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In this paper two ways to order the nodes of a graph with respect to an arbitrary node are considered, both connected to random walks on the graph. The first one is the order according to probabilities of states of a random walk of fixed length started in that arbitrary node. The walks considered here are lazy walks – instead of making a step they are allowed to stay in the same node. A class of graphs, where such order the corresponds to the weak order by geodesic distances, was found. Square and toric n-dimensional grids are shown to be instances of this class. The second way of ordering is resistance distance to a fixed node. For another class of graphs, a pair of vertices with maximal resistance distance between them is established. Grids are again shown to be an example of graphs belonging to this class.

Asymptotic properties of random walks on minimal Cayley graphs of complex reflection groups are investigated. The main result of the paper is theorem on fast mixing for random walks on Cayley graphs of complex reflection groups. Particularly, bounds of diameters and isoperimetric constants, a known result on fast fixing property for expander graphs play a crucial role to obtain the main result. A constructive way to prove a special case of Babai’s conjecture on logarithmic order of diameters for complex reflection groups is proposed. Basing on estimates of diameters and Cheeger inequality, there is obtained a non-trivial lower bound for spectral gaps of minimal Cayley graphs on complex reflection groups.

In this paper average times between elementary jumps of stock returns on the Russian market were experimentally studied. Considering the scaling of the probability density function of stock returns on different time intervals it is shown that an elementary jump in the random walks scheme for financial instrument returns is a unit price change (tick) that corresponds to a single deal on the stock market.

In this paper, we investigate the limit theorem for a number of collisions of three independent simple non-symmetric random walks on the line. We obtain the limit theorem stating that the limit distribution of the log-normal number of collisions is the exponential one. Geometrically speaking, this theorem describes the asymptotic behavior of the cover time of the 3-dimensional simple random walk on the main diagonal d :={(x,y,z) ∈ ℝ³ : x = y = z. This interpretation correlates our research results with the classical limit theorem for local and cover times of random walks. The research results can be used for statistical estimation of probability characteristics of the walks by the number of collisions observed. Specifically, it can be applied for building confidence intervals and testing statistical hypotheses for the parameter p. [ABSTRACT FROM AUTHOR]