Publications: Florian Sobieczky

Florian Sobieczky

  • F. Sobieczky: Bounds for 'the annealed return probability on large finite percolation clusters , Electronic Journal of Probability, 17 (2012), no. 79, 1-17
    Bounds for the expected return probability of the delayed random walk on finite clusters of an invariant percolation on transitive unimodular graphs are derived. They are particularly suited for the case of critical Bernoulli percolation and the associated heavy-tailed cluster size distributions. The upper bound relies on the fact that cartesian products of finite graphs with cycles of a certain minimal size are Hamiltonian. For critical Bernoulli bond percolation on the homogeneous tree this bound is sharp. The asymptotic type of the expected return probability for large times t in this case is of order of the 3/4'th power of 1/t.

  • V. Kaimanovich, F. Sobieczky: Random walks on random horospheric products (In: `Dynamical Systems and group actions', Edts. Bowen, Grigorchuk, Vorobets, Contemporary Mathematics, 2012, AMS)
    By developing the entropy theory of random walks on equivalence relations and analyzing the asymptotic geometry of horospheric products we describe the Poisson boundary for random walks on random horospheric products of trees.

  • F. Sobieczky, G. Rappitsch, E. Stadlober: Tandem queues for inventory management under random perturbations , Quality and Reliability Engineering Int. 26(8): 899-907 (2010), n/a. doi: 10.1002/qre.1161
    Using the theory of M/M/1 queues at stationarity, we provide criteria of stability (recurrence) for a stochastic inventory model with an observed selling rate and optimally chosen buying rate. Optimality is based on the maximum gain under stability, where buying and selling-prices, as well as shop- and stock-keeping costs are incorporated into the model. An important aspect is to achieve robustness of the stocking process by minimizing the fluctuation of the predicted gain. This robustness can be achieved by controlling intermediate transfer rates of the assumed stochastic tandem network. Stochastic simulations demonstrate the applicability of the stability criteria under several scenarios of differing intensities of perturbation.

  • with V. Kaimanovich Stochastic homogenization of horospheric tree products (in: Probabilistic approach to geometry, 199--229, Adv. Stud. Pure Math., 57, Math. Soc. Japan, Tokyo, 2010)
    We construct measures invariant with respect to equivalence relations which are graphed by horospheric products of trees. The construction is based on using conformal systems of boundary measures on treed equivalence relations. The existence of such an invariant measure allows us to establish amenability of horospheric products of random trees.

  • An interlacing technique for spectra of random walks and its application to finite percolation clusters JOTP, Vol. 23, No. 3, (2010), 639-670 arxive-version (v4) (2008)
    A comparison technique for finite random walks on finite graphs is introduced, using the well-known interlacing method. It yields improved return probability bounds. A key feature is the incorporation of parts of the spectrum of the transition matrix other than just the principal eigenvalue. As an application, an upper bound of the expected return probability of a random walk with symmetric transition probabilities is found. In this case, the state space is a random partial graph of a regular graph of bounded geometry and transitive automorphism group. The law of the random edge-set is assumed to be stationary with respect to some transitive subgroup of the automorphism group (`invariant percolation'). Given that this subgroup is unimodular, it is shown that stationarity strengthens the upper bound of the expected return probability, compared with standard bounds derived from the Cheeger inequality.

  • Strong amenability of horocyclic products of random trees (Submitted):
    In the context of results concerning anchored expansion and the speed of random walks on graphs in the presence of random pertubations horocyclic products of percolation trees is studied: Amenability and strong amenability (absence of anchored expansion) are proved under conditions concerning the growth of the two involved trees.

  • (Thesis:) `Bounds for the number of connected components of inverse images of stationary, ergodic fields',
    (translation of the original version:) `Zur Anzahl der Zusammenhangskomponenten von Urbildern stationaerer, ergodischer Felder', 4,2004 in Mathematica Gottingensis)
    The number of connected components `k' of inverse images of random stationary, ergodic functions with discrete two-dimensional domain is estimated, given only the functions' probability law: bounds for the number of open clusters per vertex of inverse images of an interval [-s,s] for a threshhold s of a random function on the set of nearest neighbours from ZxZ are derived. Connectedness refers to the ability to join two vertices by an open path. The existence of k in the stationary, ergodic case is proved. The method used is that of a random walk on a randomly generated graph with percolative structure. The estimates for k involve the expected n-step return probability, and the expected size of the number of bounding edges of the connected component containing the origin. The spectral properties of the Markov process allow these estimates by means of the Courant-Fischer min-max princple. The shift of the unperturbed eigenvalues of a random walk on the homogeneous, complete graph can be bounded, the pertubation is of finite rank, and the rank equals the number of edges removed. An application is the i.i.d. case of bond percolation.



    Home