We show the existence of the scaling exponent χ ∈ (0, 4[(1 + γ 2 /4) − 1 + γ 4 /16]/γ 2 ] of the graph distance associated with subcritical two-dimensional Liouville quantum gravity of paramater γ < 2 on V = [0, 1] 2 . We also show that the Liouville heat kernel satisfies, for any fixed u, v ∈ V o , the short time estimates
Let {η(v) : v ∈ V N } be a discrete Gaussian free field in a two-dimensional box V N of side length N with Dirichlet boundary conditions. We study the Liouville first passage percolation, i.e., the shortest path metric where each vertex is given a weight of e γη(v) for some γ > 0. We show that for sufficiently small but fixed γ > 0, with probability tending to 1 as N → ∞, all geodesics between vertices of macroscopic Euclidean distances simultaneously have (the conjecturally unique) length exponent strictly larger than 1.
For almost every trajectory segment over a finite time span of a finite Markov chain with any given initial distribution, the logarithm of the ratio of its probability to that of its time-reversal converges exponentially to the entropy production rate of the Markov chain. The large deviation rate function has a symmetry of Gallavotti–Cohen type, which is called the fluctuation theorem. Moreover, similar symmetries also hold for the rate functions of the joint distributions of general observables and the logarithmic probability ratio.
We consider first passage percolation (FPP) where the vertex weight is given by the exponential of two-dimensional log-correlated Gaussian fields. Our work is motivated by understanding the discrete analog for the random metric associated with Liouville quantum gravity (LQG), which roughly corresponds to the exponential of a two-dimensional Gaussian free field (GFF).The particular focus of the present paper is an aspect of universality for such FPP among the family of log-correlated Gaussian fields. More precisely, we construct a family of log-correlated Gaussian fields, and show that the FPP distance between two typically sampled vertices (according to the LQG measure) is N 1+O(ε) , where N is the side length of the box and ε can be made arbitrarily small if we tune a certain parameter in our construction. That is, the exponents can be arbitrarily close to 1. Combined with a recent work of the first author and Goswami on an upper bound for this exponent when the underlying field is a GFF, our result implies that such exponent is not universal among the family of log-correlated Gaussian fields.
This article provides two unique methodologies that may be coupled to study the dependability of multidimensional nonlinear dynamic systems. First, the structural reliability approach is well suited for multidimensional environmental and structural reactions and is either measured or numerically simulated over sufficient time, yielding lengthy ergodic time series. Second, a unique approach to predicting extreme values has technical and environmental implications. In the event of measurable environmental loads, it is also feasible to calculate the probability of system failure, as shown in this research. In addition, traditional probability approaches for time series cannot cope effectively with the system's high dimensionality and cross-correlation across dimensions. It is common knowledge that wind speeds represent a complex, nonlinear, multidimensional, and cross-correlated dynamic environmental system that is always difficult to analyze. Additionally, global warming is a significant element influencing ocean waves throughout time. This section aims to demonstrate the efficacy of the previously mentioned technique by applying a novel method to the Norwegian offshore data set for the greatest daily wind cast speeds in the vicinity of the Landvik wind station. This study aims to evaluate the state-of-the-art approach for extracting essential information about the extreme reaction from observed time histories. The approach provided in this research enables the simple and efficient prediction of failure probability for the whole nonlinear multidimensional dynamic system.
This paper demonstrates the validity of the Naess–Gadai method for extrapolating extreme value statistics of second-order Volterra series processes through application on a representative model of a deep water small size tension leg platform (TLP), with specific focus on wave sum frequency effects affecting restrained modes: heave, roll and pitch. The wave loading was estimated from a second order diffraction code WAMIT, and the stochastic TLP structural response in a random sea state was calculated exactly using Volterra series representation of the TLP corner vertical displacement, chosen as a response process. Although the wave loading was assumed to be a second order (non-linear) process, the dynamic system was modelled as a linear damped mass-spring system. Next, the mean up-crossing rate based extrapolation method (Naess–Gaidai method) was applied to calculate response levels at low probability levels. Since exact solution was available via Volterra series representation, both predictions were compared in this study, namely the exact Volterra and the approximate one. The latter gave a consistent way to estimate efficiency and accuracy of Naess–Gaidai extrapolation method. Therefore the main goal of this study was to validate Naess–Gaidai extrapolation method by available analytical-based exact solution. Moreover, this paper highlights limitations of mean up-crossing rate based extrapolation methods for the case of narrow band effects, such as clustering, typically included in the springing type of response.
Mining coal seams with high coal gas content and coal spontaneous combustion (SponCom) risk is often a challenge for mining engineers. Control measures for maintaining the gas concentration under the regulation permissible limit (Generally 1%) as well as reducing coal fire risks must be taken simultaneously in order to create safety production conditions. However, in reality, such measures either for gas or for fire problems are sometimes conflictive. For an example, the basic strategy of gas control in underground is to increase the ventilation airflow for sweeping the longwall working face and diluting the gas concentration, but it also results in leakage of an * Corresponding author. Jianwei Cheng.
As is known, the entropy production rate of a stationary Markov process vanishes if and only if the process is reversible. In this paper, we discuss the reversibility of a stationary Markov process from a functional analysis point of view. It is shown that the process is reversible if and only if it has a symmetric Markov semigroup, equivalently, a self-adjoint infinitesimal generator. Applying this fact, we prove that the Green–Kubo formula holds for reversible Markov processes. By demonstrating that the power spectrum of each reversible Markov process is Lorentz-typed, we show that it is impossible for stochastic resonance to occur in systems with zero entropy production.
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