Asymptotic Safety, Asymptotic Darkness, and the hoop conjecture in the extreme UV

Assuming the hoop conjecture in classical general relativity and quantum mechanics, any observer who attempts to perform an experiment in an arbitrarily small region will be stymied by the formation of a black hole within the spatial domain of the experiment. This behavior is often invoked in arguments for a fundamental minimum length. Extending a proof of the hoop conjecture for spherical symmetry to include higher curvature terms we investigate this minimum length argument when the gravitational couplings run with energy in the manner predicted by asymptotically safe gravity. We show that argument for the mandatory formation of a black hole within the domain of an experiment fails. Neither is there a proof that a black hole doesn't form. Instead, whether or not an observer can perform measurements in arbitrarily small regions depends on the specific numerical values of the couplings near the UV fixed point. We further argue that when an experiment is localized on a scale much smaller than the Planck length, at least one enshrouding horizon must form outside the domain of the experiment. This implies that while an observer may still be able to perform local experiments, communicating any information out to infinity is prevented by a large horizon surrounding it, and thus compatibility with general relativity can still be restored in the infrared limit.Comment: 9 pages, Matched Published Version with minor changes for clarity and emphasis of key point

Noise-Induced Stabilization of Planar Flows I

We show that the complex-valued ODE \begin{equation*} \dot z_t = a_{n+1} z^{n+1} + a_n z^n+\cdots+a_0, \end{equation*} which necessarily has trajectories along which the dynamics blows up in finite time, can be stabilized by the addition of an arbitrarily small elliptic, additive Brownian stochastic term. We also show that the stochastic perturbation has a unique invariant measure which is heavy-tailed yet is uniformly, exponentially attracting. The methods turn on the construction of Lyapunov functions. The techniques used in the construction are general and can likely be used in other settings where a Lyapunov function is needed. This is a two-part paper. This paper, Part I, focuses on general Lyapunov methods as applied to a special, simplified version of the problem. Part II of this paper extends the main results to the general setting.Comment: Part one of a two part pape

A practical criterion for positivity of transition densities

We establish a simple criterion for locating points where the transition density of a degenerate diffusion is strictly positive. Throughout, we assume that the diffusion satisfies a stochastic differential equation (SDE) on $\mathbf{R}^d$ with additive noise and polynomial drift. In this setting, we will see that it is often that case that local information of the flow, e.g. the Lie algebra generated by the vector fields defining the SDE at a point $x\in \mathbf{R}^d$, determines where the transition density is strictly positive. This is surprising in that positivity is a more global property of the diffusion. This work primarily builds on and combines the ideas of Ben Arous and L\'eandre (1991) and Jurdjevic and Kupka (1981, 1985).Comment: 24 page

Low energy bounds on Poincare violation in causal set theory

In the causal set approach to quantum gravity, Poincar\'{e} symmetry is modified by swerving in spacetime, induced by the random lattice discretization of the space-time structure. The broken translational symmetry at short distances is argued to lead to a residual diffusion in momentum space, whereby a particle can acquire energy and momentum by drift along its mass shell and a system in equilibrium can spontaneously heat up. We consider bounds on the rate of momentum space diffusion coming from astrophysical molecular clouds, nuclear stability and cosmological neutrino background. We find that the strongest limits come from relic neutrinos, which we estimate to constrain the momentum space diffusion constant by $k < 10^{-61} {\rm GeV}^3$ for neutrinos with masses $m_\nu > 0.01 {\rm eV}$, improving the previously quoted bounds by roughly 17 orders of magnitude.Comment: Additional discussion about behavior of alpha particles in nuclei added. Version matches that accepted in PR