Related papers: The Einstein Relation on Metric Measure Spaces

We study single random walks and the electrical resistance for fractals obtained as the limit of a sequence of periodic structures. In the long-scale regime, power laws describe both the mean-square displacement of a random walk as a…

We study local structure of the moduli space of compact Einstein metrics with respect to the boundary conformal metric and mean curvature. In dimension three, we confirm M. Anderson's conjecture in a strong sense, showing that the map from…

We adopt a framework where quantum-gravity's dynamical dimensional reduction of spacetime at short distances is described in terms of modified dispersion relations. We observe that by subjecting such models to a momentum-space…

We consider random walk among iid, uniformly elliptic conductances on $\mathbb Z^d$, and prove the Einstein relation (see Theorem 1). It says that the derivative of the velocity of a biased walk as a function of the bias equals the…

In a seminal paper, Abbott et al. analyzed the relationship between a particle's trajectory and the resolution of position measurements performed by an observer at fixed time intervals. They predicted that quantum paths exhibit a universal…

This paper investigates the Einstein relation; the connection between the volume growth, the resistance growth and the expected time a random walk needs to leave a ball on a weighted graph. The Einstein relation is proved under different…

We formulate an approach to the geometry of Riemann-Cartan spaces provided with nonholonomic distributions defined by generic off-diagonal and nonsymmetric metrics inducing effective nonlinear and affine connections. Such geometries can be…

The unique Nature of the Lorentz group in four dimensions is the root cause of the many remarkable properties of the Einstein spacetimes, in particular their operational structure on the 2-forms. We show how this operational structure can…

In this manuscript, a conformally invariant theory of gravitation in the context of metric measure space is studied. The proposed action is invariant under both diffeomorphism and conformal transformations. Using the variational method, a…

Recently, Ryu et al. showed that two broadened bands connected by a set of four Einstein-coefficient spectra for stimulated and spontaneous single-photon transitions will obey detailed balance at equilibrium if the spectra satisfy…

This survey deals with two closely connected topics: first, the stability of Einstein metrics under the Einstein-Hilbert functional, and second, their deformation theory and the study of the moduli space of Einstein metrics on a compact…

In this work, Einstein's view of geometry as physical geometry is taken into account in the analysis of diverse issues related to the notions of inertial motion and inertial reference frame. Einstein's physical geometry enables a…

Once the action for Einstein's equations is rewritten as a functional of an SO(3,C) connection and a conformal factor of the metric, it admits a family of ``neighbours'' having the same number of degrees of freedom and a precisely defined…

This paper presents a comprehensive introduction to the Hausdorff measure, a fundamental tool in fractal geometry and geometric measure theory. We begin by defining the Hausdorff outer measure on subsets of metric spaces, followed by a…

We study the metric structure of walks on graphs, understood as Lipschitz sequences. To this end, a weighted metric is introduced to handle sequences, enabling the definition of distances between walks based on stepwise vertex distances and…

The Einstein's equivalence principle is formulated in terms of the accuracy of measurements and its dependence of the size of the area of measurement. It is shown that different refinements of the statement 'the spacetime is locally flat'…

This work addresses problems on simultaneous Diophantine approximation on fractals, motivated by a long standing problem of Mahler regarding Cantor's middle $1/3$ set. We obtain the first instances where a complete analogue of Khintchine's…

Walk dimension is an important conception in analysis of fractals. In this paper we prove that the walk dimension of a connected compact set possessing an Alfors regular measure is an invariant under Lipschitz transforms. As an application,…

We prove the Einstein relation, relating the velocity under a small perturbation to the diffusivity in equilibrium, for certain biased random walks on Galton--Watson trees. This provides the first example where the Einstein relation is…

We give an alternative description of the physical content of general relativity that does not require a Lorentz invariant spacetime. Instead, we find that gravity admits a dual description in terms of a theory where local size is…