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Claussen, Gunnar; Apolo, Luis; Melchert, Oliver; Hartmann, Alexander K.

doi:10.1103/physreve.86.056708

Cited by 5 publications

(7 citation statements)

References 30 publications

(62 reference statements)

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“…The result for two-dimensional NWP with mixed Gaussian disorder is shown in Figure 18 . By the analysis of the distribution of loop sizes [ 66 ], two more critical exponents come into play. Right at the critical point

, the distribution exhibits a power law, governed by the exponent

.…”

Section: Results For Nwpmentioning

confidence: 99%

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From Spin Glasses to Negative-Weight Percolation

Hartmann

Melchert

Norrenbrock

2019

Entropy

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Spin glasses are prototypical random systems modelling magnetic alloys. One important way to investigate spin glass models is to study domain walls. For two dimensions, this can be algorithmically understood as the calculation of a shortest path, which allows for negative distances or weights. This led to the creation of the negative weight percolation (NWP) model, which is presented here along with all necessary basics from spin glasses, graph theory and corresponding algorithms. The algorithmic approach involves a mapping to the classical matching problem for graphs. In addition, a summary of results is given, which were obtained during the past decade. This includes the study of percolation transitions in dimension from d = 2 up to and beyond the upper critical dimension d u = 6 , also for random graphs. It is shown that NWP is in a different universality class than standard percolation. Furthermore, the question of whether NWP exhibits properties of Stochastic–Loewner Evolution is addressed and recent results for directed NWP are presented.

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, the distribution exhibits a power law, governed by the exponent

.…”

Section: Results For Nwpmentioning

confidence: 99%

“…Now, we will actually introduce the model for negative-weight percolation (NWP) [ 62 , 63 , 64 , 65 , 66 , 67 , 68 ]. This model exhibits some differences in comparison with other percolation models involving string-like objects.…”

Section: Negative-weight Percolationmentioning

confidence: 99%

From Spin Glasses to Negative-Weight Percolation

Hartmann

Melchert

Norrenbrock

2019

Entropy

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“…This leads, interestingly, to a new type of behavior giving rise to a different universality class compared to standard percolation. In a series of papers [7][8][9][10][11][12][13], NWP has been studied in different dimensions and different variants.…”

Section: Introductionmentioning

confidence: 99%

“…[6] the disorder-driven phase transition was investigated by means of finite-size scaling analyses and it turned out that the critical exponents were universal in 2D (different lattice geometries and disorder distribution were studied). Further studies regarding isotropic NWP address the influence of dilution on the critical properties [7], the upper critical dimension (d u = 6) [8], another upper critical dimension (d DPL u = 3) for densely packed loops far above the critical point [9], the meanfield behavior on a random graph with fixed connectivity [10], the Schramm-Loewner evolution properties of paths in 2D lattices [12], and loop-length distributions in several dimensions [11].…”

Section: Introductionmentioning

confidence: 99%

Directed negative-weight percolation

2019

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We consider a directed variant of the negative-weight percolation model in a two-dimensional, periodic, square lattice. The problem exhibits edge weights which are taken from a distribution that allows for both positive and negative values. Additionally, in this model variant all edges are directed. For a given realization of the disorder, a minimally weighted loop/path configuration is determined by performing a non-trivial transformation of the original lattice into a minimum weight perfect matching problem. For this problem, fast polynomial-time algorithms are available, thus we could study large systems with high accuracy. Depending on the fraction of negatively and positively weighted edges in the lattice, a continuous phase transition can be identified, whose characterizing critical exponents we have estimated by a finite-size scaling analyses of the numerically obtained data. We observe a strong change of the universality class with respect to standard directed percolation, as well as with respect to undirected negative-weight percolation. Furthermore, the relation to directed polymers in random media is illustrated.

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“…If the disorder is drawn from a distribution that allows for nonegative edge energy only, as for the canonical "directed polymer in a random medium" (DPRM), the groundstate configuration of the polymer can be computed efficiently using Dijkstra's algorithm [10,11]. However, if the disorder distribution allows for edge-energies of either sign, as for the problem of finding a minimum energy domain wall in 2D Ising spin glasses [7,8,12] (given that there is no * Electronic address: oliver.melchert@uni-oldenburg.de † Electronic address: alexander.hartmann@uni-oldenburg.de closed path with a negative energy) or more generally for the negative-weight percolation (NWP) problem [13][14][15][16][17][18], the solution of the MWP problem requires a nontrivial transformation to an auxiliary minimum-weight perfect matching problem [19]. Furthermore, the properties of MWPs with negative edges are fundamentally different from the case where all edge energies are non-negative [7,8,12].…”

Section: Introductionmentioning

confidence: 99%

Typical and large-deviation properties of minimum-energy paths on disordered hierarchical lattices

Melchert

Hartmann

2013

Eur. Phys. J. B

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We perform numerical simulations to study the optimal path problem on disordered hierarchical graphs with effective dimension d eff ≈ 2.32. Therein, edge energies are drawn from a disorder distribution that allows for positive and negative energies. This induces a behavior which is fundamentally different from the case where all energies are positive, only. Upon changing the subtleties of the distribution, the scaling of the minimum energy path length exhibits a transition from self-affine to self-similar. We analyze the precise scaling of the path length and the associated ground-state energy fluctuations in the vincinity of the disorder critical point, using a decimation procedure for huge graphs. Further, using an importance sampling procedure in the disorder we compute the negative-energy tails of the ground-state energy distribution up to 12 standard deviations away from its mean. We find that the asymptotic behavior of the negative-energy tail is in agreement with a Tracy-Widom distribution. Further, the characteristic scaling of the tail can be related to the ground-state energy flucutations, similar as for the directed polymer in a random medium.

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Analysis of the loop length distribution for the negative-weight percolation problem in dimensionsd=2throughd=6

Cited by 5 publications

References 30 publications

From Spin Glasses to Negative-Weight Percolation

From Spin Glasses to Negative-Weight Percolation

Directed negative-weight percolation

Typical and large-deviation properties of minimum-energy paths on disordered hierarchical lattices

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