We give a short account of our new approach to study models in the Kardar–Parisi–Zhang universality class by connecting them to free fermions at positive temperature. Our ideas and methods are explained mainly for the semi-discrete directed polymer model due to O’Connell and Yor.
REFERENCES
1.
2.
F.
Spitzer
, “Interaction of Markov processes
,” Adv. Math.
5
, 246
–290
(1970
).3.
T. M.
Liggett
, Interacting Particle Systems
(Springer-Verlag
, 1985
).4.
T. M.
Liggett
, Stochastic Interacting Systems: Contact, Voter, and Exclusion Processes
(Springer-Verlag
, 1999
).5.
C.
Kipnis
and C.
Landim
, Scaling Limits of Interacting Particle Systems
(Springer
, 1999
).6.
7.
A.
Borodin
and L.
Petrov
, “Lectures on Integrable probability: Stochastic vertex models and symmetric functions
,” in Lecture Notes of the Les Houches Summer School
(Oxford University Press
, 2016
), Vol. 104
; arXiv:1605.01349 [math.PR].8.
M.
Kardar
, G.
Parisi
, and Y.-C.
Zhang
, “Dynamic scaling of growing interfaces
,” Phys. Rev. Lett.
56
, 889
–892
(1986
).9.
A. L.
Barabási
and H. E.
Stanley
, Fractal Concepts in Surface Growth
(Cambridge University Press
, 1995
).10.
I.
Corwin
, “The Kardar–Parisi–Zhang equation and universality class
,” Random Matrices: Theory Appl.
1
, 1130001
(2012
).11.
T.
Sasamoto
, “The 1D Kardar–Parisi–Zhang equation: Height distribution and universality
,” Prog. Theor. Exp. Phys.
2016
, 022A01
.12.
L.
Bertini
and G.
Giacomin
, “Stochastic Burgers and KPZ equations from particle systems
,” Commun. Math. Phys.
183
, 571
–607
(1997
).13.
M.
Hairer
, “Solving the KPZ equation
,” Ann. Math.
178
, 559
–664
(2013
).14.
M.
Gubinelli
and N.
Perkowski
, “KPZ reloaded
,” Commun. Math. Phys.
349
, 165
–269
(2017
).15.
A.
Kupiainen
, “Renormalization group and stochastic PDE’s
,” Ann. Henri Poincaré
17
, 497
–535
(2016
).16.
T.
Sasamoto
and H.
Spohn
, “The crossover regime for the weakly asymmetric simple exclusion process
,” J. Stat. Phys.
140
, 209
–231
(2010
).17.
T.
Sasamoto
and H.
Spohn
, “Exact height distributions for the KPZ equation with narrow wedge initial condition
,” Nucl. Phys. B
834
, 523
–542
(2010
).18.
T.
Sasamoto
and H.
Spohn
, “One-dimensional Kardar–Parisi–Zhang equation: An exact solution and its universality
,” Phys. Rev. Lett.
104
, 230602
(2010
).19.
T.
Sasamoto
and H.
Spohn
, “The 1 + 1-dimensional Kardar–Parisi–Zhang equation and its universality class
,” J. Stat. Mech.
2010
, P11013
.20.
G.
Amir
, I.
Corwin
, and J.
Quastel
, “Probability distribution of the free energy of the continuum directed random polymer in 1 + 1 dimensions
,” Commun. Pure Appl. Math.
64
, 466
–537
(2011
).21.
C. A.
Tracy
and H.
Widom
, “Level-spacing distributions and the Airy kernel
,” Commun. Math. Phys.
159
, 151
–174
(1994
).22.
C. A.
Tracy
and H.
Widom
, “Asymptotics in ASEP with step initial condition
,” Commun. Math. Phys.
290
, 129
–154
(2009
).23.
P.
Calabrese
, P.
Le Doussal
, and A.
Rosso
, “Free-energy distribution of the directed polymer at high temperature
,” Eur. Phys. Lett.
90
, 200002
(2010
).24.
V.
Dotsenko
, “Bethe ansatz derivation of the Tracy-Widom distribution for one-dimensional directed polymers
,” Eur. Phys. Lett.
90
, 200003
(2010
).25.
T.
Imamura
, M.
Mucciconi
, and T.
Sasamoto
, “Identity between restricted Cauchy sums for the q-Whittaker and skew Schur polynomials
,” arXiv:2106.11913 [math.CO].26.
T.
Imamura
, M.
Mucciconi
, and T.
Sasamoto
, “Skew RSK dynamics: Greene invariants, affine crystals and applications to q-Whittaker polynomials
,” arXiv:2106.11922 [math.CO].27.
A.
Borodin
and I.
Corwin
, “MacDonald processes
,” Probab. Theory Relat. Fields
158
, 225
–400
(2014
).28.
A.
Borodin
, “Periodic Schur process and cylindric partitions
,” Duke J. Math.
140
(3
), 391
–468
(2007
).29.
D.
Betea
and J.
Bouttier
, “The periodic Schur process and free fermions at finite temperature
,” Math. Phys., Anal. Geom.
22
, 3
(2019
).30.
T.
Imamura
, M.
Mucciconi
, and T.
Sasamoto
, “Solvable models in the KPZ class: Approach through periodic and free boundary Schur measures
,” arXiv:2204.08420.31.
A.
Borodin
, “Stochastic higher spin six vertex model and Madconald measures
,” J. Math. Phys.
59
, 023301
(2018
).32.
N.
O’Connell
and M.
Yor
, “Brownian analogues of Burke’s theorem
,” Stochastic Processes Appl.
96
, 285
–304
(2001
).33.
A.
Borodin
, I.
Corwin
, and D.
Remenik
, “Log-gamma polymer free energy fluctuations via a Fredholm determinant identity
,” Commun. Math. Phys.
324
, 215
–232
(2013
).34.
T.
Imamura
and T.
Sasamoto
, “Free energy distribution of the stationary O’Connell–Yor directed random polymer model
,” J. Phys. A: Math. Theor.
50
, 285203
(2017
).35.
Y.
Baryshnikov
, “GUEs and queues
,” Prob. Th. Rel. Fields
119
, 256
–274
(2001
).36.
J.
Gravner
, C. A.
Tracy
, and H.
Widom
, “A growth model in a random environment
,” Ann. Probab.
30
, 1340
–1369
(2002
).37.
38.
P. J.
Forrester
, Log Gases and Random Matrices
(Princeton University Press
, 2010
).39.
G. E.
Andrews
, R.
Askey
, and R.
Roy
, Special Functions
, Encyclopedia of Mathematics and its Applications Vol. 71
(Cambridge University Press
, 1999
).40.
J.
Warren
, “Dyson’s Brownian motions, intertwining and interlacing
,” Electron. J. Probab.
12
, 573
–590
(2007
).41.
T.
Sasamoto
, “A note on a few processes related to Dyson’s Brownian motion
,” RIMS Kkyroku Bessatsu
B27
, 123
–139
(2011
).42.
J.
Warren
and P.
Windridge
, “Some examples of dynamics for Gelfand-Tsetlin patters
,” Electron. J. Probab.
14
, 1745
–1769
(2009
).43.
N.
O’Connell
, “Directed polymers and the quantum Toda lattice
,” Ann. Prob.
40
, 437
–458
(2012
).44.
45.
T.
Imamura
and T.
Sasamoto
, “Determinantal structures in the O’Connell-Yor directed random polymer model
,” J. Stat. Phys.
163
, 675
–713
(2016
).46.
A.
Borodin
, I.
Corwin
, and T.
Sasamoto
, “From duality to determinants for q-TASEP and ASEP
,” Ann. Prob.
42
, 2314
–2382
(2014
).47.
I.
Corwin
and L.
Petrov
, “Stochastic higher spin vertex models on the line
,” Commun. Math. Phys.
343
(2
), 651
–700
(2016
); arXiv:1502.07374 [math.PR].48.
K.
Matveev
and L.
Petrov
, “q-randomized Robinson–Schensted–Knuth correspondences and random polymers
,” Ann. Inst. Henri Poincaré D
4
, 1
–123
(2017
).49.
Y. B.
Sanderson
, “On the connection between MacDonald polynomials and Demazure characters
,” J. Algebraic Combinatorics
11
, 269
–275
(2000
).50.
A.
Nakayashiki
and Y.
Yamada
, “Kostka polynomials and energy functions in solvable lattice models
,” Sel. Math.
3
, 547
–599
(1997
).51.
B. E.
Sagan
and R. P.
Stanley
, “Robinson-Schensted algorithms for skew tableaux
,” J. Comb. Theory, Ser. A
55
(2
), 161
–193
(1990
).52.
D.
Takahashi
and J.
Satsuma
, “A soliton cellular automaton
,” J. Phys. Soc. Jpn.
59
(10
), 3514
–3519
(1990
).53.
R.
Inoue
, A.
Kuniba
, and T.
Takagi
, “Integrable structure of box–ball systems: Crystal, Bethe ansatz, ultradiscretization and tropical geometry
,” J. Phys. A: Math. Theor.
45
(7
), 073001
(2012
).54.
M.
Kashiwara
, “Crystalizing the q-analogue of universal enveloping algebras
,” Commun. Math. Phys.
133
(2
), 249
–260
(1990
).55.
M.
Kashiwara
, “On crystal bases of the Q-analogue of universal enveloping algebras
,” Duke Math. J.
63
(2
), 465
–516
(1991
).56.
G.
Lusztig
, “Canonical bases arising from quantized enveloping algebras
,” J. Am. Math. Soc.
3
(2
), 447
–498
(1990
).57.
A.
Borodin
, A.
Bufetov
, and M.
Wheeler
, “Between the stochastic six vertex model and Hall-Littlewood processes
,” Duke Math. J.
167
(13
), 2457
–2529
(2018
).58.
G.
Barraquand
, A.
Borodin
, I.
Corwin
, and M.
Wheeler
, “Stochastic six-vertex model in a half-quadrant and half-line open asymmetric simple exclusion process
,” Duke Math. J.
167
(13
), 2457
–2529
(2018
).59.
G.
Barraquand
, A.
Borodin
, and I.
Corwin
, “Half-space MacDonald processes
,” Forum Math.
8
, e11
(2020
).60.
J.
Baik
and E. M.
Rains
, “Limiting distributions for a polynuclear growth model with external sources
,” J. Stat. Phys.
100
, 523
–541
(2000
).61.
A.
Soshnikov
, “Determinantal random point fields
,” Russ. Math. Surv.
55
, 923
–975
(2000
).62.
T.
Shirai
and Y.
Takahashi
, “Random point fields associated with certain Fredholm determinants I: Fermion, Poisson and boson point processes
,” J. Funct. Anal.
205
, 414
–463
(2003
).63.
A.
Borodin
, “Determinantal point processes
,” in The Oxford Handbook of Random Matrix Theory
(Oxford University Press
, Oxford
, 2015
), p. 2009
.64.
65.
66.
B. E.
Sagan
, The Symmetric Group: Representations, Combinatorial Algorithm, and Symmetric Functions
, 2nd ed.
(Springer
, 2000
).67.
K.
Johansson
, “Shape fluctuations and random matrices
,” Commun. Math. Phys.
209
, 437
–476
(2000
).68.
T.
Sasamoto
, “Fluctuations of the one-dimensional asymmetric exclusion process using random matrix techniques
,” J. Stat. Mech.
2007
, P07007
.© 2023 Author(s). Published under an exclusive license by AIP Publishing.
2023
Author(s)
You do not currently have access to this content.
