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We investigate the long-time behavior of the Glauber dynamics for the random energy model below the critical temperature. We give very precise estimates on the motion of the process to and between the states of extremal energies. We show that when disregarding time, the consecutive steps of the process on these states are governed by a Markov chain that jumps uniformly on all possible states. The mean times of these jumps are also computed very precisely and are seen to be asymptotically independent of the terminal point. A first indicator of aging is the observation that the mean time of arrival in the set of states that have waiting times of order T is itself of order T . The estimates proven in this paper will furnish crucial input for a follow-up paper where aging is analysed in full detail.

We consider the parabolic Anderson model (PAM) on the n-dimensional hypercube with random i.i.d. potentials. We parametrize time by volume and study the solution at the location of the k-th largest potential. Our main result is that, for a certain class of potential distributions, the solution exhibits a phase transition: for short time scales it behaves like a system without diffusion, whereas, for long time scales the growth is dictated by the principle eigenvalue and the corresponding eigenfunction of the Anderson operator, for which we give precise asymptotics. Moreover, the transition time depends only on the difference between the largest and k-th largest potential. One of our main motivations in this article is to investigate the mutation-selection model of population genetics on a random fitness landscape, which is given by the ratio of the solution of PAM to its total mass, with the field corresponding to the fitness landscape. We show that the phase transition of the solution translates to the mutation-selection model as follows: a population initially concentrated at the site of the k-th best fitness value moves completely to the site of the best fitness on time scales where the transition of growth rates happens. The class of potentials we consider involve the Random Energy Model (REM) of statistical physics which is studied as one of the main examples of a random fitness landscape.

Recently, Bauke and Mertens conjectured that the local statistics of energies in random spin systems with discrete spin space should in most circumstances be the same as in the random energy model. This was proven in a large class of models for energies that do not grow too fast with the system size. Considering the example of the generalized random energy model, we show that the conjecture breaks down for energies proportional to the volume of the system, and describe the far more complex behavior that then sets in.

Identifying the incidence of free-ridership is significant to a range of issues relevant to program evaluation, including the calculation of net program benefits and more general assessments of political acceptability. Estimates of free-ridership in the area of energy policy frequently rely on ex-post surveys that ask program participants whether they would have behaved differently in the absence of program support. The present paper proposes an ex-ante approach to the calculation of the free-rider share using revealed preference data on home renovations from Germany's residential sector. We employ a discrete-choice model to simulate the effect of grants on renovation choices, the output from which is used to assess the extent of free-ridership under a contemporary subsidy program. Aside from its simplicity, a key advantage of the approach is that it bestows policymakers with an estimate of free-ridership prior to program implementation.

In this letter we announce rigorous results on the phenomenon of aging in the Glauber dynamics of the random energy model and their relation to Bouchaud's 'REM-like' trap model. We show that, below the critical temperature, if we consider a time-scale that diverges with the system size in such a way that equilibrium is almost, but not quite reached on that scale, a suitably defined autocorrelation function has the same asymptotic behaviour than its analog in the trap model.

We consider the random fluctuations of the free energy in the $p$-spin version of the Sherrington-Kirkpatrick model in the high temperature regime. Using the martingale approach of Comets and Neveu as used in the standard SK model combined with truncation techniques inspired by a recent paper by Talagrand on the $p$-spin version, we prove that (for $p$ even) the random corrections to the free energy are on a scale $N^-(p-2)/4$ only, and after proper rescaling converge to a standard Gaussian random variable. This is shown to hold for all values of the inverse temperature, $b$, smaller than a critical $b_p$. We also show that $b_prightarrow sqrt2ln 2$ as $puparrow +infty$. Additionally we study the formal $puparrow +infty$ limit of these models, the random energy model. Here we compute the precise limit theorem for the partition function at it all temperatures. For $b< sqrt2ln2$, fluctuations are found at an it exponentially small scale, with two distinct limit laws above and below a second critical value $sqrtln 2/2$: For $b$ up to that value the rescaled fluctuations are Gaussian, while below that there are non-Gaussian fluctuations driven by the Poisson process of the extreme values of the random energies. For $b$ larger than the critical $sqrt2ln 2$, the fluctuations of the logarithm of the partition function are on scale one and are expressed in terms of the Poisson process of extremes. At the critical temperature, the partition function divided by its expectation converges to $1/2$.