The Intermediate Disorder Regime for Directed Polymers in Dimension 1+1

Abstract

We introduce a new disorder regime for directed polymers with one space and one time dimension that is accessed by scaling the inverse temperature parameter βwith the length of the polymer n. We scale β_n := βn^{-α} for alpha non-negative. This scaling sits in between the usual weak disorder (β= 0) and strong disorder regimes (β> 0). The fluctuation exponents zeta for the polymer endpoint and χfor the free energy depend on αin this regime, with α= 0 corresponding to the usual polymer exponents ζ= 2/3, χ= 1/3 and α>= 1/4 corresponding to the simple random walk exponents ζ= 1/2, χ= 0. For 0 < α< 1/4 the exponents interpolate linearly between these two extremes. At α= 1/4 we exactly identify the limiting distribution of the free energy and the end point of the polymer.

4 pages, 1 figure. Added a more detailed description of scaling results in the critical regime. Improved simulation results.

References in corpus (10)

• Probability Distribution of the Free Energy of the Continuum Directed Random Polymer in 1+1 dimensions
• The one-dimensional KPZ equation: an exact solution and its universality
• Free-energy distribution of the directed polymer at high temperature
• Asymptotics in ASEP with Step Initial Condition
• An exact solution for the KPZ equation with flat initial conditions
• Scaling for a one-dimensional directed polymer with boundary conditions
• Exact height distributions for the KPZ equation with narrow wedge initial condition
• A Fredholm Determinant Representation in ASEP
• The Continuum Directed Random Polymer
• Diffusivity of Rescaled Random Polymer in Random Environment in dimensions 1 and 2

Cited by in corpus (74)

• The one-dimensional KPZ equation and its universality class
• The KPZ fixed point
• KPZ reloaded
• Exact solution for the stationary Kardar-Parisi-Zhang equation
• The intermediate disorder regime for directed polymers in dimension $1+1$
• The Continuum Directed Random Polymer
• Log-Gamma polymer free energy fluctuations via a Fredholm determinant identity
• Stationary correlations for the 1D KPZ equation
• Renormalization fixed point of the KPZ universality class
• The 1+1-dimensional Kardar-Parisi-Zhang equation and its universality class
• KPZ equation limit of higher-spin exclusion processes
• Airy processes and variational problems
• A multi-layer extension of the stochastic heat equation
• Shift-invariance for vertex models and polymers
• Macdonald processes, quantum integrable systems and the Kardar-Parisi-Zhang universality class
• Multidimensional SDE with distributional drift and Lévy noise
• Crossover distributions at the edge of the rarefaction fan
• Intermediate disorder limits for multi-layer semi-discrete directed polymers
• Kardar-Parisi-Zhang equation and large deviations for random walks in weak random environments
• Determinantal structures in the O'Connell-Yor directed random polymer model
• Universality for the pinning model in the weak coupling regime
• On global solutions of the random Hamilton-Jacobi equations and the KPZ problem
• A central limit theorem for the KPZ equation
• Large Deviations For Sticky Brownian Motions
• Anomalous Fluctuations of Extremes in Many-Particle Diffusion
• Localization of directed polymers with general reference walk
• A forward-backward SDE from the 2D nonlinear stochastic heat equation
• ASEP(q,j) converges to the KPZ equation
• The KPZ equation in a half space with flat initial condition and the unbinding of a directed polymer from an attractive wall
• The scaling limit of the directed polymer with power-law tail disorder
• Space-time fluctuation of the Kardar-Parisi-Zhang equation in $d\ge 3$ and the Gaussian free field
• Tightness of discrete Gibbsian line ensembles with exponential interaction Hamiltonians
• Brownian Polymers in Poissonian Environment: a survey
• Brownian Gibbs property for Airy line ensembles
• Local Brownian property of the narrow wedge solution of the KPZ equation
• The stationary AKPZ equation: logarithmic superdiffusivity
• Law of large numbers and fluctuations in the sub-critical and $L^2$ regions for SHE and KPZ equation in dimension $d\ge 3$
• Stochastic growth in time dependent environments
• On comparison principle and strict positivity of solutions to the nonlinear stochastic fractional heat equations
• Universality of deterministic KPZ
• From Directed Polymers in Spatial-correlated Environment to Stochastic Heat Equations Driven by Fractional Noise in 1 + 1 Dimensions
• Local KPZ behavior under arbitrary scaling limits
• The KPZ equation and moments of random matrices
• Continuum directed random polymers on disordered hierarchical diamond lattices
• Moments of the SHE under delta initial measure
• Scaling limits of disordered systems and disorder relevance
• Diffusivity of Rescaled Random Polymer in Random Environment in dimensions 1 and 2
• The scaling relation chi = 2 xi - 1 for directed polymers in a random environment
• The scaling limits for Wiener sausages in random environments
• Analytical Methods and Field Theory for Disordered Systems
• The intermediate disorder regime for a directed polymer model on a hierarchical lattice
• The stochastic heat equation with multiplicative Lévy noise: Existence, moments, and intermittency
• Polynomial chaos and scaling limits of disordered systems
• A stationary model of non-intersecting directed polymers
• A note about Domination and monotonicity in disordered systems
• On the (strict) positivity of solutions of the stochastic heat equation
• Fluctuation exponents for directed polymers in the intermediate disorder regime
• Discrete time q-TASEPs
• Central limit theorems for the (2+1)-dimensional directed polymer in the weak disorder limit
• Another look into the Wong-Zakai Theorem for Stochastic Heat Equation
• Scaling limit of a directed polymer among a Poisson field of independent walks
• Non-directed polymers in heavy-tail random environment in dimension $d\ge 2$
• On the scaling properties of (2+1) directed polymers in the high temperature limit
• High temperature limits for $(1+1)$-dimensional directed polymer with heavy-tailed disorder
• Localization and free energy asymptotics in disordered statistical mechanics and random growth models
• Localization of directed polymers in continuous space
• Two-sided bounds on free energy of directed polymers on strongly recurrent graphs
• An invariance principle for the stochastic heat equation
• The near-critical scaling window for directed polymers on disordered trees
• Quenched Central Limit Theorem in a Corner Growth Setting
• Free energy of directed polymers in random environment in $1+1$-dimension at high temperature
• The continuum disordered pinning model
• Weak coupling limits for directed polymers in tube environments
• Rate of convergence for polymers in a weak disorder