TRACY, CRAIG A. / Mathematics / Researchers

International Center for Scientific Research

Researchers

Mathematics / TRACY, CRAIG A.

tracymath.ucdavis.edu

Position

Distinguished Professor at the Department of Mathematics, University of California Davis, USA.

Research interests

Craig Tracy’s research focuses on statistical physics, integrable systems and probability theory.

Craig Tracy and Harold Widom have done deep and original work on Random Matrix Theory, a subject which has remarkable applications across the scientific spectrum, from the scattering of neutrons off large nuclei to the behavior of the zeros of the Riemannzeta-function.

The contributions of Tracy and Widom center around a connection between a class of Fredholm determinants associated with random matrix ensembles on the one hand, and Painlevé functions on the other. Most notably, they have introduced a new class of distributions, the so-called Tracy-Widom distributions, which have a universal character and which have applications, in particular, to Ulam’s longest increasing subsequence problem in combinatorics, tiling problems, the airline boarding problem of Bachmat et al., various random walker and statistical mechanical growth models in the KPZ class, and principal component analysis in statistics when the number of variables is comparable to the sample size.

Prizes and awards

2002 George Pólya Prize, with Harold Widom, for their remarkable work on random matrix theory, a subject with multiple connections to complex analysis, orthogonal polynomials, probability theory and integral systems.

2007 Norbert Wiener Prize

American Academy of Arts & Sciences

Publications

S. Krinsky, C. Tracy, and M. Blume, Variational approximation to the Ising model in a magneticfield, Phys. Rev. Letts. 30 (1973), 750–753.

C. A. Tracy and B. M. McCoy, Neutron scattering and the correlation functions of the Ising modelnear Tc, Phys. Rev. Letts. 31 (1973), 1500–1504.

C. A. Tracy and B. M. McCoy, On the maximum of the k-dependent susceptibility for fixed k andT Tc, Phys. Letts. 46A (1974), 371–372.

C. A. Tracy, On the decay rate of order-parameter fluctuations, Phys. Letts. 48A (1974), 9–10.

S. Krinsky, C. Tracy, and M. Blume, Variational approximation to a ferromagnet in a magneticfield, Phys. Rev. B9 (1974), 4808–4815.

C. A. Tracy and B. M. McCoy, Examination of the phenomenological scaling functions for criticalscattering, Phys. Rev. B12 (1975), 368–387.

C. A. Tracy, Critical scattering scaling functions and the measurement of ?, in Magnetism andMagnetic Materials–1975, eds. J. J. Becker, G. H. Lander, and J. J. Rhyne, American Instituteof Physics Conference Proceedings 29, 1975, pp. 483–487.

T. T. Wu, B. M. McCoy, C. A. Tracy, E. Barouch, Spin-spin correlation functions for the two-dimensional Ising model: Exact theory in the scaling region, Phys. Rev. B13 (1976), 316–374.

B. M. McCoy, C. A. Tracy, and T. T. Wu, Painleve functions of the third kind, J. Math. Phys.18 (1977), 1058–1092.

B. M. McCoy, C. A. Tracy, and T. T. Wu, Spin-spin correlation functions for the two-dimensionalIsing model, in Statistical Mechanics and Statistical Methods in Theory and Application, ed. U. Land-man, Plenum Publ. Corp., 1977, pp. 83–97.

B. M. McCoy, C. A. Tracy, and T. T. Wu, Two-dimensional Ising model as an exactly solvablerelativistic quantum field theory: Explicit formulas for n-point functions, Phys. Rev. Letts. 38(1977), 793–796.

B. M. McCoy, C. A. Tracy, and T. T. Wu, Connection between the KdV equation and the two-dimensional Ising model, Phys. Letts. 61A (1977), 283–284

H. G. Vaidya and C. A. Tracy, Transverse time-dependent spin correlation functions for the one-dimensional XY model at zero temperature, Physica 92A (1978), 1–41.

C. A. Tracy, Painleve transcendents and scaling functions of the two-dimensional Ising model, inNonlinear Equations in Physics and Mathematics, ed. A. O. Barut, D. Reidel Publ. Co., Dordrecht,Holland, 1978, 221–237.

H. G. Vaidya and C. A. Tracy, Crossover scaling function for the one-dimensional XY model atzero temperature, Phys. Letts. 68A (1978), 378–380.

H. G. Vaidya and C. A. Tracy, One-particle reduced density matrix of impenetrable bosons in onedimension at zero temperature, Phys. Rev. Letts. 42 (1979), 3–6.

H. G. Vaidya and C. A. Tracy, One-particle reduced density matrix of impenetrable bosons in onedimension at zero temperature, J. Math. Phys. 20 (1979), 2291–2313.

J. Palmer and C. Tracy, Two-dimensional Ising correlation functions: Convergence of the scalinglimit, Adv. in Applied Math. 2 (1981), 329–388.

J. Palmer and C. Tracy, Two-dimensional Ising correlation functions: The SMJ analysis, Adv. inApplied Math. 4 (1983), 46–102.

C. A. Tracy, Complete integrability in statistical mechanics and the Yang-Baxter equations, Phys-ica 14D (1984), 253–264.

C. A. Tracy, Embedded elliptic curves and the Yang-Baxter equations, Physica 16D (1985), 203–220.

M. P. Richey and C. A. Tracy, The ZnBaxter model: Symmetries and the Belavin parametrization,J. Statistical Phys. 42 (1986), 311–348.

C. A. Tracy, ZnBaxter model: Critical behavior, J. Statistical Phys. 44 (1986), 183–191.

C. A. Tracy, The emerging role of number theory in exactly solvable models in lattice statisticalmechanics, Physica 25D (1987), 1–19.

M. P. Richey and C. A. Tracy, Symmetry group for a completely symmetric vertex model, J. Phys.A.: Math. Gen. 20 (1987), 2667–2677.

C. A. Tracy, L. Grove, and M. F. Newman, Modular properties of the hard hexagon model, J.Statistical Phys. 48 (1987), 477–502.

M. P. Richey and C. A. Tracy, Equation of state and isothermal compressibility for the hardhexagon model in the disordered regime, J. Phys. A.: Math. Gen. 20 (1987), L1121–L1126.

C. A. Tracy, Universality class of a Fibonacci Ising model, J. Statistical Phys. 51 (1988), 481–490.

C. A. Tracy, Universality classes of some aperiodic Ising models, J. Phys. A.: Math. Gen. 21(1988), L603–L605.

C. A. Tracy, Monodromy preserving deformation of the Klein-Gordon equation in the hyperbolicplane, Physica 34D (1989), 347–365.

C. A. Tracy, Introduction to exactly solvable models in statistical mechanics, Proc. Symposia inPure Math. 49 (1989), Part I, 355–375.

M. P. Richey and C. A. Tracy, Algorithms for the computation of polynomial relationships for thehard hexagon model, Nuclear Physics B330 (1990), 681–704.

C. A. Tracy, Monodromy preserving deformation of linear ordinary and partial differential equa-tions, in Solitons in Physics, Mathematics, and Nonlinear Optics, eds. P. J. Olver and D. H. Sat-tinger, Springer-Verlag, New York, 1990, pp. 165–174.

J. Palmer and C. A. Tracy, Monodromy preserving deformation of the Dirac operator acting on thehyperbolic plane, in Mathematics of Nonlinear Science ed. M. S. Berger, American MathematicalSociety, Providence, 1990, pp. 119–131.

R. Narayanan and C. A. Tracy, Holonomic quantum field theory of bosons in the Poincare diskand the zero curvature limit, Nuclear Physics B340 (1990), 568–594

E. L. Basor and C. A. Tracy, The Fisher-Hartwig conjecture and its generalizations, Physica 177A(1991), 167–173.

C. A. Tracy, Asymptotics of a ?-function arising in the two-dimensional Ising model, Commun.Math. Phys. 142 (1991), 297–311.

E. L. Basor and C. A. Tracy, Some problems associated with the asymptotics of ?-functions,Surikagaku (Mathematical Sciences) 30, no. 3 (1992), 71–76 [English translation appears in RIMS–845 preprint].

B. Temple and C. A. Tracy, From Newton to Einstein, American Math. Monthly 99 (1992),507–521.

E. L. Basor and C. A. Tracy, Asymptotics of a tau-function and Toeplitz determinants withsingular generating functions, Int. J. Modern Physics A 7, Suppl. 1A (1992), 83–107.

E. L. Basor, C. A. Tracy and H. Widom, Asymptotics of level spacing distributions for randommatrices, Phys. Rev. Letts. 69 (1992), 5–8.

R. Narayanan, J. Palmer and C. A. Tracy, Some isomonodromy problems in hyperbolic space,in Proceedings of the NATO Advanced Workshop on Painleve Transcendents, Their Asymptotics,and Physical Applications, eds. D. Levi and P. Winternitz, Plenum, New York, 1992, pp. 407–424.

C. A. Tracy and H. Widom, Level-spacing distributions and the Airy kernel, Phys. Letts. B 305(1993), 115–118.

C. A. Tracy and H. Widom, Introduction to random matrices, in Geometric and Quantum Aspectsof Integrable Systems, ed. G. F. Helminck, Lecture Notes in Physics, Vol. 424, Springer-Verlag,Berlin, 1993, pp. 103–130.

E. L. Basor and C. A. Tracy, Variance calculations and the Bessel kernel, J. Statistical Phys. 73(1993), 415–421.

J. Harnad, C. A. Tracy and H. Widom, Hamiltonian structure of equations appearing in randommatrices, in Low-Dimensional Topology and Quantum Field Theory, ed. H. Osborn, NATO ASISeries B, Vol. 314, Plenum Press, New York, 1993, pp. 231–245.

C. A. Tracy and H. Widom, Level-spacing distributions and the Airy kernel, Commun. Math.Physics 159 (1994), 151–174.

C. A. Tracy and H. Widom, Level-spacing distributions and the Bessel kernel, Commun. Math.Phys. 161 (1994), 289–309.

C. A. Tracy and H. Widom, Fredholm determinants, differential equations and matrix models,Commun. Math. Phys. 163 (1994), 33–72.

J. Palmer, M. Beatty, and C. A. Tracy, Tau-functions for the Dirac operator on the Poincare disk,Commun. Math. Phys. 165 (1994), 97–173.

Y. Chen, K. J. Eriksen and C. A. Tracy, Largest eigenvalue distributions in the double scalinglimit of matrix models: a Coulomb fluid approach, J. Phys. A.: Math. Gen. 28 (1995), L207–L211.

C. A. Tracy and H. Widom, Systems of partial differential equations for a class of operatordeterminants, in Partial Differential Operators and Mathematical Physics, eds. M. Demuth andB.-W. Schulz, Operator Theory: Advances and Applications, Vol. 78, Birkhauser Verlag, Berlin,1995, pp. 381–388.

C. A. Tracy and H. Widom, On orthogonal and symplectic matrix ensembles, Commun. Math.Phys. 177 (1996), 727–754.

C. A. Tracy and H. Widom, Fredholm determinants and the mKdV/sinh-Gordon hierarchies,Commun. Math. Phys. 179 (1996), 1–10.

C. A. Tracy and H. Widom, Proofs of two conjectures related to the thermodynamic Bethe Ansatz,Commun. Math. Phys. 179 (1996), 667–680.

C. A. Tracy and H. Widom, The thermodynamic Bethe Ansatz and a connection with Painleveequations, Int. J. Mod. Physics B 11 (1997), 69–74.

C. A. Tracy and H. Widom, On exact solutions to the cylindrical Poisson-Boltzmann equationwith application to polyelectrolytes, Physica 244A (1997), 402–413.

C. A. Tracy and H. Widom, Asymptotics of a class of solutions to the cylindrical Toda equations,Commun. Math. Phys. 190 (1998), 697–721.

C. A. Tracy and H. Widom, Correlation functions, cluster functions and spacing distributions forrandom matrices, J. Statistical Phys. 92 (1998), 809–835.

C. A. Tracy and H. Widom, Universality of the distribution functions of random matrix theory,in Statistical Physics on the Eve of the 21st Century: In Honour of J B McGuire on the Occasionof His 65th Birthday, eds. M. T. Batchelor and L. T. Wille, World Scientific Publishing, 1999, pp.230–239.

C. A. Tracy and H. Widom, Asymptotics of a Class of Fredholm Determinants, in Spectral Problemsin Geometry and Arithmetic, ed. T. Branson, American Mathematical Society, Providence, 1999,pp. 167–174.

C. A. Tracy and H. Widom, Random unitary matrices, permutations and Painleve, Commun.Math. Phys. 207 (1999), 665–685.

C. A. Tracy and H. Widom, The distribution of the largest eigenvalue in the Gaussian ensembles:? = 1, 2, 4, in Calogero-Moser-Sutherland Models, eds. J. F. van Diejen and L. Vinet, CRM Seriesin Mathematical Physics 4, Springer-Verlag, New York, 2000, pp. 461–472.

C. A. Tracy and H. Widom, Universality of the distribution functions of random matrix theory. II,in Integrable Systems: From Classical to Quantum, CRM Proceedings & Lecture Notes, Vol. 26,eds. J. Harnad, G. Sabidussi, and P. Winternitz, American Mathematical Society, Providence,2000, pp. 251–264.

C. A. Tracy and H. Widom, On the distributions of the lengths of the longest monotone subse-quences in random words, Probab. Theo. Related Fields 119 (2001), 350–380.

J. Gravner, C. A. Tracy and H. Widom, Limit theorems for height fluctuations in a class of discretespace and time growth models, J. Statistical Physics 102 (2001), 1085–1132.

A. R. Its, C. A. Tracy and H. Widom, Random words, Toeplitz determinants and integrablesystems, I., Random Matrix Models and their Applications, eds. P. Bleher and A. Its, Math. Sci.Res. Inst. Publications 40, Cambridge University Press, New York, 2001, pp. 245–258.

A. R. Its, C. A. Tracy and H. Widom, Random words, Toeplitz determinants and integrablesystems, II., Physica 152–153D (2001), 199–224.

C. A. Tracy and H. Widom, On the limit of some Toeplitz-like determinants, SIAM J. MatrixAnal. Appl. 23 (2002), 1194–1196.

C. A. Tracy and H. Widom, Airy kernel and Painleve II, in Isomonodromic Deformations andApplications in Physics, eds. A. Its and J. Harnad, CRM Proceedings & Lecture Notes, Vol. 31,Amer. Math. Soc., Providence, 2002, pp. 85–98.

C. A. Tracy and H. Widom, On a distribution function arising in computational biology, in Math-Phys Odyssey 2002: Integrable Models and Beyond, eds. M. Kashiwara and T. Miwa, Birkhauser,2002, pgs. 467–474.

J. Gravner, C. A. Tracy and H. Widom, A growth model in a random environment, Ann. Probab.30 (2002), 1340–1368.

J. Gravner, C. A. Tracy and H. Widom, Fluctuations in the composite regime of a disorderedgrowth model, Commun. Math. Phys. 229 (2002), 433–458.

C. A. Tracy and H. Widom, Distribution functions for largest eigenvalues and their applications, inProceedings of the International Congress of Mathematicians, Beijing 2002, Vol. I, ed. LI Tatsien,Higher Education Press, Beijing, 2002, pgs. 587–596.

C. A. Tracy and H. Widom, A system of differential equations for the Airy process, Elect. Comm.in Probab. 8 (2003), 93–98

C. A. Tracy and H. Widom, A limit theorem for shifted Schur measures, Duke Math. J. 123(2004), 171–208.

C. A. Tracy and H. Widom, Differential equations for Dyson processes, Commun. Math. Phys.252 (2004), 7–41.

C. A. Tracy and H. Widom, Matrix kernels for the Gaussian orthogonal and symplectic ensembles,Ann. Inst. Fourier, Grenoble 55 (2005), 2197–2207.

C. A. Tracy and H. Widom, The Pearcey process, Commun. Math. Phys. 263 (2006), 381–400.

C. A. Tracy and H. Widom, Nonintersecting Brownian excursions, The Annals of Applied Prob-ability 17 (2007), 953–979.

C. A. Tracy and H. Widom, Integral formulas for the asymmetric simple exclusion process, Com-mun. Math. Phys. 279 (2008), 815–844

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