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The gossip problem (telephone problem) is an information dissemination problem in which each of $n$ nodes of a communication network has a unique piece of information that must be transmitted to all the other nodes using two-way communications (telephone calls) between the pairs of nodes. During a call between the given two nodes, they exchange the whole information known to them at that moment. In this paper we investigate the $k$-fault-tolerant gossip problem, which is a generalization of the gossip problem, where at most $k$ arbitrary faults of calls are allowed. The problem is to find the minimal number of calls $\tau(n,k)$ needed to guarantee the $k$-fault-tolerance. We construct two classes of $k$-fault-tolerant gossip schemes (sequences of calls) and found two upper bounds of $\tau(n,k)$, which improve the previously known results. The first upper bound for general even $n$ is $\tau(n,k) \leq 1/2 n \lceil\log_2 n\rceil + 1/2 n k$. This result is used to obtain the upper bound for general odd $n$. From the expressions for the second upper bound it follows that $\tau(n,k) \leq 2/3 n k + O(n)$ for large $n$. Assuming that the calls can take place simultaneously, it is also of interest to find $k$-fault-tolerant gossip schemes, which can spread the full information in minimal time. For even $n$ we showed that the minimal time is $T(n,k)=\lceil\log_2 n\rceil + k$., Comment: 19 pages, 5 figures

The two-dimensional dense O(n) loop model for $n=1$ is equivalent to the bond percolation and for $n=0$ to the dense polymers or spanning trees. We consider the boundary correlations on the half space and calculate the probability $P_b$ that a cluster of bonds has a single common point with the boundary. In the limit $n\rightarrow 0$, we find an analytical expression for $P_b$ using the generalized Kirchhoff theorem., 23 pages, 14 figures

Single site height probabilities in the Abelian sandpile model, and the corresponding mean height $$, are directly related to the probability $P_{\rm ret}$ that a loop erased random walk passes through a nearest neighbour of the starting site (return probability). The exact values of these quantities on the square lattice have been conjectured, in particular $ = 25/8$ and $P_{\rm ret} = 5/16$. We provide a rigourous proof of this conjecture by using a {\it local} monomer-dimer formulation of these questions., Comment: 12 pages, 11 figures

We consider the totally asymmetric exclusion process on a ring in discrete time with the backward-ordered sequential update and particle-dependent hopping probabilities. Using a combinatorial treatment of the Bethe ansatz, we derive the determinant expression for the non-stationary probability of transitions between particle configurations. In the continuous-time limit, we find a generalization of the recent result, obtained by A. R\'akos and G.M. Sch\"utz for infinite lattice, to the case of ring geometry., Comment: 4 pages, RevTeX4, 2 EPS figures