Polymers on disordered trees, spin glasses, and traveling waves

Derrida, Bernard; Spohn, Herbert

doi:10.1007/bf01014886

Cited by 457 publications

(821 citation statements)

References 25 publications

(8 reference statements)

Supporting

Mentioning

794

Contrasting

Unclassified

Order By: Relevance

“…The sk-trace of the many-dimensional DPRM is, at the start, distinct, but then rises precipitously at a different rate. Indeed, this suggestion is consistent with independent work [134] investigating directed polymers on Cayley trees [135], effectively an infinite dimensional implementation of the KPZ problem.…”

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

confidence: 75%

A KPZ Cocktail-Shaken, not Stirred...

2015

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

confidence: 75%

A KPZ Cocktail-Shaken, not Stirred...

2015

View full text Add to dashboard Cite

show abstract

“…Here, we are interested especially in classes of problems in which the FKPP equation is obtained either in the large-scale limit (N → ∞) of many-particle systems or in the mean-field limit of physical problems that are discrete at a microscopic level [6,7,11,12,22,23].…”

Section: Introductionmentioning

confidence: 99%

The critical wave speed for the Fisher–Kolmogorov–Petrowskii–Piscounov equation with cut-off

2007

View full text Add to dashboard Cite

The Fisher-Kolmogorov-Petrowskii-Piscounov (FKPP) equation with cut-off was introduced in (Brunet and Derrida 1997 Shift in the velocity of a front due to a cut-off Phys. Rev. E 56 2597-604) to model N-particle systems in which concentrations less than ε = 1/N are not attainable. It was conjectured that the cut-off function, which sets the reaction terms to zero if the concentration is below the small threshold ε, introduces a substantial shift in the propagation speed of the corresponding travelling waves. In this paper, we prove the conjecture of Brunet and Derrida, showing that the speed of propagation is given by c crit (ε) = 2 − π 2 /(ln ε) 2 + O((ln ε) −3 ), as ε → 0, for a large class of cut-off functions. Moreover, we extend this result to a more general family of scalar reaction-diffusion equations with cut-off. The main mathematical techniques used in our proof are the geometric singular perturbation theory and the blow-up method, which lead naturally to the identification of the reasons for the logarithmic dependence of c crit on ε as well as for the universality of the corresponding leading-order coefficient (π 2 ).

show abstract

“…These results were obtained by mapping the problem to a Gaussian field theory in an ultrametric space (a Cayley tree). The calculation of multifractal scaling properties was then reduced to the computation of thermodynamic functions of a special generalized random energy model (GREM), which are known exactly [9][10][11][12]. The field theory in the localization problem is defined in Euclidean space.…”

Section: Introductionmentioning

confidence: 99%

Exact calculation of multifractal exponents of the critical wave function of Dirac fermions in a random magnetic field

et al. 1997

View full text Add to dashboard Cite

The multifractal scaling exponents are calculated for the critical wave function of a twodimensional Dirac fermion in the presence of a random magnetic field. It is shown that the problem of calculating the multifractal spectrum maps into the thermodynamics of a static particle in a random potential. The multifractal exponents are simply given in terms of thermodynamic functions, such as free energy and entropy, which are argued to be self-averaging in the thermodynamic limit. These thermodynamic functions are shown to coincide exactly with those of a Generalized Random Energy Model, in agreement with previous results obtained using Gaussian field theories in an ultrametric space.

show abstract

Polymers on disordered trees, spin glasses, and traveling waves

Cited by 457 publications

References 25 publications

A KPZ Cocktail-Shaken, not Stirred...

A KPZ Cocktail-Shaken, not Stirred...

The critical wave speed for the Fisher–Kolmogorov–Petrowskii–Piscounov equation with cut-off

Exact calculation of multifractal exponents of the critical wave function of Dirac fermions in a random magnetic field

Contact Info

Product

Resources

About