(<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mn>2</mml:mn><mml:mo mathvariant="bold">+</mml:mo><mml:mn>1</mml:mn></mml:math>)-Dimensional Directed Polymer in a Random Medium: Scaling Phenomena and Universal Distributions

Halpin-Healy, Timothy

doi:10.1103/physrevlett.109.170602

Cited by 120 publications

(150 citation statements)

References 35 publications

Supporting

Mentioning

126

Contrasting

Order By: Relevance

“…Here, we perform a 2+1 KPZ Euler integration on expanding substrates with Ω = 10, complementing our earlier efforts on the 3d pt-pt SHE with multiplicative noise. Results are indicated here in Figure 4, with insets demonstrating that the variance, found to be 0.326, is dead-on, the skewness and kurtosis well within accepted values [84,85,86,89], while subleading corrections render extraction of the universal mean a bit more challenging. Nevertheless, successive fit lines monotonically approach a well-defined boundary (i.e., envelope), with intercepts converging to ξ 2 ≈-2.32, a quite decent value.…”

Section: Universal Limit Distribution: 3d Radial Kpz Classmentioning

confidence: 99%

“…In previous works [84,85], attention was focussed on large-scale numerical simulations of multiple models (RSOS, EDEN, driven-dimer, DPRM), as well as a direct Euler integration of 2+1 KPZ equation itself, establishing universality of the 2+1 flat KPZ Class, manifest in a limit distribution P 2+1 KP Z (ξ) possessing universal moments ( ξ 1 , ξ 2 1 , s, k)= (-0.85,0.235,0.424,0.346), with ξ = (h − v ∞ t)/(θt) β . We draw particular attention to [85]-Section III, Figs.…”

Section: Universal Limit Distribution: 3d Radial Kpz Classmentioning

confidence: 99%

“…• Recalling recent 2+1 DPRM results [84], which have isolated the higher-dimensional analog of KPZ/TW-GOE, measuring s=0.424 and k=0.346 for the generic pt-plane case, and pinned down the key index ω 2+1 =0.241 [20,124] via a multi-model study [85] of 2+1 KPZ stationary-state statistics, we notice quite similar values for the b=2.3 ♦DPRM, where the exponent, 0.274, might be a little high, but the skewness, 0.423, and kurtosis, 0.372, certainly close to the mark. Given our findings above for the ♦DPRM sk relation, one cannot resist looking at Euclidean KPZ from this same vantage point.…”

Section: The Many-dimensional Dprm and Fate Of D=∞ Kpzmentioning

confidence: 99%

“…Fortunately, recent numerics have helped chart the course, framing the essential features and providing a target for future analytical efforts. Halpin-Healy [84] simulated 3d extremal paths connecting far corners of a cube filled with exponentially distributed site energies; the underlying pt-pt energy fluctuation limit distribution was measured to possess universal variance, skewness and kurtosis: 0.291, 0.331, and 0.212, respectively. Later authors [86], studying three distinct 3d pt-pt KPZ growth models, among them single-step (SSC) and on-lattice version of Takeuchi Eden D, find values in the range s 2 ≈ 0.32-0.34 and k 2 ≈ 0.20-0.22, providing additional evidence for these characteristic, universal quantities.…”

Section: Universal Limit Distribution: 3d Radial Kpz Classmentioning

confidence: 99%

“…Importantly, the present KPZ Renaissance has inspired a renewed interest in the 2+1 KPZ problem, as well as other dimension-dependent issues, such as the existence of a finite UCD, and using Wilsonian renormalization group methods, hopes of elucidating the full KPZ phase diagram [83]. Universality of the 2+1 KPZ Class and its associated limit distributions (i.e., higher-dimensional analogs of TW-GUE, TW-GOE and BR-F 0 ) have been well-characterized [84,85,86,87], along with universal spatial and temporal correlators [88,89], supplemented by secondary KPZ "patch" PDFs [88,90]. On the mathematical side, the transmission has shifted into hyperdrive; a rapidly accelerating moveable feast-e.g., [91,92,93,94,95,96,97,98,99,100].…”

mentioning

confidence: 99%

See 4 more Smart Citations

A KPZ Cocktail-Shaken, not Stirred...

2015

Self Cite

View full text Add to dashboard Cite

The stochastic partial differential equation proposed nearly three decades ago by Kardar, Parisi and Zhang (KPZ) continues to inspire, intrigue and confound its many admirers. Here, we i) pay debts to heroic predecessors, ii) highlight additional, experimentally relevant aspects of the recently solved 1+1 KPZ problem, iii) use an expanding substrates formalism to gain access to the 3d radial KPZ equation and, lastly, iv) examining extremal paths on disordered hierarchical lattices, set our gaze upon the fate of d=∞ KPZ. Clearly, there remains ample unexplored territory within the realm of KPZ and, for the hearty, much work to be done, especially in higher dimensions, where numerical and renormalization group methods are providing a deeper understanding of this iconic equation.Keywords Nonequilibrium Growth · Extremal Paths · Universal Limit Distributions In a NutshellThe history of physics has been punctuated at seminal moments by the appearance of certain fundamental equations (and associated models) which have vigorously propelled the enterprise forward, serving as an explosively rich departure point, generating a myriad of alternative perspectives, creative insights, surprising connections and, given its sustained impregnability, often remain for many years a sacred object of fascination and obsession to its dedicated disciples. Recent, obvious suspects in this regard include the quantum mechanical Schrödinger equation (equally well, its flipside-Feynman's path integral formulation), or the wonderfully elusive Navier-Stokes equation governing fluid mechanics, by which may be gleaned the scaling secrets of turbulent flow, dynamically encoded in the whirling eddies drawn by da Vinci centuries ago. Within the domain of equilibrium statistical physics, the 2d Ising Model, with its tour-de-force algebraic solution by Onsager, followed by combinatoric, graphical, Grassmannian, Monte Carlo, as well as full-blown field-theoretic, scaling, and renormalization group treatments, represents an extraordinary legacy that continues unabated to this very day. Arguably, a non-equilibrium statistical mechanical analogue to Ising/Onsager is the iconic equation [1] proposed a generation ago by Kardar, Parisi, and Zhang (KPZ); it captures the statistical fluctuations of a kinetically-roughened scalar height field h(x, t):

show abstract

Section: Universal Limit Distribution: 3d Radial Kpz Classmentioning

confidence: 99%

Section: Universal Limit Distribution: 3d Radial Kpz Classmentioning

confidence: 99%

Section: The Many-dimensional Dprm and Fate Of D=∞ Kpzmentioning

confidence: 99%

Section: Universal Limit Distribution: 3d Radial Kpz Classmentioning

confidence: 99%

mentioning

confidence: 99%

See 3 more Smart Citations

A KPZ Cocktail-Shaken, not Stirred...

2015

Self Cite

View full text Add to dashboard Cite

show abstract

Lifshitz tails at spectral edge and holography with a finite cutoff

Gorsky

Nechaev

Valov

2021

J. High Energ. Phys.

View full text Add to dashboard Cite

We propose the holographic description of the Lifshitz tail typical for one-particle spectral density of bounded disordered system in D = 1 space. To this aim the “polymer representation” of the Jackiw-Teitelboim (JT) 2D dilaton gravity at a finite cutoff is used and the corresponding partition function is considered as the weighted sum over paths of fixed length in an external magnetic field. We identify the regime of small loops, responsible for emergence of a Lifshitz tail in the Gaussian disorder, and relate the strength of disorder to the boundary value of the dilaton. The geometry corresponding to the Poisson disorder in the boundary theory involves random paths fluctuating in the vicinity of the hard impenetrable cut-off disc in a 2D plane. It is shown that the ensemble of “stretched” paths evading the disc possesses the Kardar-Parisi-Zhang (KPZ) scaling for fluctuations, which is the key property that ensures the dual description of the Lifshitz tail in the spectral density for the Poisson disorder.

show abstract

Local KPZ Behavior Under Arbitrary Scaling Limits

Chatterjee

2022

Commun. Math. Phys.

View full text Add to dashboard Cite

(2+1)-Dimensional Directed Polymer in a Random Medium: Scaling Phenomena and Universal Distributions

Cited by 120 publications

References 35 publications

A KPZ Cocktail-Shaken, not Stirred...

A KPZ Cocktail-Shaken, not Stirred...

Lifshitz tails at spectral edge and holography with a finite cutoff

Local KPZ Behavior Under Arbitrary Scaling Limits

Contact Info

Product

Resources

About