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Scott Robertson

Publications and source records attributed to Scott Robertson.

49 records · Page 3Linked to original sources
arXiv2015

The theory of Hawking radiation in laboratory analogues

Hawking radiation, despite being known to theoretical physics for nearly forty years, remains elusive and undetected. It also suffers, in its original context of gravitational black holes, from practical and conceptual difficulties. Of particular note is the trans-Planckian problem, which is concerned with the apparent origin of the radiation in absurdly high frequencies. In order to gain better theoretical understanding and, it is hoped, experimental verification of Hawking radiation, much study is being devoted to laboratory systems which use moving media to model the spacetime geometry of black holes, and which, by analogy, are also thought to emit Hawking radiation. These analogue systems typically exhibit dispersion, which regularizes the wave behaviour at the horizon at the cost of a more complicated theoretical framework. This tutorial serves as an introduction to Hawking radiation and its analogues, developing the moving medium analogy for black holes and demonstrating how dispersion can be incorporated into this generalized framework.

gr-qc↗
arXiv2015

Hawking radiation in the presence of high-momentum dissipation

We study the Hawking radiation in field theories which break Lorentz invariance via dissipative effects above a certain energy scale. We assume that the additional degrees of freedom which cause dissipation are Gaussian and freely falling. The asymptotic spectrum and the correlations are extracted from the anticommutator of the radiation field. The singular behavior of the Green function found for relativistic fields as one point crosses the horizon is completely suppressed by dissipation. Yet, when the dissipative frequency scale is much larger than the surface gravity of the black hole, we show that the asymptotic observables acquire their standard (relativistic) vacuum expectation values. We explicitly compute the effects of dissipation on the spectrum and on the nonseparable character of the correlations when varying the dissipative scale, the extension of the near-horizon geometry, and the temperature of the environment.

gr-qc↗
arXiv2014

Long Term Optimal Investment in Matrix Valued Factor Models

Long term optimal investment problems are studied in a factor model with matrix valued state variables. Explicit parameter restrictions are obtained under which, for an isoelastic investor, the finite horizon value function and optimal strategy converge to their long-run counterparts as the investment horizon approaches infinity. This convergence also yields portfolio turnpikes for general utilities. By using results on large time behaviour of semi-linear partial differential equations, our analysis extends affine models, where the Wishart process drives investment opportunities, to a non-affine setting. Furthermore, in the affine setting, an example is constructed where the value function is not exponentially affine, in contrast to models with vector-valued state variables.

q-fin.MF↗
arXiv2014

Quantum entanglement due to modulated Dynamical Casimir Effect

We study the creation and entanglement of quasiparticle pairs due to a periodic variation of the mode frequencies of a homogeneous quantum system. Depending on the values of the parameters describing the periodic modulation, the number of created pairs either oscillates or, in a narrow resonant frequency interval, grows exponentially in time. For a system initially in a thermal state, we determine in which cases the final state is quantum mechanically entangled, i.e., where the bipartite state is nonseparable. We include some weak dissipation, expected to be found in any experimental setup, and study the corresponding reduction of the quantum entanglement. Our findings are used to interpret the results of two recent experiments.

cond-mat.quant-gas↗
arXiv2014

Large time behavior of solutions to semi-linear equations with quadratic growth in the gradient

This paper studies the large time behavior of solutions to semi-linear Cauchy problems with quadratic nonlinearity in gradients. The Cauchy problem considered has a general state space and may degenerate on the boundary of the state space. Two types of large time behavior are obtained: i) pointwise convergence of the solution and its gradient; ii) convergence of solutions to associated backward stochastic differential equations. When the state space is R^d or the space of positive definite matrices, both types of convergence are obtained under growth conditions on model coefficients. These large time convergence results have direct applications in risk sensitive control and long term portfolio choice problems.

math.AP↗
arXiv2014

Integral method for the calculation of Hawking radiation in dispersive media I. Symmetric asymptotics

Hawking radiation has become experimentally testable thanks to the many analogue systems which mimic the effects of the event horizon on wave propagation. These systems are typically dominated by dispersion, and give rise to a numerically soluble and stable ODE only if the rest-frame dispersion relation $Ω^{2}(k)$ is a polynomial of relatively low degree. Here we present a new method for the calculation of wave scattering in a one-dimensional medium of arbitrary dispersion. It views the wave equation as an integral equation in Fourier space, which can be solved using standard and efficient numerical techniques.

gr-qc↗
arXiv2014

Integral method for the calculation of Hawking radiation in dispersive media II. Asymmetric asymptotics

Analogue gravity experiments make feasible the realisation of black hole spacetimes in a laboratory setting and the observational verification of Hawking radiation. Since such analogue systems are typically dominated by dispersion, efficient techniques for calculating the predicted Hawking spectrum in the presence of strong dispersion are required. In the preceding paper, an integral method in Fourier space is proposed for stationary $1+1$-dimensional backgrounds which are asymptotically symmetric. Here, this method is generalised to backgrounds which are different in the asymptotic regions to the left and right of the scattering region.

gr-qc↗
arXiv2013

Pricing for Large Positions in Contingent Claims

Approximations to utility indifference prices are provided for a contingent claim in the large position size limit. Results are valid for general utility functions on the real line and semi-martingale models. It is shown that as the position size approaches infinity, the utility function's decay rate for large negative wealths is the primary driver of prices. For utilities with exponential decay, one may price like an exponential investor. For utilities with a power decay, one may price like a power investor after a suitable adjustment to the rate at which the position size becomes large. In a sizable class of diffusion models, limiting indifference prices are explicitly computed for an exponential investor. Furthermore, the large claim limit is seen to endogenously arise as the hedging error for the claim vanishes.

q-fin.PR↗
arXiv2012

Robust maximization of asymptotic growth

This paper addresses the question of how to invest in a robust growth-optimal way in a market where the instantaneous expected return of the underlying process is unknown. The optimal investment strategy is identified using a generalized version of the principal eigenfunction for an elliptic second-order differential operator, which depends on the covariance structure of the underlying process used for investing. The robust growth-optimal strategy can also be seen as a limit, as the terminal date goes to infinity, of optimal arbitrages in the terminology of Fernholz and Karatzas [Ann. Appl. Probab. 20 (2010) 1179-1204].

q-fin.PM↗
arXiv2012

Portfolios and risk premia for the long run

This paper develops a method to derive optimal portfolios and risk premia explicitly in a general diffusion model for an investor with power utility and a long horizon. The market has several risky assets and is potentially incomplete. Investment opportunities are driven by, and partially correlated with, state variables which follow an autonomous diffusion. The framework nests models of stochastic interest rates, return predictability, stochastic volatility and correlation risk. In models with several assets and a single state variable, long-run portfolios and risk premia admit explicit formulas up the solution of an ordinary differential equation which characterizes the principal eigenvalue of an elliptic operator. Multiple state variables lead to a quasilinear partial differential equation which is solvable for many models of interest. The paper derives the long-run optimal portfolio and the long-run optimal pricing measures depending on relative risk aversion, as well as their finite-horizon performance.

math.PR↗
arXiv2012

Abstract, Classic, and Explicit Turnpikes

Portfolio turnpikes state that, as the investment horizon increases, optimal portfolios for generic utilities converge to those of isoelastic utilities. This paper proves three kinds of turnpikes. In a general semimartingale setting, the abstract turnpike states that optimal final payoffs and portfolios converge under their myopic probabilities. In diffusion models with several assets and a single state variable, the classic turnpike demonstrates that optimal portfolios converge under the physical probability; meanwhile the explicit turnpike identifies the limit of finite-horizon optimal portfolios as a long-run myopic portfolio defined in terms of the solution of an ergodic HJB equation.

q-fin.PM↗
arXiv2008

Fiber-optical analogue of the event horizon

The physics at the event horizon resembles the behavior of waves in moving media. Horizons are formed where the local speed of the medium exceeds the wave velocity. We use ultrashort pulses in microstructured optical fibers to demonstrate the formation of an artificial event horizon in optics. We observed a classical optical effect, the blue-shifting of light at a white-hole horizon. We also show by theoretical calculations that such a system is capable of probing the quantum effects of horizons, in particular Hawking radiation.

gr-qc↗