## 2006 |

Taloni, Alessandro; Cagliori, Emanuele; Loreto, Vittorio; Pietronero, Luciano Conformal approach to cylindrical DLA Journal Article JOURNAL OF STATISTICAL MECHANICS: THEORY AND EXPERIMENT, Art. n. P09004 , 2006. Abstract | Links | BibTeX | Tag: caglioti, DLA, fractals, loreto, pietronero, self_organization, statistical_physics, taloni @article{b, title = {Conformal approach to cylindrical DLA}, author = {Alessandro Taloni and Emanuele Cagliori and Vittorio Loreto and Luciano Pietronero}, url = {http://www.scopus.com/inward/record.url?eid=2-s2.0-37649026579&partnerID=65&md5=18e07a1e17966902ff4a2c6e9b9dadfc http://gateway.webofknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&KeyUT=000241413300004&DestLinkType=FullRecord&DestApp=ALL_WOS&UsrCustomerID=0c7ff228ccbaaa74236f48834a34396a}, year = {2006}, date = {2006-01-01}, journal = {JOURNAL OF STATISTICAL MECHANICS: THEORY AND EXPERIMENT}, volume = {Art. n. P09004}, abstract = {We extend the conformal mapping approach elaborated for the radial diffusion limited aggregation model (DLA) to cylindrical geometry. We introduce in particular a complex function which allows a cylindrical cluster to be grown using as an intermediate step a radial aggregate. The aggregate grown exhibits the same self-affine features as the original cylindrical DLA. The specific choice of the transformation allows us to study the relationship between the radial and the cylindrical geometry. In particular the cylindrical aggregate can be seen as a radial aggregate with particles of size increasing with the radius. On the other hand, the radial aggregate can be seen as a cylindrical aggregate with particles of size decreasing with the height. This framework, which shifts the point of view from the geometry to the size of the particles, can open the way to more quantitative studies on the relationship between radial and cylindrical DLA.}, keywords = {caglioti, DLA, fractals, loreto, pietronero, self_organization, statistical_physics, taloni}, pubstate = {published}, tppubtype = {article} } We extend the conformal mapping approach elaborated for the radial diffusion limited aggregation model (DLA) to cylindrical geometry. We introduce in particular a complex function which allows a cylindrical cluster to be grown using as an intermediate step a radial aggregate. The aggregate grown exhibits the same self-affine features as the original cylindrical DLA. The specific choice of the transformation allows us to study the relationship between the radial and the cylindrical geometry. In particular the cylindrical aggregate can be seen as a radial aggregate with particles of size increasing with the radius. On the other hand, the radial aggregate can be seen as a cylindrical aggregate with particles of size decreasing with the height. This framework, which shifts the point of view from the geometry to the size of the particles, can open the way to more quantitative studies on the relationship between radial and cylindrical DLA. |

## 2002 |

Ciliberti, Sergio; Caldarelli, Guido; Loreto, Vittorio; Pietronero, Luciano Local Rigidity in sandpile models Journal Article PHYSICAL REVIEW E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS, 66 , pp. 016133:1–016133:7, 2002. Links | BibTeX | Tag: caldarelli, ciliberti, DLA, loreto, pietronero, self_organization, statistical_physics @article{b, title = {Local Rigidity in sandpile models}, author = {Sergio Ciliberti and Guido Caldarelli and Vittorio Loreto and Luciano Pietronero}, url = {http://www.scopus.com/inward/record.url?eid=2-s2.0-41349104849&partnerID=65&md5=aacf73720b482187890036fa4974915f http://gateway.webofknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&KeyUT=000177200600046&DestLinkType=FullRecord&DestApp=ALL_WOS&UsrCustomerID=0c7ff228ccbaaa74236f48834a34396a}, year = {2002}, date = {2002-01-01}, journal = {PHYSICAL REVIEW E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS}, volume = {66}, pages = {016133:1--016133:7}, keywords = {caldarelli, ciliberti, DLA, loreto, pietronero, self_organization, statistical_physics}, pubstate = {published}, tppubtype = {article} } |

## 1997 |

Vespignani, Alessandro; Zapperi, Stefano; Loreto, Vittorio Dynamically driven renormalization group Journal Article JOURNAL OF STATISTICAL PHYSICS, 88 , pp. 47–79, 1997. Abstract | Links | BibTeX | Tag: DDRG, loreto, self_organization, statistical_physics, vespignani, zapperi @article{b, title = {Dynamically driven renormalization group}, author = {Alessandro Vespignani and Stefano Zapperi and Vittorio Loreto}, url = {http://www.springerlink.com/content/d1215122128712g1/, http://gateway.webofknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&KeyUT=A1997XT83300003&DestLinkType=FullRecord&DestApp=ALL_WOS&UsrCustomerID=0c7ff228ccbaaa74236f48834a34396a}, year = {1997}, date = {1997-01-01}, journal = {JOURNAL OF STATISTICAL PHYSICS}, volume = {88}, pages = {47--79}, abstract = {We present a detailed discussion of a novel dynamical renormalization group scheme: the dynamically driven renormalization group (DDRG). This is a general renormalization method developed for dynamical systems with non-equilibrium critical steady state. The method is based on a real-space renormalization scheme driven by a dynamical steady-state condition which acts as a feedback on the transformation equations. This approach has been applied to open nonlinear systems such as self-organized critical phenomena, and it allows the analytical evaluation of scalling dimensions and critical exponents. Equilibrium models at the critical point can also be considered. The explicit application to some models and the corresponding results are discussed.}, keywords = {DDRG, loreto, self_organization, statistical_physics, vespignani, zapperi}, pubstate = {published}, tppubtype = {article} } We present a detailed discussion of a novel dynamical renormalization group scheme: the dynamically driven renormalization group (DDRG). This is a general renormalization method developed for dynamical systems with non-equilibrium critical steady state. The method is based on a real-space renormalization scheme driven by a dynamical steady-state condition which acts as a feedback on the transformation equations. This approach has been applied to open nonlinear systems such as self-organized critical phenomena, and it allows the analytical evaluation of scalling dimensions and critical exponents. Equilibrium models at the critical point can also be considered. The explicit application to some models and the corresponding results are discussed. |

Hallgass, Riccardo; Loreto, Vittorio; Mazzella, Orfeo; Paladin, Giovanni Earthquakes statistics and fractal faults Journal Article PHYSICAL REVIEW E, 56 , pp. 1346–1356, 1997. Abstract | Links | BibTeX | Tag: earthquakes, hallgass, loreto, mazzella, paladin, self_organization, statistical_physics @article{b, title = {Earthquakes statistics and fractal faults}, author = {Riccardo Hallgass and Vittorio Loreto and Orfeo Mazzella and Giovanni Paladin}, url = {http://pre.aps.org/abstract/PRE/v56/i2/p1346_1, http://gateway.webofknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&KeyUT=A1997XR16000015&DestLinkType=FullRecord&DestApp=ALL_WOS&UsrCustomerID=0c7ff228ccbaaa74236f48834a34396a}, year = {1997}, date = {1997-01-01}, journal = {PHYSICAL REVIEW E}, volume = {56}, pages = {1346--1356}, publisher = {AMERICAN PHYSICAL SOC}, abstract = {We introduce a Self-affine Asperity Model (SAM) for the seismicity that mimics the fault friction by means of two fractional Brownian profiles (fBm) that slide one over the other. An earthquake occurs when there is an overlap of the two profiles representing the two fault faces and its energy is assumed proportional to the overlap surface. The SAM exhibits the Gutenberg-Richter law with an exponent $beta$ related to the roughness index of the profiles. Apart from being analytically treatable, the model exhibits a non-trivial clustering in the spatio-temporal distribution of epicenters that strongly resembles the experimentally observed one. A generalized and more realistic version of the model exhibits the Omori scaling for the distribution of the aftershocks. The SAM lies in a different perspective with respect to usual models for seismicity. In this case, in fact, the critical behaviour is not Self-Organized but stems from the fractal geometry of the faults, which, on its turn, is supposed to arise as a consequence of geological processes on very long time scales with respect to the seismic dynamics. The explicit introduction of the fault geometry, as an active element of this complex phenomenology, represents the real novelty of our approach.}, keywords = {earthquakes, hallgass, loreto, mazzella, paladin, self_organization, statistical_physics}, pubstate = {published}, tppubtype = {article} } We introduce a Self-affine Asperity Model (SAM) for the seismicity that mimics the fault friction by means of two fractional Brownian profiles (fBm) that slide one over the other. An earthquake occurs when there is an overlap of the two profiles representing the two fault faces and its energy is assumed proportional to the overlap surface. The SAM exhibits the Gutenberg-Richter law with an exponent $beta$ related to the roughness index of the profiles. Apart from being analytically treatable, the model exhibits a non-trivial clustering in the spatio-temporal distribution of epicenters that strongly resembles the experimentally observed one. A generalized and more realistic version of the model exhibits the Omori scaling for the distribution of the aftershocks. The SAM lies in a different perspective with respect to usual models for seismicity. In this case, in fact, the critical behaviour is not Self-Organized but stems from the fractal geometry of the faults, which, on its turn, is supposed to arise as a consequence of geological processes on very long time scales with respect to the seismic dynamics. The explicit introduction of the fault geometry, as an active element of this complex phenomenology, represents the real novelty of our approach. |

## 1996 |

Hur, Asa Ben; Hallgass, Riccardo; Loreto, Vittorio A renormalization procedure for directed sandpile models Journal Article PHYSICAL REVIEW E, 54 , pp. 1426–1432, 1996. Abstract | Links | BibTeX | Tag: ben-hur, DDRG, hallgass, loreto, self_organization, statistical_physics @article{b, title = {A renormalization procedure for directed sandpile models}, author = {Asa Ben Hur and Riccardo Hallgass and Vittorio Loreto}, url = {http://pre.aps.org/abstract/PRE/v54/i2/p1426_1, http://gateway.webofknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&KeyUT=A1996VD67400057&DestLinkType=FullRecord&DestApp=ALL_WOS&UsrCustomerID=0c7ff228ccbaaa74236f48834a34396a}, year = {1996}, date = {1996-01-01}, journal = {PHYSICAL REVIEW E}, volume = {54}, pages = {1426--1432}, publisher = {AMERICAN PHYSICAL SOC}, abstract = {Directed models of self-organized criticality are studied in the framework of a real-space renormalization group of a different type. The identification of a suitable phase space in which to define the renormalization transformation and the coupling with the stationarity condition enables us to clarify the nature of the critical state. The renormalization equations are found to have an attractive fixed point, as expected from the self-critical nature of the model. The values of the critical exponents obtained by this procedure are in excellent agreement with exact results.}, keywords = {ben-hur, DDRG, hallgass, loreto, self_organization, statistical_physics}, pubstate = {published}, tppubtype = {article} } Directed models of self-organized criticality are studied in the framework of a real-space renormalization group of a different type. The identification of a suitable phase space in which to define the renormalization transformation and the coupling with the stationarity condition enables us to clarify the nature of the critical state. The renormalization equations are found to have an attractive fixed point, as expected from the self-critical nature of the model. The values of the critical exponents obtained by this procedure are in excellent agreement with exact results. |

Caglioti, Emanuele; Loreto, Vittorio Dynamical properties and predictability in a class of self-organized critical models Journal Article PHYSICAL REVIEW E, 53 , pp. R2953–R2956, 1996. Abstract | Links | BibTeX | Tag: caglioti, dynamical_systems, loreto, self_organization, statistical_physics @article{b, title = {Dynamical properties and predictability in a class of self-organized critical models}, author = {Emanuele Caglioti and Vittorio Loreto}, url = {http://pre.aps.org/abstract/PRE/v53/i3/p2953_1 http://gateway.webofknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&KeyUT=A1996UA37100116&DestLinkType=FullRecord&DestApp=ALL_WOS&UsrCustomerID=0c7ff228ccbaaa74236f48834a34396a}, year = {1996}, date = {1996-01-01}, journal = {PHYSICAL REVIEW E}, volume = {53}, pages = {R2953--R2956}, publisher = {AMERICAN PHYSICAL SOC}, abstract = {We consider a particular class of self-organized critical models. For these systems we show that the Lyapunov exponent is strictly lower than zero. That allows us to describe the dynamics in terms of a piecewise linear contractive map. We describe the physical mechanisms underlying the approach to the recurrent set in the configuration space and we discuss the structure of the attractor for the dynamics. Finally the problem of the chaoticity of these systems and the definition of a predictability are addressed.}, keywords = {caglioti, dynamical_systems, loreto, self_organization, statistical_physics}, pubstate = {published}, tppubtype = {article} } We consider a particular class of self-organized critical models. For these systems we show that the Lyapunov exponent is strictly lower than zero. That allows us to describe the dynamics in terms of a piecewise linear contractive map. We describe the physical mechanisms underlying the approach to the recurrent set in the configuration space and we discuss the structure of the attractor for the dynamics. Finally the problem of the chaoticity of these systems and the definition of a predictability are addressed. |

Vespignani, Alessandro; Zapperi, Stefano; Loreto, Vittorio Renormalization of non-equilibrium systems with critical stationary state Journal Article PHYSICAL REVIEW LETTERS, 77 , pp. 4560–4563, 1996. Abstract | Links | BibTeX | Tag: DDRG, loreto, self_organization, statistical_physics, vespignani, zapperi @article{b, title = {Renormalization of non-equilibrium systems with critical stationary state}, author = {Alessandro Vespignani and Stefano Zapperi and Vittorio Loreto}, url = {http://prl.aps.org/abstract/PRL/v77/i22/p4560_1 http://gateway.webofknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&KeyUT=A1996VU50200020&DestLinkType=FullRecord&DestApp=ALL_WOS&UsrCustomerID=0c7ff228ccbaaa74236f48834a34396a}, year = {1996}, date = {1996-01-01}, journal = {PHYSICAL REVIEW LETTERS}, volume = {77}, pages = {4560--4563}, abstract = {We introduce the general formulation of a renormalization method suitable to study the critical properties of non-equilibrium systems with steady-states: the Dynamically Driven Renormalization Group. We renormalize the time evolution operator by computing the rescaled time transition rate between coarse grained states. The obtained renormalization equations are coupled to a stationarity condition which provides the approximate non-equilibrium statistical weights of steady-state configurations to be used in the calculations. In this way we are able to write recursion relations for the parameters evolution under scale change, from which we can extract numerical values for the critical exponents. This general framework allows the systematic analysis of several models showing self-organized criticality in terms of usual concepts of phase transitions and critical phenomena.}, keywords = {DDRG, loreto, self_organization, statistical_physics, vespignani, zapperi}, pubstate = {published}, tppubtype = {article} } We introduce the general formulation of a renormalization method suitable to study the critical properties of non-equilibrium systems with steady-states: the Dynamically Driven Renormalization Group. We renormalize the time evolution operator by computing the rescaled time transition rate between coarse grained states. The obtained renormalization equations are coupled to a stationarity condition which provides the approximate non-equilibrium statistical weights of steady-state configurations to be used in the calculations. In this way we are able to write recursion relations for the parameters evolution under scale change, from which we can extract numerical values for the critical exponents. This general framework allows the systematic analysis of several models showing self-organized criticality in terms of usual concepts of phase transitions and critical phenomena. |

Loreto, Vittorio; Vespignani, Alessandro; Zapperi, Stefano Renormalization scheme for forest-fire models Journal Article JOURNAL OF PHYSICS. A, MATHEMATICAL AND GENERAL, 29 , pp. 2981–3004, 1996. Links | BibTeX | Tag: DDRG, loreto, self_organization, statistical_physics, vespignani, zapperi @article{b, title = {Renormalization scheme for forest-fire models}, author = {Vittorio Loreto and Alessandro Vespignani and Stefano Zapperi}, url = {http://www.scopus.com/inward/record.url?eid=2-s2.0-0040108749&partnerID=65&md5=f34393a56c2dfc0797f394df2fda943a http://gateway.webofknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&KeyUT=A1996UU80300008&DestLinkType=FullRecord&DestApp=ALL_WOS&UsrCustomerID=0c7ff228ccbaaa74236f48834a34396a}, year = {1996}, date = {1996-01-01}, journal = {JOURNAL OF PHYSICS. A, MATHEMATICAL AND GENERAL}, volume = {29}, pages = {2981--3004}, publisher = {IOP Publishing Limited:Dirac House, Temple Back, Bristol BS1 6BE United Kingdom:011 44 117 9297481, EMAIL: custserv@iop.org, INTERNET: http://www.iop.org, Fax: 011 44 117 9294318}, keywords = {DDRG, loreto, self_organization, statistical_physics, vespignani, zapperi}, pubstate = {published}, tppubtype = {article} } |

## 1995 |

Cafiero, Raffaele; Loreto, Vittorio; Pietronero, Luciano; Vespignani, Alessandro; Zapperi, Stefano Local rigidity and self-organized criticality for avalanches Journal Article EUROPHYSICS LETTERS, 29 , pp. 111–116, 1995. Abstract | Links | BibTeX | Tag: cafiero, loreto, pietronero, self_organization, statistical_physics, vespignani, zapperi @article{b, title = {Local rigidity and self-organized criticality for avalanches}, author = {Raffaele Cafiero and Vittorio Loreto and Luciano Pietronero and Alessandro Vespignani and Stefano Zapperi}, url = {http://iopscience.iop.org/0295-5075/29/2/001;jsessionid=8822A74DE3C96358510326242C4AF4E3.c3 http://gateway.webofknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&KeyUT=A1995QC36900001&DestLinkType=FullRecord&DestApp=ALL_WOS&UsrCustomerID=0c7ff228ccbaaa74236f48834a34396a}, year = {1995}, date = {1995-01-01}, journal = {EUROPHYSICS LETTERS}, volume = {29}, pages = {111--116}, publisher = {EDITIONS PHYSIQUE}, abstract = {The general conditions for a sandpile system to evolve spontaneously into a critical state characterized by a power law distribution of avalanches or bursts are identified as: a) the existence of a stationary state with a global conservation law; b) long-range correlations in the continuum limit (i.e. Laplacian diffusive field); c) the existence of a local rigidity for the microscopic dynamics. These conditions permit a classification of the models that have been considered up to now and the identification of the local rigidity as a new basic parameter that can lead to various possible scenarios ranging continuously from SOC behaviour to standard diffusion.}, keywords = {cafiero, loreto, pietronero, self_organization, statistical_physics, vespignani, zapperi}, pubstate = {published}, tppubtype = {article} } The general conditions for a sandpile system to evolve spontaneously into a critical state characterized by a power law distribution of avalanches or bursts are identified as: a) the existence of a stationary state with a global conservation law; b) long-range correlations in the continuum limit (i.e. Laplacian diffusive field); c) the existence of a local rigidity for the microscopic dynamics. These conditions permit a classification of the models that have been considered up to now and the identification of the local rigidity as a new basic parameter that can lead to various possible scenarios ranging continuously from SOC behaviour to standard diffusion. |

Loreto, Vittorio; Pietronero, Luciano; Vespignani, Alessandro; Zapperi, Stefano Renormalization group approach to the crtical behavior of the forest-fire model Journal Article PHYSICAL REVIEW LETTERS, 75 , pp. 465–468, 1995. Abstract | Links | BibTeX | Tag: DDRG, loreto, pietronero, self_organization, statistical_physics, vespignani, zapperi @article{b, title = {Renormalization group approach to the crtical behavior of the forest-fire model}, author = {Vittorio Loreto and Luciano Pietronero and Alessandro Vespignani and Stefano Zapperi}, url = {http://prl.aps.org/abstract/PRL/v75/i3/p465_1, http://www.scopus.com/inward/record.url?eid=2-s2.0-0000977225&partnerID=65&md5=4f03991047dc2820ef6c00a2e81b3c7d http://gateway.webofknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&KeyUT=A1995RK33000028&DestLinkType=FullRecord&DestApp=ALL_WOS&UsrCustomerID=0c7ff228ccbaaa74236f48834a34396a}, year = {1995}, date = {1995-01-01}, journal = {PHYSICAL REVIEW LETTERS}, volume = {75}, pages = {465--468}, abstract = {We introduce a renormalization scheme for the one- and two-dimensional forest-fire model in order to characterize the nature of the critical state and its scale invariant dynamics. We show the existence of a relevant scaling field associated with a repulsive fixed point. This model is therefore critical in the usual sense because the control parameter has to be tuned to its critical value in order to get criticality. It turns out that this is not just the condition for a time scale separation. The critical exponents are computed analytically and we obtain ν = 1.0, τ = 1.0 and ν = 0.65, τ = 1.16, respectively, for the one- and two-dimensional cases, in very good agreement with numerical simulations.}, keywords = {DDRG, loreto, pietronero, self_organization, statistical_physics, vespignani, zapperi}, pubstate = {published}, tppubtype = {article} } We introduce a renormalization scheme for the one- and two-dimensional forest-fire model in order to characterize the nature of the critical state and its scale invariant dynamics. We show the existence of a relevant scaling field associated with a repulsive fixed point. This model is therefore critical in the usual sense because the control parameter has to be tuned to its critical value in order to get criticality. It turns out that this is not just the condition for a time scale separation. The critical exponents are computed analytically and we obtain ν = 1.0, τ = 1.0 and ν = 0.65, τ = 1.16, respectively, for the one- and two-dimensional cases, in very good agreement with numerical simulations. |

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