| Grant number: | 16/03988-5 |
| Support type: | Scholarships in Brazil - Post-Doctorate |
| Effective date (Start): | June 01, 2016 |
| Effective date (End): | May 31, 2017 |
| Field of knowledge: | Physical Sciences and Mathematics - Probability and Statistics - Probability |
| Principal Investigator: | Jefferson Antonio Galves |
| Grantee: | Achillefs Tzioufas |
| Home Institution: | Instituto de Matemática e Estatística (IME). Universidade de São Paulo (USP). São Paulo , SP, Brazil |
| Associated research grant: | 13/07699-0 - Research, Innovation and Dissemination Center for Neuromathematics - NeuroMat, AP.CEPID |
Abstract Overview Chains with memory of variable length is a family of nite non-Markovian processes have been of great interest for probabilists and have found many applications; we refer to [2] for a review and background on this topic. Interacting particle systems is a class of Markovian processes that has received profound interest as a eld of probability; we refer to [5, 6] for background. Recently an extension of both classes of processes was issued in [4], where the existence of stationary versions of innite non-Markovian interacting particle systems processes has been established. Furthermore, in the case that the interactions between components are given by a critical directed Erdos-Renyi-type random graph, explicit upper-bound for the correlation between successive inter-spike intervals have been provided. We propose to investigate the existence of a phase transition, that is, the uniqueness or not of the invariant measure as a function of the summability or not of the synaptic weights of the model. This is a natural follow-up in the study of the Galves-Locherbach model, which could provide a microscopic interpretation of the existence of dierent macroscopic states. We propose investigating the extension of the existence result for non stationary under strong additional assumptions for a special subclass of these processes. In addition, we examine the related aspect that the interactions between components are given by random regular graphs. Starting with the seminal work of [1] metastability results for interacting particle systems have been of major importance, and metastability for a process on this class of graphs has been investigated recently [7]. We shall investigate the extension of these results for Galves-Locherbach models. (AU) | |