The quantum Einstein gravity is treated by the functional renormalization group method using the Einstein-Hilbert action. The ultraviolet non-Gaussian fixed point is determined and its corresponding exponent of the correlation length is calculated for a wide range of regulators. It is shown that the exponent provides a minimal sensitivity to the parameters of the regulator which correspond to the Litim's regulator.

In this paper we provide a numerical approximation of bifurcation branches from nodal radial solutions of the Lane Emden Dirichlet problem in the unit ball in ℝ2, as the exponent of the nonlinearity varies. We consider solutions with two or three nodal regions. In the first case our numerical results complement the analytical ones recently obtained in [11]. In the case of solutions with three nodal regions, for which no analytical results are available, our analysis gives numerical evidence of the existence of bifurcation branches. We also compute additional approximations indicating presence of an unexpected branch of solutions with six nodal regions. In all cases the numerical results allow to formulate interesting conjectures.

We performed a functional renormalization group analysis for the quantum Einstein gravity including a quadratic term in the curvature. The ultraviolet non-gaussian fixed point and its critical exponent for the correlation length are identified for different forms of regulators in case of dimension 3. We searched for that optimized regulator where the physical quantities show the least regulator parameter dependence. It is shown that the Litim regulator satisfies this condition. The infrared fixed point has also been investigated, it is found that the exponent is insensitive to the third coupling introduced by the R 2 term.

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We study the length of T -contaminated runs of heads in the well-known coin tossing experiment. A T -contaminated run of heads is a sequence of consecutive heads interrupted by T tails. For T = 1 and T = 2 we find the asymptotic distribution for the first hitting time of the T contaminated run of heads having length m; furthermore, we obtain a limit theorem for the length of the longest T -contaminated head run. We prove that the rate of the approximation of our accompanying distribution for the length of the longest T -contaminated head run is considerably better than the previous ones. For the proof we use a powerful lemma by Csáki, Földes and Komlós, see [1].