KPZ方程与KPZ普适性简介
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摘要
本文首先介绍KPZ (Kardar-Parisi-Zhang)普适类的物理背景,其中, Eden模型、黏性落体模型和KPZ方程这几类物理模型将被提及;其次,将考察一维KPZ方程的Cole-Hopf解以及几类收敛到一维KPZ方程的离散模型(如角落生长模型和定向聚合物模型等).
In this survey, we shall ?rst introduce the physical background of KPZ universality class. The Eden model, sticky block model and KPZ equation will be mentioned. Then we shall focus on the Cole-Hopf solution to 1-dimensional KPZ equation and present some discrete models like the corner growth model and directed polymer model, which could converge to the KPZ equation in some sense.
In this survey, we shall ?rst introduce the physical background of KPZ universality class. The Eden model, sticky block model and KPZ equation will be mentioned. Then we shall focus on the Cole-Hopf solution to 1-dimensional KPZ equation and present some discrete models like the corner growth model and directed polymer model, which could converge to the KPZ equation in some sense.
引文
[1] Kardar M,Parisi G,Zhang Y C.Dynamic scaling of growing interfaces.Phys Rev Lett,1986,56:889-892
2 Eden M.A two-dimensional growth process.In:Proceedings of the Fourth Berkeley Symposium on Mathematical Statistics and Probability,vol.4.Berkeley:University of California,1961
3苏汝铿.统计物理学(第二版).北京:高等教育出版社,2004
4刘玉鑫.热学(第二版).北京:北京大学出版社,2013
5 Kolmogoroff A.The local structure of turbulence in incompressible viscous fluid for very large Reynold’s numbers(in Russian).Dokl Akad Nauk SSSR,1941,30:376-387
6 Frisch U,Kolmogorov A N.Turbulence.Cambridge:Cambridge University Press,1995
7 Falconer K.Fractal Geometry.Chichester:John Wiley&Sons,2014
8刘式达,刘式适.物理学中的分形.北京:北京大学出版社,2014
9 Walsh J B.An introduction to stochastic partial differential equations.In:′Ecole d’′Et′e de Probabilit′es de Saint Flour XIV-1984.Berlin-Heidelberg:Springer,1986,265-439
10 Gouyet J F.Physics and Fractal Structures.New York:Springer-Verlag,1996
11 Jullien R,Botet R.Surface thickness in the Eden model.Phys Rev Lett,1985,54:2055-2055
12 Plischke M,R′acz Z.Dynamic scaling and the surface structure of Eden clusters.Phys Rev A(3),1985,32:3825-3828
13 Corwin I.Kardar-Parisi-Zhang universality.Notices Amer Math Soc,2016,63:230-239
14 Vold M J.A numerical approach to the problem of sediment volume.J Colloid Sci,1959,14:168-174
15 Edwards S F,Wilkinson D R.The surface statistics of a granular aggregate.Proc R Soc Lond Ser A Math Phys Eng Sci,1982,381:17-31
16 Corwin I.The Kardar-Parisi-Zhang equation and universality class.Random Matrices Theor Appl,2012,01:1130001-1130076
17 Forster D,Nelson D R,Stephen M J.Large-distance and long-time properties of a randomly stirred fluid.Phys Rev A(3),1977,16:732-749
18 Funaki T,Quastel J.KPZ equation,its renormalization and invariant measures.Stoch PDE Anal Comp,2015,3:159-220
19 Bertini L,Giacomin G.Stochastic burgers and KPZ equations from particle systems.Comm Math Phys,1997,183:571-607
20李灵峰.量子场论.北京:科学出版社,2015
21 Cole J D.On a quasi-linear parabolic equation occurring in aerodynamics.Quart Appl Math,1951,9:225-236
22 Hopf E.The partial differential equation ut+uux=μuxx.Comm Pure Appl Math,1950,3:201-230
23 Bertini L,Cancrini N.The stochastic heat equation:Feynman-Kac formula and intermittence.J Stat Phys,1995,78:1377-1401
24 Quastel J.Introduction to KPZ.In:Current Developments in Mathematics,vol.2011.Somerville:International Press,2012,125-194
25 Mueller C.On the support of solutions to the heat equation with noise.Stochastics Stochastic Rep,1991,37:225-245
26 Bal′azs M,Quastel J,Sepp¨al¨ainen T.Fluctuation exponent of the KPZ/stochastic Burgers equation.J Amer Math Soc,2011,24:683-708
27 Amir G,Corwin I,Quastel J.Probability distribution of the free energy of the continuum directed random polymer in 1+1 dimensions.Comm Pure Appl Math,2011,64:466-537
28 Takeuchi K A,Sano M.Universal fluctuations of growing interfaces:Evidence in turbulent liquid crystals.Phys Rev Lett,2010,104:230601
29 Sasamoto T,Spohn H.Exact height distributions for the KPZ equation with narrow wedge initial condition.Nuclear Phys B,2010,834:523-542
30 Hairer M.Solving the KPZ equation.Ann of Math(2),2013,178:559-664
31 Chan T.Scaling limits of Wick ordered KPZ equation.Comm Math Phys,2000,209:671-690
32 Holden H,Lindstrm T,Ksendal B,et al.The stochastic Wick-type Burgers equation.In:Stochastic Partial Differential Equations(Edinburgh,1994).Cambridge:Cambridge University Press,1995,141-161
33 Lindstr?m T,?ksendal B,Ub?e J.Wick multiplication and It?o-Skorohod stochastic differential equations.In:Ideas and Methods in Mathematical Analysis,Stochastics,and Applications(Oslo,1988).Cambridge:Cambridge University Press,1992,183-206
34 Hairer M.A theory of regularity structures.Invent Math,2014,198:269-504
35 McKean H P Jr.An exponential formula for solving Boltzmann’s equation for a Maxwellian gas.J Combin Theory,1967,2:358-382
36 Wild E.On Boltzmann’s equation in the kinetic theory of gases.Math Proc Cambridge Philos Soc,1951,47:602-609
37 Hairer M.Rough stochastic PDEs.Comm Pure Appl Math,2011,64:1547-1585
38 Corwin I,Quastel J.Crossover distributions at the edge of the rarefaction fan.Ann Probab,2013,41:1243-1314
39 Liggett T M.Interacting Particle Systems.Berlin:Springer-Verlag,2005
40 Rost H.Non-equilibrium behaviour of a many particle process:Density profile and local equilibria.Z Wahrsch Verw Gebiete,1981,58:41-53
41 Johansson K.Shape fluctuations and random matrices.Comm Math Phys,2000,209:437-476
42 Mehta M L.Random Matrices.Amsterdam:Elsevier/Academic Press,2004
43 Borodin A,Ferrari P L.Large time asymptotics of growth models on space-like paths,I:Push ASEP.Electron JProbab,2008,13:1380-1418
44 Corwin I,Ferrari P L,P′ech′e S.Limit processes for TASEP with shocks and rarefaction fans.J Stat Phys,2010,140:232-267
45 Tracy C,Widom H.Asymptotics in ASEP with step initial condition.Comm Math Phys,2009,290:129-154
46 Tracy C A,Widom H.On ASEP with step Bernoulli initial condition.J Stat Phys,2009,137:825-838
47 Tracy C A,Widom H.Formulas for ASEP with two-sided Bernoulli initial condition.J Stat Phys,2010,140:619-634
48 Tracy C A,Widom H.Formulas for joint probabilities for the asymmetric simple exclusion process.J Math Phys,2010,51:063302
49 G¨artner J U.Convergence towards Burgers’equation and propagation of chaos for weakly asymmetric exclusion processes.Stochastic Process Appl,1988,27:233-260
50 Huse D A,Henley C L.Pinning and roughening of domain walls in Ising systems due to random impurities.Phys Rev Lett,1985,54:2708-2711
51 Alberts T,Khanin K,Quastel J.The intermediate disorder regime for directed polymers in dimension 1+1.Ann Probab,2014,42:1212-1256
52 Alberts T,Khanin K,Quastel J.The continuum directed random polymer.J Stat Phys,2014,154:305-326
53 Borodin A,Corwin I,Ferrari P.Free energy fluctuations for directed polymers in random media in 1+1 dimension.Comm Pure Appl Math,2014,67:1129-1214
54 Corwin I,Quastel J,Remenik D.Renormalization fixed point of the KPZ universality class.J Stat Phys,2015,160:815-834
55 Matetski K,Quastel J,Remenik D.The KPZ fixed point.ArXiv:1701.00018,2017
56 Wick G C.The evaluation of the collision matrix.Phys Rev,1950,80:268-272
57 Simon B.The P(?)2 Euclidean(Quantum)Field Theory.Princeton:Princeton University Press,1974
58 Avram F,Taqqu M S.Noncentral limit theorems and Appell polynomials.Ann Probab,1987,15:767-775
59 Nualart D.The Malliavin Calculus and Related Topics.Berlin:Springer-Verlag,2006
2 Eden M.A two-dimensional growth process.In:Proceedings of the Fourth Berkeley Symposium on Mathematical Statistics and Probability,vol.4.Berkeley:University of California,1961
3苏汝铿.统计物理学(第二版).北京:高等教育出版社,2004
4刘玉鑫.热学(第二版).北京:北京大学出版社,2013
5 Kolmogoroff A.The local structure of turbulence in incompressible viscous fluid for very large Reynold’s numbers(in Russian).Dokl Akad Nauk SSSR,1941,30:376-387
6 Frisch U,Kolmogorov A N.Turbulence.Cambridge:Cambridge University Press,1995
7 Falconer K.Fractal Geometry.Chichester:John Wiley&Sons,2014
8刘式达,刘式适.物理学中的分形.北京:北京大学出版社,2014
9 Walsh J B.An introduction to stochastic partial differential equations.In:′Ecole d’′Et′e de Probabilit′es de Saint Flour XIV-1984.Berlin-Heidelberg:Springer,1986,265-439
10 Gouyet J F.Physics and Fractal Structures.New York:Springer-Verlag,1996
11 Jullien R,Botet R.Surface thickness in the Eden model.Phys Rev Lett,1985,54:2055-2055
12 Plischke M,R′acz Z.Dynamic scaling and the surface structure of Eden clusters.Phys Rev A(3),1985,32:3825-3828
13 Corwin I.Kardar-Parisi-Zhang universality.Notices Amer Math Soc,2016,63:230-239
14 Vold M J.A numerical approach to the problem of sediment volume.J Colloid Sci,1959,14:168-174
15 Edwards S F,Wilkinson D R.The surface statistics of a granular aggregate.Proc R Soc Lond Ser A Math Phys Eng Sci,1982,381:17-31
16 Corwin I.The Kardar-Parisi-Zhang equation and universality class.Random Matrices Theor Appl,2012,01:1130001-1130076
17 Forster D,Nelson D R,Stephen M J.Large-distance and long-time properties of a randomly stirred fluid.Phys Rev A(3),1977,16:732-749
18 Funaki T,Quastel J.KPZ equation,its renormalization and invariant measures.Stoch PDE Anal Comp,2015,3:159-220
19 Bertini L,Giacomin G.Stochastic burgers and KPZ equations from particle systems.Comm Math Phys,1997,183:571-607
20李灵峰.量子场论.北京:科学出版社,2015
21 Cole J D.On a quasi-linear parabolic equation occurring in aerodynamics.Quart Appl Math,1951,9:225-236
22 Hopf E.The partial differential equation ut+uux=μuxx.Comm Pure Appl Math,1950,3:201-230
23 Bertini L,Cancrini N.The stochastic heat equation:Feynman-Kac formula and intermittence.J Stat Phys,1995,78:1377-1401
24 Quastel J.Introduction to KPZ.In:Current Developments in Mathematics,vol.2011.Somerville:International Press,2012,125-194
25 Mueller C.On the support of solutions to the heat equation with noise.Stochastics Stochastic Rep,1991,37:225-245
26 Bal′azs M,Quastel J,Sepp¨al¨ainen T.Fluctuation exponent of the KPZ/stochastic Burgers equation.J Amer Math Soc,2011,24:683-708
27 Amir G,Corwin I,Quastel J.Probability distribution of the free energy of the continuum directed random polymer in 1+1 dimensions.Comm Pure Appl Math,2011,64:466-537
28 Takeuchi K A,Sano M.Universal fluctuations of growing interfaces:Evidence in turbulent liquid crystals.Phys Rev Lett,2010,104:230601
29 Sasamoto T,Spohn H.Exact height distributions for the KPZ equation with narrow wedge initial condition.Nuclear Phys B,2010,834:523-542
30 Hairer M.Solving the KPZ equation.Ann of Math(2),2013,178:559-664
31 Chan T.Scaling limits of Wick ordered KPZ equation.Comm Math Phys,2000,209:671-690
32 Holden H,Lindstrm T,Ksendal B,et al.The stochastic Wick-type Burgers equation.In:Stochastic Partial Differential Equations(Edinburgh,1994).Cambridge:Cambridge University Press,1995,141-161
33 Lindstr?m T,?ksendal B,Ub?e J.Wick multiplication and It?o-Skorohod stochastic differential equations.In:Ideas and Methods in Mathematical Analysis,Stochastics,and Applications(Oslo,1988).Cambridge:Cambridge University Press,1992,183-206
34 Hairer M.A theory of regularity structures.Invent Math,2014,198:269-504
35 McKean H P Jr.An exponential formula for solving Boltzmann’s equation for a Maxwellian gas.J Combin Theory,1967,2:358-382
36 Wild E.On Boltzmann’s equation in the kinetic theory of gases.Math Proc Cambridge Philos Soc,1951,47:602-609
37 Hairer M.Rough stochastic PDEs.Comm Pure Appl Math,2011,64:1547-1585
38 Corwin I,Quastel J.Crossover distributions at the edge of the rarefaction fan.Ann Probab,2013,41:1243-1314
39 Liggett T M.Interacting Particle Systems.Berlin:Springer-Verlag,2005
40 Rost H.Non-equilibrium behaviour of a many particle process:Density profile and local equilibria.Z Wahrsch Verw Gebiete,1981,58:41-53
41 Johansson K.Shape fluctuations and random matrices.Comm Math Phys,2000,209:437-476
42 Mehta M L.Random Matrices.Amsterdam:Elsevier/Academic Press,2004
43 Borodin A,Ferrari P L.Large time asymptotics of growth models on space-like paths,I:Push ASEP.Electron JProbab,2008,13:1380-1418
44 Corwin I,Ferrari P L,P′ech′e S.Limit processes for TASEP with shocks and rarefaction fans.J Stat Phys,2010,140:232-267
45 Tracy C,Widom H.Asymptotics in ASEP with step initial condition.Comm Math Phys,2009,290:129-154
46 Tracy C A,Widom H.On ASEP with step Bernoulli initial condition.J Stat Phys,2009,137:825-838
47 Tracy C A,Widom H.Formulas for ASEP with two-sided Bernoulli initial condition.J Stat Phys,2010,140:619-634
48 Tracy C A,Widom H.Formulas for joint probabilities for the asymmetric simple exclusion process.J Math Phys,2010,51:063302
49 G¨artner J U.Convergence towards Burgers’equation and propagation of chaos for weakly asymmetric exclusion processes.Stochastic Process Appl,1988,27:233-260
50 Huse D A,Henley C L.Pinning and roughening of domain walls in Ising systems due to random impurities.Phys Rev Lett,1985,54:2708-2711
51 Alberts T,Khanin K,Quastel J.The intermediate disorder regime for directed polymers in dimension 1+1.Ann Probab,2014,42:1212-1256
52 Alberts T,Khanin K,Quastel J.The continuum directed random polymer.J Stat Phys,2014,154:305-326
53 Borodin A,Corwin I,Ferrari P.Free energy fluctuations for directed polymers in random media in 1+1 dimension.Comm Pure Appl Math,2014,67:1129-1214
54 Corwin I,Quastel J,Remenik D.Renormalization fixed point of the KPZ universality class.J Stat Phys,2015,160:815-834
55 Matetski K,Quastel J,Remenik D.The KPZ fixed point.ArXiv:1701.00018,2017
56 Wick G C.The evaluation of the collision matrix.Phys Rev,1950,80:268-272
57 Simon B.The P(?)2 Euclidean(Quantum)Field Theory.Princeton:Princeton University Press,1974
58 Avram F,Taqqu M S.Noncentral limit theorems and Appell polynomials.Ann Probab,1987,15:767-775
59 Nualart D.The Malliavin Calculus and Related Topics.Berlin:Springer-Verlag,2006