On the extreme gap in the renewal process
On the large deviations of a class of modulated additive processes
We prove that the large deviation principle holds for a class of processes inspired by semi-Markov additive processes. For the processes we consider, the sojourn times in the phase process need not be independent and identically distributed. Moreover the state selection process need not be independent of the sojourn times. We assume that the phase process takes values in a finite set and that the order in which elements in the set, called states, are visited is selected stochastically. The sojourn...
On the large deviations of a class of modulated additive processes
We prove that the large deviation principle holds for a class of processes inspired by semi-Markov additive processes. For the processes we consider, the sojourn times in the phase process need not be independent and identically distributed. Moreover the state selection process need not be independent of the sojourn times. We assume that the phase process takes values in a finite set and that the order in which elements in the set, called states, are visited is selected stochastically. The sojourn...
On the level crossing of multi-dimensional delayed renewal processes.
On the limiting velocity of random walks in mixing random environment
We consider random walks in strong-mixing random Gibbsian environments in , . Based on regeneration arguments, we will first provide an alternative proof of Rassoul-Agha’s conditional law of large numbers (CLLN) for mixing environment (Electron. Commun. Probab.10(2005) 36–44). Then, using coupling techniques, we show that there is at most one nonzero limiting velocity in high dimensions ().
On the M/G/1 retrial queue subjected to breakdowns
Retrial queueing systems are characterized by the requirement that customers finding the service area busy must join the retrial group and reapply for service at random intervals. This paper deals with the M/G/1 retrial queue subjected to breakdowns. We use its stochastic decomposition property to approximate the model performance in the case of general retrial times.
On the M/G/1 retrial queue subjected to breakdowns
Retrial queueing systems are characterized by the requirement that customers finding the service area busy must join the retrial group and reapply for service at random intervals. This paper deals with the M/G/1 retrial queue subjected to breakdowns. We use its stochastic decomposition property to approximate the model performance in the case of general retrial times.
On the mixed even-spin Sherrington–Kirkpatrick model with ferromagnetic interaction
We study a spin system with both mixed even-spin Sherrington–Kirkpatrick (SK) couplings and Curie–Weiss (CW) interaction. Our main results are: (i) The thermodynamic limit of the free energy is given by a variational formula involving the free energy of the SK model with a change in the external field. (ii) In the presence of a centered Gaussian external field, the positivity of the overlap and the extended Ghirlanda–Guerra identities hold on a dense subset of the temperature parameters. (iii) We...
On the multiple overlap function of the SK model.
In this note, we prove an asymptotic expansion and a central limit theorem for the multiple overlap R1, ..., s of the SK model, defined for given N, s ≥ 1 by R1, ..., s = N-1Σi≤N σ1i ... σsi. These results are obtained by a careful analysis of the terms appearing in the cavity derivation formula, as well as some graph induction procedures. Our method could hopefully be applied to other spin glasses models.
On the non-convexity of the time constant in first-passage percolation.
On the non-equivalence of two criteria of comparability of stationary point processes
On the number of collisions in lambda-coalescents.
On the number of ground states of the Edwards–Anderson spin glass model
Ground states of the Edwards–Anderson (EA) spin glass model are studied on infinite graphs with finite degree. Ground states are spin configurations that locally minimize the EA Hamiltonian on each finite set of vertices. A problem with far-reaching consequences in mathematics and physics is to determine the number of ground states for the model on for any . This problem can be seen as the spin glass version of determining the number of infinite geodesics in first-passage percolation or the number...
On the number of word occurrences in a semi-Markov sequence of letters
Let a finite alphabet Ω. We consider a sequence of letters from Ω generated by a discrete time semi-Markov process We derive the probability of a word occurrence in the sequence. We also obtain results for the mean and variance of the number of overlapping occurrences of a word in a finite discrete time semi-Markov sequence of letters under certain conditions.
On the preemptive priority queues
On the proof of the Parisi formula by Guerra and Talagrand
The Parisi formula is an expression for the limiting free energy of the Sherrington-Kirkpatrick spin glass model, which had first been derived by Parisi using a non-rigorous replica method together with an hierarchical ansatz for the solution of the variational problem. It had become quickly clear that behind the solution, if correct, lies an interesting mathematical structure. The formula has recently been proved by Michel Talagrand based partly on earlier ideas and results by Francesco Guerra....
On the Queueing Problem of Correlated Arrivals with Many Servers in a Finite Waiting Space.
On the queue-size distribution in the multi-server system with bounded capacity and packet dropping
A multi-server -type queueing system with a bounded total volume and finite queue size is considered. An AQM algorithm with the “accepting” function is being used to control the arrival process of incoming packets. The stationary queue-size distribution and the loss probability are derived. Numerical examples illustrating theoretical results are attached as well.
On the range of the simple random walk bridge on groups.
On the small maximal flows in first passage percolation
We consider the standard first passage percolation on : with each edge of the lattice we associate a random capacity. We are interested in the maximal flow through a cylinder in this graph. Under some assumptions Kesten proved in 1987 a law of large numbers for the rescaled flow. Chayes and Chayes established that the large deviations far away below its typical value are of surface order, at least for the Bernoulli percolation and cylinders of certain height. Thanks to another approach we extend...