scholarly journals Average properties of combinatorial problems and thermodynamics of spin models on graphs

2003 ◽  
Vol DMTCS Proceedings vol. AC,... (Proceedings) ◽  
Author(s):  
Alessandro Vezzani ◽  
Davide Cassi ◽  
Raffaella Burioni
Keyword(s):  
Random Walks ◽  
Spontaneous Magnetization ◽  
Discrete Symmetry ◽  
Combinatorial Problems ◽  
Infinite Graphs ◽  
Continuous Symmetry ◽  
Spin Models ◽  
Graph Topology ◽  
International Audience ◽  
Classical Spin Models

International audience The study of thermodynamic properties of classical spin models on infinite graphs naturally leads to consider the new combinatorial problems of random-walks and percolation on the average. Indeed, spinmodels with O(n) continuous symmetry present spontaneous magnetization only on transient on the average graphs, while models with discrete symmetry (Ising and Potts) are spontaneously magnetized on graphs exhibiting percolation on the average. In this paper we define the combinatorial problems on the average, showing that they give rise to classifications of graph topology which are different from the ones obtained in usual (local) random-walks and percolation. Furthermore, we illustrate the theorem proving the correspondence between Potts model and average percolation.

scholarly journals Discrete Random Walks on One-Sided ``Periodic'' Graphs

2003 ◽  
Vol DMTCS Proceedings vol. AC,... (Proceedings) ◽  
Author(s):  
Michael Drmota
Keyword(s):  
Brownian Motion ◽  
Random Walk ◽  
Random Walks ◽  
Generating Functions ◽  
Connected Graph ◽  
Infinite Graphs ◽  
Reflected Brownian Motion ◽  
Strongly Connected ◽  
International Audience ◽  
Null Recurrence

International audience In this paper we consider discrete random walks on infinite graphs that are generated by copying and shifting one finite (strongly connected) graph into one direction and connecting successive copies always in the same way. With help of generating functions it is shown that there are only three types for the asymptotic behaviour of the random walk. It either converges to the stationary distribution or it can be approximated in terms of a reflected Brownian motion or by a Brownian motion. In terms of Markov chains these cases correspond to positive recurrence, to null recurrence, and to non recurrence.

scholarly journals Classical spin models with broken symmetry: Random-field-induced order and persistence of spontaneous magnetization in the presence of a random field

2014 ◽  
Vol 90 (17) ◽  
Author(s):  
Anindita Bera ◽  
Debraj Rakshit ◽  
Maciej Lewenstein ◽  
Aditi Sen(De) ◽  
Ujjwal Sen ◽  
...  
Keyword(s):  
Random Field ◽  
Spontaneous Magnetization ◽  
Broken Symmetry ◽  
Spin Models ◽  
Classical Spin ◽  
Classical Spin Models

scholarly journals Evidence of exactness of the mean-field theory in the nonextensive regime of long-range classical spin models

2000 ◽  
Vol 61 (17) ◽  
pp. 11521-11528 ◽  
Author(s):  
Sergio A. Cannas ◽  
A. C. N. de Magalhães ◽  
Francisco A. Tamarit
Keyword(s):  
Field Theory ◽  
Long Range ◽  
Mean Field Theory ◽  
Mean Field ◽  
Spin Models ◽  
Classical Spin ◽  
The Mean ◽  
Classical Spin Models

scholarly journals Nanolaser-based emulators of spin Hamiltonians

Nanophotonics ◽  
2020 ◽  
Vol 9 (13) ◽  
pp. 4193-4198 ◽  
Author(s):  
Midya Parto ◽  
William E. Hayenga ◽  
Alireza Marandi ◽  
Demetrios N. Christodoulides ◽  
Mercedeh Khajavikhan
Keyword(s):  
Large Scale ◽  
Optimization Problems ◽  
Exchange Interactions ◽  
Np Hard ◽  
Spin Models ◽  
Geometric Frustration ◽  
Spin Hamiltonians ◽  
Hard Problems ◽  
On Chip ◽  
Classical Spin Models

AbstractFinding the solution to a large category of optimization problems, known as the NP-hard class, requires an exponentially increasing solution time using conventional computers. Lately, there has been intense efforts to develop alternative computational methods capable of addressing such tasks. In this regard, spin Hamiltonians, which originally arose in describing exchange interactions in magnetic materials, have recently been pursued as a powerful computational tool. Along these lines, it has been shown that solving NP-hard problems can be effectively mapped into finding the ground state of certain types of classical spin models. Here, we show that arrays of metallic nanolasers provide an ultra-compact, on-chip platform capable of implementing spin models, including the classical Ising and XY Hamiltonians. Various regimes of behavior including ferromagnetic, antiferromagnetic, as well as geometric frustration are observed in these structures. Our work paves the way towards nanoscale spin-emulators that enable efficient modeling of large-scale complex networks.

scholarly journals Area of Brownian Motion with Generatingfunctionology

2003 ◽  
Vol DMTCS Proceedings vol. AC,... (Proceedings) ◽  
Author(s):  
Michel Nguyên Thê
Keyword(s):  
Brownian Motion ◽  
Random Walks ◽  
Generating Functions ◽  
Limit Distributions ◽  
Dyck Paths ◽  
Different Types ◽  
International Audience

International audience This paper gives a survey of the limit distributions of the areas of different types of random walks, namely Dyck paths, bilateral Dyck paths, meanders, and Bernoulli random walks, using the technology of generating functions only.

scholarly journals Infinite limits and folding

2005 ◽  
Vol DMTCS Proceedings vol. AE,... (Proceedings) ◽  
Author(s):  
Anthony Bonato ◽  
Jeannette Janssen
Keyword(s):  
Biological Networks ◽  
Infinite Graphs ◽  
Isomorphism Type ◽  
Connected Component ◽  
Induced Subgraph ◽  
Pursuit Games ◽  
Graph Homomorphisms ◽  
International Audience

International audience We study infinite limits of graphs generated by the duplication model for biological networks. We prove that with probability 1, the sole nontrivial connected component of the limits is unique up to isomorphism. We describe certain infinite deterministic graphs which arise naturally from the model. We characterize the isomorphism type and induced subgraph structure of these infinite graphs using the notion of dismantlability from the theory of vertex pursuit games, and graph homomorphisms.

scholarly journals Hypergroups derived from random walks on some infinite graphs

2019 ◽  
Vol 189 (2) ◽  
pp. 321-353
Author(s):  
Tomohiro Ikkai ◽  
Yusuke Sawada
Keyword(s):  
Random Walks ◽  
Infinite Graphs
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