The KPZ Limit of ASEP with Boundary
It was recently proved in Corwin and Shen (CPAM, [CS16]) that under weakly asymmetric scaling, the height functions for ASEP with sources and sinks converges to the Hopf–Cole solution of the KPZ equation with inhomogeneous Neumann boundary conditions. In their assumptions [CS16] chose positive values for the Neumann boundary condition, and they assumed initial data which is close to stationarity. By developing more extensive heat-kernel estimates, we clarify and extend their results to negative values of the Neumann boundary parameters, and we also show how to generalize their results to empty initial data (which is very far from stationarity). Combining our result with Barraquand et al. (Duke Math J, [BBCW17]), we obtain the Laplace transform of the one-point distribution for half-line KPZ, and use this to confirm t1/3-scale GOE Tracy–Widom long-time fluctuations at the origin.
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The authorwishes to thank Ivan Corwin for suggesting the problem, for providing helpful discussions about various issues which came up during the writing of the paper, for suggesting the free energy heuristic to obtain the limit and T 1/2-fluctuations in the low-density regime, and also for thoroughly reading the first four preliminary drafts of this paper. We also wish to thank Hao Shen and Li-Cheng Tsai, who provided some very useful discussions. The author was partially supported by the Fernholz Foundation’s “Summer Minerva Fellows” program, as well as summer support from Ivan Corwin’s NSF Grant DMS:1811143.
- AKQ13.Alberts, T., Khanin, K., Quastel, J.: The continuum directed random polymer. J. Stat. Phys. 154(1–2), (2014)Google Scholar
- BBC16.Borodin, A., Bufetov, A., Corwin, I.: Directed random polymers via nested contour integrals (2016). arXiv preprint. arXiv:1511.07324
- BBC18.Barraquand, G., Borodin, A., Corwin, I.: Half-space Macdonald processes (2018). arXiv preprint arXiv:1802.08210
- BBCS16.Baik, J., Barraquand, G., Corwin, I., Suidan, T.: Pfaffian Schur proceses and last passage percolation in a half-quadrant (2016). arXiv preprint. arXiv:1606.00525
- BBCW17.Barraquand, G., Borodin, A., Corwin, I., Wheeler, M.: Stochastic six-vertex model in a half-quadrant and half-line open ASEP. Duke Math. J. (2018). arXiv:1704.04309v2
- BR01.Baik, J., Rains, E.: The asymptotics of monotone subsequences of involutions. Duke Math. J. 109(2), 205–281 (2001)Google Scholar
- CG18.Corwin, I., Ghosal, P.: Lower tail of the KPZ equation (2018). arXiv preprint arXiv:1802.03273
- CS16.Corwin, I., Shen, H.: Open ASEP in the weakly asymmetric regime. CPAM (2018)Google Scholar
- CST18.Corwin, I., Shen, H., Tsai, L.C.: ASEP (q, j) converges to the KPZ equation. Ann. Inst. H. Poincaré Prob. Stat. 54(2), 995–1012 (2018)Google Scholar
- DELO05.Derrida, B., Enaud, C., Landim, C., Olla, S.: Fluctuations in the weakly asymmetric exclusion process with open boundary conditions. J. Stat. Phys. 118(5–6), 795–811 (2005)Google Scholar
- Der06.Derrida, B.: Matrix Ansatz and large deviations of the density in exclusion processes. In: Proceedings of the ICM, Madrid, pp. 367–382 (2006)Google Scholar
- GH17.Gerencsér, M., Hairer, M.: Singular SPDEs in domains with boundaries (2017). arXiv preprint arXiv:1702.06522
- GIP15.Gubinelli, M., Imkeller, P., Perkowski, N.: Paracontrolled distributions and singular PDEs. Forum Math. Pi 3 (e6), p. 75 (2015)Google Scholar
- GLM15.Gonçalves, P., Landim, C., Milanés, A.: Nonequilibrium fluctuations of one-dimensional boundary driven weakly asymmetric exclusion processes. Ann. Appl. Prob. 27(1), 140–177 (2017)Google Scholar
- GPS17.Gonçalves, P., Perkowski, N., Simon, M.: Derivation of the stochastic Burgers equation with Dirichlet boundary conditions from WASEP (2017). arXiv preprint arXiv:1710.11011
- Gro04.Grossinsky, S.: Phase transitions in nonequilibrium stochastic particle systems with local conservation laws. PhD Thesis. TU Munich (2004)Google Scholar
- Hai09.Hairer, M.: An introduction to SPDEs (July 2009). arXiv eprint arXiv:0907.4178
- HQ15.Hairer, M., Quastel, J.: A class of growth models rescaling to KPZ (2015). arXiv preprint arXiv:1512.07845
- Wu18.Wu, X.: Intermediate disorder regime for half-space directed polymers (2018). arXiv preprint arXiv:1804.09815