The Experts below are selected from a list of 93 Experts worldwide ranked by ideXlab platform
Alexandre Souto Martinez - One of the best experts on this subject based on the ideXlab platform.
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Analytical calculation for the percolation crossover in deterministic partially self-avoiding walks in one-dimensional random media.
Physical Review E, 2007Co-Authors: César Augusto Sangaletti Terçariol, Rodrigo Silva González, Alexandre Souto MartinezAbstract:Consider N Points randomly distributed along a line segment of unitary length. A walker explores this disordered medium, moving according to a partially self-avoiding deterministic walk. The walker, with memory mu , leaves from the Leftmost Point and moves, at each discrete time step, to the nearest Point that has not been visited in the preceding mu steps. Using open boundary conditions, we have calculated analytically the probability P{N}(mu)=(1-2{-mu}){N-mu-1} that all N Points are visited, with N>>mu>>1 . This approximated expression for P{N}(mu) is reasonable even for small N and mu values, as validated by Monte Carlo simulations. We show the existence of a critical memory mu{1}=lnNln2 . For mu mu{1}+e(2ln2) , the walker explores the whole system. Since the intermediate region increases as lnN and its width is constant, a sharp transition is obtained for one-dimensional large systems. This means that the walker need not have full memory of its trajectory to explore the whole system. Instead, it suffices to have memory of order log{2}N .
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Exact analytical calculation for the percolation crossover in deterministic partially self-avoiding walks in one-dimensional random media
arXiv: Disordered Systems and Neural Networks, 2007Co-Authors: César Augusto Sangaletti Terçariol, Rodrigo Silva González, Alexandre Souto MartinezAbstract:Consider $N$ Points randomly distributed along a line segment of unitary length. A walker explores this disordered medium moving according to a partially self-avoiding deterministic walk. The walker, with memory $\mu$, leaves from the Leftmost Point and moves, at each discrete time step, to the nearest Point which has not been visited in the preceding $\mu$ steps. Using open boundary conditions, we have calculated analytically the probability $P_N(\mu) = (1 - 2^{-\mu})^{N - \mu - 1}$ that all $N$ Points are visited, with $N \gg \mu \gg 1$. This approximated expression for $P_N(\mu)$ is reasonable even for small $N$ and $\mu$ values, as validated by Monte Carlo simulations. We show the existence of a critical memory $\mu_1 = \ln N/\ln 2$. For $\mu \mu_1 + e/(2\ln2)$ the walker explores the whole system. Since the intermediate region increases as $\ln N$ and its width is constant, a sharp transition is obtained for one-dimensional large systems. This means that the walker needs not to have full memory of its trajectory to explore the whole system. Instead, it suffices to have memory of order $\log_{2} N$.
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Optimum exploration memory and anomalous diffusion in deterministic partially self-avoiding walks in one-dimensional random media
arXiv: Disordered Systems and Neural Networks, 2006Co-Authors: César Augusto Sangaletti Terçariol, Rodrigo Silva González, Alexandre Souto MartinezAbstract:Consider $N$ Points randomly distributed along a line segment of unitary length. A walker explores this disordered medium moving according to a partially self-avoiding deterministic walk. The walker, with memory $\mu$, leaves from the Leftmost Point and moves, at each discrete time step, to the nearest Point, which has not been visited in the preceding $\mu$ steps. We have obtained analytically the probability $P_N(\mu) = (1 - 2^{-\mu})^{N - \mu - 1}$ that all $N$ Points are visited in this open system, with $N \gg \mu \gg 1$. The expression for $P_N(\mu)$ evaluated in the mentioned limit is valid even for small $N$ and leads to a transition region centered at $\mu_1 = \ln N/\ln 2$ and with width $\epsilon = e/\ln2$. For $\mu \mu_1 + \epsilon/2$ the walker explores the whole system. In both cases the walker presents diffusive behavior. Nevertheless, in the intermediate regime $\mu \sim \mu_1 \pm \epsilon/2$, the walker presents anoumalous diffusion behavior. Since the intermediate region increases as $\ln N$ and its width is constant, a sharp transition is obtained for one-dimensional large systems. The walker does not need to have full memory of its trajectory to explore the whole system, it suffices to have memory of order $\mu_1$.
Toninelli C. - One of the best experts on this subject based on the ideXlab platform.
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Universality for one-dimensional hierarchical coalescence processes with double and triple merges
Institute of Mathematical Statistics (IMS), 2014Co-Authors: Faggionato A., Roberto Cyril, Toninelli C.Abstract:41 pagesInternational audienceWe consider one--dimensional hierarchical coalescence processes (in short HCP) where two or three neighbouring domains can merge. An HCP consists of an infinite sequence of stochastic coalescence processes: each process occurs in a different "epoch" and evolves for an infinite time, while the evolutions in subsequent epochs are linked in such a way that the initial distribution of epoch $n+1$ coincides with the final distribution of epoch $n$. Inside each epoch a domain can incorporate one of its neighbouring domains or both of them if its length belongs to a certain epoch-dependent finite range. Assuming that the distribution at the beginning of the first epoch is described by a renewal simple Point process, we prove limit theorems for the domain length and for the position of the Leftmost Point (if any). Our analysis extends the results obtained in \cite{FMRT0} to a larger family of models, including relevant examples from the physics literature \cite{BDG}, \cite{SE}. It reveals the presence of a common abstract structure behind models which are apparently very different, thus leading to very similar limit theorems. Finally, we give here a full characterization of the infinitesimal generator for the dynamics inside each epoch, thus allowing to describe the time evolution of the expected value of regular observables in terms of an ordinary differential equation
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Universality for one-dimensional hierarchical coalescence processes with double and triple merges
'Institute of Mathematical Statistics', 2014Co-Authors: Faggionato A., Roberto C., Toninelli C.Abstract:We consider one-dimensional hierarchical coalescence processes (in short HCPs) where two or three neighboring domains can merge. An HCP consists of an infinite sequence of stochastic coalescence processes: each process occurs in a different "epoch" and evolves for an infinite time, while the evolutions in subsequent epochs are linked in such a way that the initial distribution of epoch $n+1$ coincides with the final distribution of epoch $n$. Inside each epoch a domain can incorporate one of its neighboring domains or both of them if its length belongs to a certain epoch-dependent finite range. Assuming that the distribution at the beginning of the first epoch is described by a renewal simple Point process, we prove limit theorems for the domain length and for the position of the Leftmost Point (if any). Our analysis extends the results obtained in [Ann. Probab. 40 (2012) 1377-1435] to a larger family of models, including relevant examples from the physics literature [Europhys. Lett. 27 (1994) 175-180, Phys. Rev. E (3) 68 (2003) 031504]. It reveals the presence of a common abstract structure behind models which are apparently very different, thus leading to very similar limit theorems. Finally, we give here a full characterization of the infinitesimal generator for the dynamics inside each epoch, thus allowing us to describe the time evolution of the expected value of regular observables in terms of an ordinary differential equation.Comment: Published in at http://dx.doi.org/10.1214/12-AAP917 the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org
César Augusto Sangaletti Terçariol - One of the best experts on this subject based on the ideXlab platform.
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Analytical calculation for the percolation crossover in deterministic partially self-avoiding walks in one-dimensional random media.
Physical Review E, 2007Co-Authors: César Augusto Sangaletti Terçariol, Rodrigo Silva González, Alexandre Souto MartinezAbstract:Consider N Points randomly distributed along a line segment of unitary length. A walker explores this disordered medium, moving according to a partially self-avoiding deterministic walk. The walker, with memory mu , leaves from the Leftmost Point and moves, at each discrete time step, to the nearest Point that has not been visited in the preceding mu steps. Using open boundary conditions, we have calculated analytically the probability P{N}(mu)=(1-2{-mu}){N-mu-1} that all N Points are visited, with N>>mu>>1 . This approximated expression for P{N}(mu) is reasonable even for small N and mu values, as validated by Monte Carlo simulations. We show the existence of a critical memory mu{1}=lnNln2 . For mu mu{1}+e(2ln2) , the walker explores the whole system. Since the intermediate region increases as lnN and its width is constant, a sharp transition is obtained for one-dimensional large systems. This means that the walker need not have full memory of its trajectory to explore the whole system. Instead, it suffices to have memory of order log{2}N .
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Exact analytical calculation for the percolation crossover in deterministic partially self-avoiding walks in one-dimensional random media
arXiv: Disordered Systems and Neural Networks, 2007Co-Authors: César Augusto Sangaletti Terçariol, Rodrigo Silva González, Alexandre Souto MartinezAbstract:Consider $N$ Points randomly distributed along a line segment of unitary length. A walker explores this disordered medium moving according to a partially self-avoiding deterministic walk. The walker, with memory $\mu$, leaves from the Leftmost Point and moves, at each discrete time step, to the nearest Point which has not been visited in the preceding $\mu$ steps. Using open boundary conditions, we have calculated analytically the probability $P_N(\mu) = (1 - 2^{-\mu})^{N - \mu - 1}$ that all $N$ Points are visited, with $N \gg \mu \gg 1$. This approximated expression for $P_N(\mu)$ is reasonable even for small $N$ and $\mu$ values, as validated by Monte Carlo simulations. We show the existence of a critical memory $\mu_1 = \ln N/\ln 2$. For $\mu \mu_1 + e/(2\ln2)$ the walker explores the whole system. Since the intermediate region increases as $\ln N$ and its width is constant, a sharp transition is obtained for one-dimensional large systems. This means that the walker needs not to have full memory of its trajectory to explore the whole system. Instead, it suffices to have memory of order $\log_{2} N$.
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Optimum exploration memory and anomalous diffusion in deterministic partially self-avoiding walks in one-dimensional random media
arXiv: Disordered Systems and Neural Networks, 2006Co-Authors: César Augusto Sangaletti Terçariol, Rodrigo Silva González, Alexandre Souto MartinezAbstract:Consider $N$ Points randomly distributed along a line segment of unitary length. A walker explores this disordered medium moving according to a partially self-avoiding deterministic walk. The walker, with memory $\mu$, leaves from the Leftmost Point and moves, at each discrete time step, to the nearest Point, which has not been visited in the preceding $\mu$ steps. We have obtained analytically the probability $P_N(\mu) = (1 - 2^{-\mu})^{N - \mu - 1}$ that all $N$ Points are visited in this open system, with $N \gg \mu \gg 1$. The expression for $P_N(\mu)$ evaluated in the mentioned limit is valid even for small $N$ and leads to a transition region centered at $\mu_1 = \ln N/\ln 2$ and with width $\epsilon = e/\ln2$. For $\mu \mu_1 + \epsilon/2$ the walker explores the whole system. In both cases the walker presents diffusive behavior. Nevertheless, in the intermediate regime $\mu \sim \mu_1 \pm \epsilon/2$, the walker presents anoumalous diffusion behavior. Since the intermediate region increases as $\ln N$ and its width is constant, a sharp transition is obtained for one-dimensional large systems. The walker does not need to have full memory of its trajectory to explore the whole system, it suffices to have memory of order $\mu_1$.
Elena E. Dyakonova - One of the best experts on this subject based on the ideXlab platform.
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Waves in Reduced Branching Processes in a Random Environment
Theory of Probability & Its Applications, 2009Co-Authors: Vladimir Vatutin, Elena E. DyakonovaAbstract:Let $Z(n)$, $n=0,1\ldots,$ be a branching process evolving in the random environment generated by a sequence of independent identically distributed generating functions $f_{0}(s),f_{1}(s),\ldots,$ and let $S_{0}=0$, $S_{k}=X_{1}+\cdots+X_{k}$, $k\ge1,$ be the associated random walk with $X_{i}=\log f_{i-1}'(1),$ and $\tau (n)$ be the Leftmost Point of the minimum of $\{ S_{k}$,$k\ge0\} $ on the interval $[0,n].$ Denoting by $Z(k,m)$ the number of particles existing in the branching process at the time moment $k\le m$ which have nonempty offspring at the time moment m, and assuming that the associated random walk satisfies the Doney condition ${\bf P}( S_{n}>0)\to \rho \in (0,1)$, $n\to \infty,$ we prove (under the quenched approach) conditional limit theorems, as $n\to \infty $, for the distribution of $Z(nt_{1},nt_{2})$, $0 0$. It is shown that the form of the limit distributions essentially depends on the position of $\tau (n)$ with respect to the interval $[nt_{1},nt_{2}].$
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Limit Theorems for Reduced Branching Processes in a Random Environment
Theory of Probability & Its Applications, 2008Co-Authors: Vladimir Vatutin, Elena E. DyakonovaAbstract:Let $Z(n)$, $n=0,1,\ldots,$ be a branching process evolving in the random environment generated by a sequence of independent identically distributed generating functions $f_{0}(s),f_{1}(s),\ldots,$ and let $S_{0}=0$, $S_{k}=X_{1}+\cdots +X_{k}$, $k\geq 1$, be the associated random walk with $X_{i}=\log f_{i-1}^{\prime }(1)$, and let $\tau (n)$ be the Leftmost Point of minimum of $\{S_{k}\}_{k\geq 0}$ on the interval $[0,n]$. Denoting by $Z(k,n)$ the number of particles existing in the branching process at moment $k\leq n$ and having nonempty offspring at time n and assuming that the associated random walk satisfies the Spitzer–Doney condition ${\bf P}\{S_n>0\}\to \rho\in (0,1)$, $n\to \infty$, we show (under the quenched approach) that for each fixed ${m=0,\pm 1,\pm 2,\ldots}$ the distribution of $Z(\tau (n)+m,n)$ given $Z(n)>0$ converges as $n\to \infty $ to a (random) discrete distribution which is proper with probability 1. On the other hand, if $m=m(n)\to \infty $ as $n\to \infty$, then to prove a con...
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Branching Processes in a Random Environment and Bottlenecks in the Evolution of Populations
Theory of Probability & Its Applications, 2007Co-Authors: Vladimir Vatutin, Elena E. DyakonovaAbstract:A branching process $Z(n)$, $n=0,1,\ldots,$ is considered which evolves in a random environment generated by a sequence of independent identically distributed generating functions $f_0(s),f_{1}(s),\ldots\;$. Let $S_0=0$, $S_k=\log f_0^{\prime}(1)+\cdots +\log f_{k-1}^{\prime }(1)$, $k\ge 1$, be the associated random walk and let $\tau (n)$ be the Leftmost Point of minimum of $\{S_k\}_{k\ge 0}$ on the interval $[0,n]$. Assuming that the random walk satisfies the Spitzer condition $n^{-1}\sum_{k=1}^{n}{\bf P}\{S_k>0\}\to\rho\in(0,1)$, $n\to \infty$, we show (under the quenched approach) that for each fixed $t\in (0,1]$ and $m=0,\pm 1,\pm 2,\ldots,$ the distribution of $Z(\tau (nt)+m)$ given $Z(n)>0$ converges as $n\to \infty$ to a (random) discrete distribution. Thus, in contrast to fixed Points of the form $nt$, where the size of the population is large (even exponentially large; see [V. A. Vatutin and E. E. Dyakonova, Theory Probab. Appl., 49 (2005), pp. 275–309]), the size of the population at (random) p...
Faggionato A. - One of the best experts on this subject based on the ideXlab platform.
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Universality for one-dimensional hierarchical coalescence processes with double and triple merges
Institute of Mathematical Statistics (IMS), 2014Co-Authors: Faggionato A., Roberto Cyril, Toninelli C.Abstract:41 pagesInternational audienceWe consider one--dimensional hierarchical coalescence processes (in short HCP) where two or three neighbouring domains can merge. An HCP consists of an infinite sequence of stochastic coalescence processes: each process occurs in a different "epoch" and evolves for an infinite time, while the evolutions in subsequent epochs are linked in such a way that the initial distribution of epoch $n+1$ coincides with the final distribution of epoch $n$. Inside each epoch a domain can incorporate one of its neighbouring domains or both of them if its length belongs to a certain epoch-dependent finite range. Assuming that the distribution at the beginning of the first epoch is described by a renewal simple Point process, we prove limit theorems for the domain length and for the position of the Leftmost Point (if any). Our analysis extends the results obtained in \cite{FMRT0} to a larger family of models, including relevant examples from the physics literature \cite{BDG}, \cite{SE}. It reveals the presence of a common abstract structure behind models which are apparently very different, thus leading to very similar limit theorems. Finally, we give here a full characterization of the infinitesimal generator for the dynamics inside each epoch, thus allowing to describe the time evolution of the expected value of regular observables in terms of an ordinary differential equation
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Universality for one-dimensional hierarchical coalescence processes with double and triple merges
'Institute of Mathematical Statistics', 2014Co-Authors: Faggionato A., Roberto C., Toninelli C.Abstract:We consider one-dimensional hierarchical coalescence processes (in short HCPs) where two or three neighboring domains can merge. An HCP consists of an infinite sequence of stochastic coalescence processes: each process occurs in a different "epoch" and evolves for an infinite time, while the evolutions in subsequent epochs are linked in such a way that the initial distribution of epoch $n+1$ coincides with the final distribution of epoch $n$. Inside each epoch a domain can incorporate one of its neighboring domains or both of them if its length belongs to a certain epoch-dependent finite range. Assuming that the distribution at the beginning of the first epoch is described by a renewal simple Point process, we prove limit theorems for the domain length and for the position of the Leftmost Point (if any). Our analysis extends the results obtained in [Ann. Probab. 40 (2012) 1377-1435] to a larger family of models, including relevant examples from the physics literature [Europhys. Lett. 27 (1994) 175-180, Phys. Rev. E (3) 68 (2003) 031504]. It reveals the presence of a common abstract structure behind models which are apparently very different, thus leading to very similar limit theorems. Finally, we give here a full characterization of the infinitesimal generator for the dynamics inside each epoch, thus allowing us to describe the time evolution of the expected value of regular observables in terms of an ordinary differential equation.Comment: Published in at http://dx.doi.org/10.1214/12-AAP917 the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org