Probability distribution of the free energy of the continuum directed random polymer in 1 + 1 dimensions

By Gideon Amir, Ivan Corwin, Jeremy Quastel. 2010

652 citations recorded by the source corpus.

We consider the solution of the stochastic heat equation $$\partial_T {\cal Z} = {{1}\over{2}} \partial_X^2 {\cal Z} - {\cal Z} \dot{\cal{W}}$$ with delta function initial condition $${\cal Z} (T=0,X)= \delta_{X=0}$$ whose logarithm, with appropriate normalization, is the free energy of the continuum directed polymer, or the Hopf‐Cole solution of the Kardar‐Parisi‐Zhang equation with narrow wedge initial conditions.

Subfields: Condensed Matter Physics, Mathematical Physics, Statistics and Probability

View Paper: https://www.semanticscholar.org/paper/bf83ea90e48c5261c256e64448e6b50a38160cfc

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Citations in the corpus

Evidence for Geometry-Dependent Universal Fluctuations of the Kardar-Parisi-Zhang Interfaces in Liquid-Crystal Turbulence (2012, background): “dimensional KPZ class, namely the totally and partially asymmetric simple exclusion processes (TASEP and PASEP) [43, 97], the polynuclear growth (PNG) model [71, 72]4 and also the KPZ equation [2, 14, 23, 84, 85].”
An appetizer to modern developments on the Kardar–Parisi–Zhang universality class (2017, methodology): “The combination of this correspondence and the developments by Tracy and Widom on ASEP with the step initial condition, with other mathematical elaborations, led Sasamoto and Spohn [44, 45] and independently Amir et al.[46] to establish the exact solution of the 1D KPZ equation for the narrow-wedge ”
Free Energy Fluctuations for Directed Polymers in Random Media in 1 + 1 Dimension (2012, methodology): “The : exp : is the Wick exponential which accounts for the fact that one can not näıvely integrate white noise along a Brownian motion (various equivalent ways exist to define this exponential [5, 13, 23]).”
Rough paths and 1d SDE with a time dependent distributional drift: application to polymers (2014, background): “Proof For N ∈ N\{0}, we consider a smooth function φN : R → [0, 1], symmetric, equal to 1 on [0, N ] and to 0 on [2N ,+∞), non-increasing on [N , 2N ], satisfying ‖dpφN/dx p‖∞ ≤ cp/N p for some cp ≥ 1, independent of N , for any integer p ≥ 1.”
Tropical Combinatorics and Whittaker functions (2011, methodology): “ng statistics for the KPZ equation. The ﬁrst relied on the work of Tracy and Widom [90, 91, 92, 93] on the solvability of the partially asymmetry simple exclusion process. Via an approximation scheme [16, 5], this solvability was employed by [5, 31] to rigorously compute the one-point statistics of ”
Macdonald processes (2011, methodology): “It follows from the exact formulas for the probability distribution for the solution to the KPZ equation which was simultaneously and independently discovered in [4, 88] and proved rigorously in [4].”
Stochastic six-vertex model (2014, methodology): “sults were obtained for more complicated systems including Asymmetric Simple Exclusion Process (ASEP) [TW4], [BCS], q{TASEP [BC], [FV], and the solution to KPZ stochastic partial dierential equation [ACQ], [SS], [D], [CDR], [BCF]; see also [BG2], [BP], [Cor3] for reviews. The process Xb 1;b 2(t) has”
Non-equilibrium statistical mechanics: from a paradigmatic model to biological transport (2011, neutral): “The distribution of the height of the surface at 1388 any finite time is now known exactly (starting from some special initial conditions), 1389 solving a problem that remained open for 25 years: a description of the historical 1390 development of these results and the contributions of the various g”
Introduction to KPZ (2011, methodology): “This was first proved by Bertini and Giacomin [BG97] in the case of data close to equilibrium, and extended to the case of step initial data (Dirac initial data for the stochastic heat equation) in [ACQ11].”
Directed polymers and the quantum Toda lattice (2009, background): “It will be interesting to relate this, in a suitable scaling limit, to the “crossover distributions” recently introduced in the context of KPZ and the stochastic heat equation by Sasamoto and Spohn [49–52] and Amir, Corwin and Quastel [1], building on recent work of Tracy and Widom [58–61] on the as”
Moments and growth indices for the nonlinear stochastic heat equation with rough initial conditions (2013, background): “See [2, 3, 4, 5, 8, 9, 10, 17, 20] for the intermittency problem, [15, 16] for probabilistic potential theory, [31, 32] for regularity of the solution, and [26, 27, 29, 33] for several other properties.”
Operator Spreading in Random Unitary Circuits (2017, methodology): “For an initially localized operator, this is expressed in terms of the Gaussian unitary ensemble (GUE) Tracy-Widom distribution [67,70] (which describes the extremal eigenvalue statistics for the GUE of Hermitian matrices [90,91]).”
Nonlinear Fluctuations of Weakly Asymmetric Interacting Particle Systems (2013, methodology): “Recent advances in combinatoric properties of the asymmetric simple exclusion process [43] led to an intense research activity based upon [5] (see [38], [1], and see [10] for a recent review).”
Height Fluctuations for the Stationary KPZ Equation (2014, background): “The physically relevant solution to the KPZ equation [17] is defined indirectly via the well-posed stochastic heat equation (SHE) with multiplicative noise [2, 5, 16], ∂tZ(t, x) = 1 2∂ 2 xZ(t, x) + Z(t, x)ξ(t, x), with initial condition Z(0, x) = Z0(x) = e .”