonlinear delay differential equation with piecewise linearity and identify several interesting features in the nature of bifurcations and chaos associated with it as a function of the delay time and external forcing parameters. In particular, we point out that the fixed point solution exhibits a stability island in the two parameter space of time delay and strength of nonlinearity. Significant role played by transients in attaining steady state solutions is pointed out. Various routes to chaos and existence of hyperchaos even for low values of time delay which is evidenced by multiple positive Lyapunov exponents are brought out. The study is extended to the case of two coupled systems, one with delay and the other one without delay.Comment: 34 Pages, 14 Figure

Active Spanning Trees with Bending Energy on Planar Maps and SLE-Decorated Liouville Quantum Gravity for κ&gt; 8

We consider the Peano curve separating a spanning tree from its dual spanning tree on an embedded planar graph, where the tree and dual tree are weighted by $y$ to the number of active edges, and "active" is in the sense of the Tutte polynomial. When the graph is a portion of the square grid approximating a simply connected domain, it is known ($y=1$ and $y=1+\sqrt{2}$) or believed ($1<y<3$) that the Peano curve converges to a space-filling SLE$_{\kappa}$ loop, where $y=1-2\cos(4\pi/\kappa)$, corresponding to $4<\kappa\leq 8$. We argue that the same should hold for $0\le y<1$, which corresponds to $8<\kappa\leq 12$

Guidance on Design and Construction of the Built Environment Against Wildland Urban Interface Fire Hazard: A Review

Wildland-Urban Interface (WUI) fires, a worldwide problem, are gaining more importance over time due to climate change and increased urbanization in WUI areas. Some jurisdictions have provided standards, codes and guidelines, which may greatly help planning, prevention and protection against wildfires. This work presents a wide systematic review of standards, codes and guidelines for the design and construction of the built environment against WUI fire hazard from North American, European, Oceanic countries, alongside with trans-national codes. The main information reviewed includes: the definition of WUI hazards, risk areas and related severity classes, the influence of land and environmental factors, the requirements for building materials, constructions, utilities, fire protection measures and road access. Some common threads among the documents reviewed have been highlighted. They include similar attempts at: (a) defining WUI risk areas and severity classes, (b) considering land factors including the defensible space (also known as ignition zones), (c) prescribing requirements for buildings and access. The main gaps highlighted in the existing standards/guidelines include lacks of detailed and widespread requirements for resources, fire protection measures, and lacks of taking into account environmental factors in detail. The main design and construction principles contained in the reviewed documents are largely based on previous research and/or good practices. Hence, the main contributions of this paper consist in: (a) systematically disseminate these guidance concepts, (b) setting a potential basis for the development of standards/guidelines in other jurisdictions lacking dedicated WUI fire design guidance, (c) highlighting gaps in existing standards/guidelines to be addressed by current and future research

Anomalous diffusion of random walk on random planar maps

Funder: University of CambridgeAbstractWe prove that the simple random walk on the uniform infinite planar triangulation (UIPT) typically travels graph distance at most $$n^{1/4 + o_n(1)}$$ n 1 / 4 + o n ( 1 ) in n units of time. Together with the complementary lower bound proven by Gwynne and Miller (2017) this shows that the typical graph distance displacement of the walk after n steps is $$n^{1/4 + o_n(1)}$$ n 1 / 4 + o n ( 1 ) , as conjectured by Benjamini and Curien (Geom Funct Anal 2(2):501–531, 2013. arXiv:1202.5454). More generally, we show that the simple random walks on a certain family of random planar maps in the $$\gamma$$ γ -Liouville quantum gravity (LQG) universality class for $$\gamma \in (0,2)$$ γ ∈ ( 0 , 2 ) —including spanning tree-weighted maps, bipolar-oriented maps, and mated-CRT maps—typically travels graph distance $$n^{1/d_\gamma + o_n(1)}$$ n 1 / d γ + o n ( 1 ) in n units of time, where $$d_\gamma$$ d γ is the growth exponent for the volume of a metric ball on the map, which was shown to exist and depend only on $$\gamma$$ γ by Ding and Gwynne (Commun Math Phys 374:1877–1934, 2018. arXiv:1807.01072). Since d_\gamma > 2 d γ &gt; 2 , this shows that the simple random walk on each of these maps is subdiffusive. Our proofs are based on an embedding of the random planar maps under consideration into $${\mathbb {C}}$$ C wherein graph distance balls can be compared to Euclidean balls modulo subpolynomial errors. This embedding arises from a coupling of the given random planar map with a mated-CRT map together with the relationship of the latter map to SLE-decorated LQG. </jats:p

An open multi-physics framework for modelling wildland-urban interface fire evacuations

Fire evacuations at wildland-urban interfaces (WUI) pose a serious challenge to the emergency services, and are a global issue affecting thousands of communities around the world. This paper presents a multi-physics framework for the simulation of evacuation in WUI wildfire incidents, including three main modelling layers: wildfire, pedestrians, and traffic. Currently, these layers have been mostly modelled in isolation and there is no comprehensive model which accounts for their integration. The key features needed for system integration are identified, namely: consistent level of refinement of each layer (i.e. spatial and temporal scales) and their application (e.g. evacuation planning or emergency response), and complete data exchange. Timelines of WUI fire events are analysed using an approach similar to building fire engineering (available vs. required safe egress times for WUI fires, i.e. WASET/WRSET). The proposed framework allows for a paradigm shift from current wildfire risk assessment and mapping tools towards dynamic fire vulnerability mapping. This is the assessment of spatial and temporal vulnerabilities based on the wildfire threat evolution along with variables related to the infrastructure, population and network characteristics. This framework allows for the integration of the three main modelling layers affecting WUI fire evacuation and aims at improving the safety of WUI communities by minimising the consequences of wildfire evacuations

Mutual Information for the Detection of Crush

Fatal crush conditions occur in crowds with tragic frequency. Event organizers and architects are often criticised for failing to consider the causes and implications of crush, but the reality is that both the prediction and prevention of such conditions offer a significant technical challenge. Full treatment of physical force within crowd simulations is precise but often computationally expensive; the more common method of human interpretation of results is computationally “cheap” but subjective and time-consuming. This paper describes an alternative method for the analysis of crowd behaviour, which uses information theory to measure crowd disorder. We show how this technique may be easily incorporated into an existing simulation framework, and validate it against an historical event. Our results show that this method offers an effective and efficient route towards automatic detection of the onset of crush