Critical behavior of the<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mrow><mml:mi>X</mml:mi><mml:mi>Y</mml:mi></mml:mrow></mml:math>model in complex topologies

Ibáñez-Berganza, Miguel; Leuzzi, Luca

doi:10.1103/physrevb.88.144104

Cited by 29 publications

(36 citation statements)

References 74 publications

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“…For the LR percolation model under exam we expect the existence of a SR effective dimension d eff such that the universal quantities of the LR model can be related to the ones of the corresponding SR in d eff dimensions [30,34,46]. Such effective dimension relation reproduces standard scaling arguments and it reproduces correctly the qualitative features of phase diagrams [30].…”

Section: Critical Exponentsmentioning

confidence: 68%

One-dimensional long-range percolation: A numerical study

et al. 2017

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In this paper we study bond percolation on a one-dimensional chain with power-law bond probability C/r 1+σ , where r is the distance length between distinct sites. We introduce and test an order N Monte Carlo algorithm and we determine as a function of σ the critical value Cc at which percolation occurs. The critical exponents in the range 0 < σ < 1 are reported and compared with mean-field and ε-expansion results. Our analysis is in agreement, up to a numerical precision ≈ 10 −3 , with the mean field result for the anomalous dimension η = 2 − σ, showing that there is no correction to η due to correlation effects.

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Section: Critical Exponentsmentioning

confidence: 68%

One-dimensional long-range percolation: A numerical study

et al. 2017

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“…Indeed, in ref. 24 , for the XY model in d = 2 the three regimes (mean field, ρdependent non-mean field, short range) have been found, with a strong numerical evidence for ρ mf = 3d/2 and a moderate numerical evidence for the more subtle estimation ρ sr 3.75.…”

Section: The Long-range Diluted Xy Model {Sec:contexmentioning

confidence: 86%

“…However, the question of the presence of spontaneous magnetisation has not been directly addressed in ref. 24 .…”

Section: The Long-range Diluted Xy Model {Sec:contexmentioning

confidence: 99%

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Analysis of the low-temperature phase in the two-dimensional long-range diluted XY model

et al. 2019

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The critical behaviour of statistical models with long-range interactions exhibits distinct regimes as a function of ρ, the power of the interaction strength decay. For ρ large enough, ρ > ρ sr , the critical behaviour is observed to coincide with that of the short-range model. However, there are controversial aspects regarding this picture, one of which is the value of the short-range threshold ρ sr in the case of the long-range XY model in two dimensions. We study the 2d XY model on the diluted graph, a sparse graph obtained from the 2d lattice by rewiring links with probability decaying with the Euclidean distance of the lattice as |r| −ρ , which is expected to feature the same critical behavior of the long range model. Through Monte Carlo sampling and finite-size analysis of the spontaneous magnetisation and of the Binder cumulant, we present numerical evidence that ρ sr = 4. According to such a result, one expects the model to belong to the Berezinskii-Kosterlitz-Thouless (BKT) universality class for ρ ≥ 4, and to present a 2 nd -order transition for ρ < 4. 1 arXiv:1905.06688v1 [cond-mat.stat-mech]

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“…It was shown previously (40,41,69) that, when α is greater than or equal to four, the system described by Eq. 10 belongs to the Kosterlitz-Thouless universality class (37,38).…”

Section: [10]mentioning

confidence: 99%

Universal transition from unstructured to structured neural maps

Weigand

Sartori

Cuntz

2017

Proc. Natl. Acad. Sci. U.S.A.

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Neurons sharing similar features are often selectively connected with a higher probability and should be located in close vicinity to save wiring. Selective connectivity has, therefore, been proposed to be the cause for spatial organization in cortical maps. Interestingly, orientation preference (OP) maps in the visual cortex are found in carnivores, ungulates, and primates but are not found in rodents, indicating fundamental differences in selective connectivity that seem unexpected for closely related species. Here, we investigate this finding by using multidimensional scaling to predict the locations of neurons based on minimizing wiring costs for any given connectivity. Our model shows a transition from an unstructured salt-and-pepper organization to a pinwheel arrangement when increasing the number of neurons, even without changing the selectivity of the connections. Increasing neuronal numbers also leads to the emergence of layers, retinotopy, or ocular dominance columns for the selective connectivity corresponding to each arrangement. We further show that neuron numbers impact overall interconnectivity as the primary reason for the appearance of neural maps, which we link to a known phase transition in an Ising-like model from statistical mechanics. Finally, we curated biological data from the literature to show that neural maps appear as the number of neurons in visual cortex increases over a wide range of mammalian species. Our results provide a simple explanation for the existence of salt-and-pepper arrangements in rodents and pinwheel arrangements in the visual cortex of primates, carnivores, and ungulates without assuming differences in the general visual cortex architecture and connectivity.neural maps | optimal wiring | visual cortex | orientation preference | pinwheels M odels assuming short cables and fast signal propagation in the circuit predict the precise placement of neurons and brain areas (1-4), the existence of topographic maps (5), and the existence of ocular dominance (OD) columns and orientation preference (OP) maps in the visual cortex (6, 7). The latter examples have become model systems to study structured neural maps because of the combination of striking striped patterns of OD and the intricate arrangement of OPs in a radial symmetry around pinwheel-like structures (8-11). A number of modeling approaches have been shown to predict different map properties and their possible biological origin (12-15). Examples are the link between the shape of the visual cortex and the overall stripe pattern of OD columns (16, 17) as well as the link between monocular deprivation and stripe thickness (16,18). In accordance with these observations, the enlargement of specific brain areas has been predicted by competitive Hebbian models (Kohonen maps) in regions with increased input (19) and has been found in monkeys and cats (20,21). Furthermore, the order of OD and OP map development has been linked to the ratio between OD and OP wavelength (22), and a constant density of pinwheels relative to the ...

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Critical behavior of theXYmodel in complex topologies

Cited by 29 publications

References 74 publications

One-dimensional long-range percolation: A numerical study

One-dimensional long-range percolation: A numerical study

Analysis of the low-temperature phase in the two-dimensional long-range diluted XY model

Universal transition from unstructured to structured neural maps

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