APPLICATIONS OF RANDOM MATRICES IN PHYSICS
CONTENTS INTRODUCTION 2 + IT) AND LOG ?( 2 + IT) CHARACTERISTIC POLYNOMIALS OF RANDOM UNITARY MATRICES ASYMPTOTIC EXPANSIONS REFERENCES 2D QUANTUM GRAVITY, MATRIX MODELS AND GRAPH COMBINATORICS ION MATRIX MODELS FOR 2D QUANTUM GRAVITY THE ONE-MATRIX MODEL I: LARGE N LIMIT AND THE ENUMERATION OF PLANAR GRAPHS THE TREES BEHIND THE GRAPHS THE ONE-MATRIX MODEL II: TOPOLOGICAL EXPANSIONS AND QUANTUM GRAVITY THE COMBINATORICS BEYOND MATRIX MODELS: GEODESIC DISTANCE IN PLANAR GRAPHS PLANAR GRAPHS AS SPATIAL BRANCHING PROCESSES CONCLUSION REFERENCES REFERENCES SUPERSYMMETRY METHOD WAVE FUNCTIONS ?UCTUATIONS IN A ?NITE VOLUME MULTIFRACTALITY RECENT AND POSSIBLE FUTURE DEVELOPMENTS SUMMARY RANDOM MATRICES AND NUMBER THEORY OTHER COMPACT GROUPS EIGENVALUE DYNAMICS, FOLLYTONS AND LARGE N LIMITS OF MATRICES REFERENCES FAMILIES OF L-FUNCTIONS AND SYMMETRY ACKNOWLEDGEMENTS RANDOM MATRICES AND SUPERSYMMETRY IN DISORDERED SYSTEMS INSTANTON OR RARE ?UCTUATION METHOD HYDRODYNAMIC APPROACH LINEARIZED HYDRODYNAMICS OR BOSONIZATION EFP THROUGH AN ASYMPTOTICS OF THE SOLUTION CONCLUSION REFERENCES SUMMARY QCD CONCLUSIONS REFERENCES BASIC DE?NITIONS PHYSICAL MOTIVATIONS FIELD THEORY THE SIMPLEST CASE PHONONS HYDRODYNAMICS OF CORRELATED SYSTEMS QCD, CHIRAL RANDOM MATRIX THEORY AND INTEGRABILITY EUCLIDEAN RANDOM MATRICES: SOLVED AND OPEN PROBLEMS THE DIRAC SPECTRUM IN QCD LOW ENERGY LIMIT OF QCD INTEGRABILITY AND THE QCD PARTITION FUNCTION CHIRAL RMT AND THE QCD DIRAC SPECTRUM QCD AT FINITE BARYON DENSITY FULL QCD AT NONZERO CHEMICAL POTENTIAL REFERENCES FREE FERMIONS CALOGERO-SUTHERLAND MODEL FREE FERMIONS ON THE LATTICE SOME ENSEMBLES OF RANDOM MATRICES WITH COMPLEX EIGENVALUES MATRIX MODELS AND GROWTH PROCESSES ACKNOWLEDGEMENTS ACKNOWLEDGEMENTS EXACT RESULTS AT ?NITE N LARGE N LIMIT THE MATRIX MODEL AS A GROWTH PROBLEM REFERENCES MATRIX MODELS TYPE B TOPOLOGICAL STRINGS AND MATRIX MODELS TYPE A TOPOLOGICAL STRINGS, CHERN-SIMONS THEORY AND MATRIX MODELS REFERENCES MATRIX MODELS OF MODULI SPACE THE PENNER MODEL THE KONTSEVICH MODEL CONCLUSIONS REFERENCES MATRIX MODELS AND 2D STRING THEORY AN OVERVIEW OF STRING THEORY STRINGS IN D-DIMENSIONAL SPACETIME DISCRETIZED SURFACES AND 2D STRING THEORY AN OVERVIEW OF OBSERVABLES SAMPLE CALCULATION: THE DISK ONE-POINT FUNCTION WORLDSHEET DESCRIPTION OF MATRIX EIGENVALUES FURTHER RESULTS OPEN PROBLEMS MATRIX MODELS AND TOPOLOGICAL STRINGS QUADRATIC DIFFERENTIALS AND FATGRAPHS PENNER MODEL AND MATRIX GAMMA FUNCTION APPLICATIONS TO STRING THEORY REFERENCES MATRIX MODELS AS CONFORMAL FIELD THEORIES INTRODUCTION AND HISTORICAL NOTES HERMITIAN MATRIX INTEGRAL: SADDLE POINTS AND HYPERELLIPTIC CURVES THE HERMITIAN MATRIX MODEL AS A CHIRAL CFT QUASICLASSICAL EXPANSION: CFT ON A HYPERELLIPTIC RIEMANN SURFACE GENERALIZATION TO CHAINS OF RANDOM MATRICES REFERENCES MODULI SPACE OF RIEMANN SURFACES AND ITS TOPOLOGY DE?NITIONS ORTHOGONAL POLYNOMIALS RIEMANN-HILBERT PROBLEMS AND ISOMONODROMIES WKB–LIKE ASYMPTOTICS AND SPECTRAL CURVE ORTHOGONAL POLYNOMIALS AS MATRIX INTEGRALS (0) SOLUTION OF THE SADDLEPOINT EQUATION ASYMPTOTICS OF ORTHOGONAL POLYNOMIALS DIFFERENTIAL EQUATIONS AND INTEGRABILITY SADDLE POINT METHOD LARGE N ASYMPTOTICS OF ORTHOGONAL POLYNOMIALS FROM INTEGRABILITY TO ALGEBRAIC COMPUTATION OF DERIVATIVES OF F GEOMETRY RANDOM MATRICES AND NUMBER THEORY 1 INTRODUCTION MY PURPOSE IN THESE LECTURE NOTES IS TO REVIEW AND EXPLAIN SOME RECENT RE- SULTS CONCERNING CONNECTIONS BETWEEN RANDOM MATRIX THEORY AND NUMBER THE- ORY SPECI?CALLY, I WILL FOCUS ON HOW RANDOM MATRIX THEORY HAS BEEN USED TO SHED NEW LIGHT ON SOME CLASSICAL PROBLEMS RELATING TO THE VALUE DISTRIBUTIONS OF THE RIEMANN ZETA-FUNCTION AND OTHER L-FUNCTIONS, AND ON APPLICATIONS TO MODULAR FORMS AND ELLIPTIC CURVES THIS MAY ALL SEEM RATHER FAR FROM PHYSICS, BUT, AS I HOPE TO MAKE CLEAR, THE QUESTIONS I SHALL BE REVIEWING ARE RATHER NATURAL FROM THE RANDOM-MATRIX POINT OF VIEW, AND ATTEMPTS TO ANSWER THEM HAVE STIMULATED SIGNI?CANT DEVELOPMENTS WITHIN THAT SUBJECT MOREOVER, ANALOGIES BETWEEN PROPERTIES OF THE RIEMANN ZETA FUNCTION, RANDOM MATRIX THEORY, AND THE SEMICLASSICAL THEORY OF QUANTUM CHAOTIC SYSTEMS HAVE BEEN THE SUBJECT OF CONSIDERABLE INTEREST OVER THE PAST (1) FOR RES > 1, WHERE P LABELS THE PRIMES, AND THEN BY ANALYTIC CONTINUATION TO THE REST OF THE COMPLEX PLANE IT HAS A SINGLE SIMPLE POLE AT S = 1, ZEROS AT S = ?2, ?4, ?6, ETC , AND IN?NITELY MANY ZEROS, CALLED THE NON-TRIVIAL ZEROS, APPLICATIONS OF RANDOM MATRICES IN PHYSICS IN THE CRITICAL STRIP 0 < RES < 1 IT SATIS?ES THE FUNCTIONAL EQUATION ??S/2? S ?(S) = ??(1?S)/2? 1 ?S ?(1 ?S) (2) THE RIEMANN HYPOTHESIS STATES THAT ALL OF THE NON-TRIVIAL ZEROS LIE ON THE CRITICAL LINE RES = 1/2 (I E ON THE SYMMETRY LINE OF THE FUNCTIONAL EQUATION); THAT IS, ?(1/2 + IT) = 0 HAS NON-TRIVIAL SOLUTIONS ONLY WHEN T = TN ?R [33] THIS IS KNOWN TO BE TRUE FOR AT LEAST 40% OF THE NON-TRIVIAL ZEROS [6], FOR THE ?RST 100 BILLION OF THEM [36], AND FOR BATCHES LYING MUCH HIGHER [29] IN THESE NOTES I WILL, FOR EASE OF PRESENTATION, ASSUME THE RIEMANN HYPOTH- ESIS TO BE TRUE THIS IS NOT STRICTLY NECESSARY – IT SIMPLY MAKES SOME OF THE FORMULAE MORE TRANSPARENT THE MEAN DENSITY OF THE NON-TRIVIAL ZEROS INCREASES LOGARITHMICALLY WITH HEIGHT T UP THE CRITICAL LINE SPECI?CALLY, THE UNFOLDED ZEROS WN = TN 2? LOG |TN| 2? (3) SATISFY LIM W ?? RPR/2 SIN(RT LOG P), (5) WHERE N(T ) = T 2? LOG T 2? ?T 2? + 8 + O T (6) AS T ?? THIS FOLLOWS FROM INTEGRATING THE LOGARITHMIC DERIVATIVE OF ?(S) AROUND A RECTANGLE, POSITIONED SYMMETRICALLY WITH RESPECT TO THE CRITICAL LINE AND PASSING THROUGH THE POINTS S = 1/2 AND S = 1/2+IT, USING THE FUNCTIONAL EQUATION (FORMULAE LIKE THIS CAN BE MADE TO CONVERGE BY INTEGRATING BOTH SIDES AGAINST A SMOOTH FUNCTION WITH SUF?CIENTLY FAST DECAY AS |T | ?? ) IF P ?D AND X2 ?D (MOD P) SOLVABLE HISTORY THE CONNECTION BETWEEN RANDOM MATRIX THEORY AND NUMBER THEORY WAS ?RST MADE IN 1973 IN THE WORK OF MONTGOMERY [28], WHO CONJECTURED THAT LIM W ?? DX (9) THIS CONJECTURE WAS MOTIVATED BY A THEOREM MONTGOMERY PROVED IN THE SAME PAPER THAT MAY BE RESTATED AS FOLLOWS: LIM N?? DX (10) RANDOM MATRICES AND NUMBER THEORY APPLICATIONS OF RANDOM MATRICES IN PHYSICS FOR ALL TEST FUNCTIONS F(X) WHOSE FOURIER TRANSFORMS F(?) = ? ?? F(X) EXP(2?IX?)DX (11) HAVE SUPPORT IN THE RANGE (-1, 1) AND ARE SUCH THAT THE SUM AND INTEGRAL IN (10) CONVERGE THE GENERALIZED FORM OF THE MONTGOMERY CONJECTURE IS THAT (10) HOLDS FOR ALL TEST FUNCTIONS SUCH THAT THE SUM AND INTEGRAL CONVERGE, WITHOUT ANY RESTRICTION ON THE SUPPORT OF F(?) THE FORM OF THE CONJECTURE (9) THEN CORRESPONDS TO THE PARTICULAR CASE IN WHICH F(X) IS TAKEN TO BE THE INDICATOR FUNCTION ON THE INTERVAL [?, ?) (AND SO DOES NOT FALL WITHIN THE CLASS OF TEST FUNCTIONS COVERED BY THE THEOREM) THE LINK WITH RANDOM MATRIX THEORY FOLLOWS FROM THE OBSERVATION THAT THE PAIR CORRELATION OF THE NONTRIVIAL ZEROS CONJECTURED BY MONTGOMERY COINCIDES PRECISELY WITH THAT WHICH HOLDS FOR THE EIGENVALUES OF RANDOM MATRICES TAKEN FROM EITHER THE CIRCULAR UNITARY ENSEMBLE (CUE) OR THE GAUSSIAN UNITARY EN- SEMBLE (GUE) OF RANDOM MATRICES [27] (I E RANDOM UNITARY OR HERMITIAN MA- TRICES) IN THE LIMIT OF LARGE MATRIX SIZE FOR EXAMPLE, LET A BE AN N × N UNITARY MATRIX, SO THAT A(AT )?= AA† = I THE EIGENVALUES OF A LIE ON THE UNIT CIRCLE; THAT IS, THEY MAY BE EXPRESSED IN THE FORM EI?N, ?N ?R SCALING THE EIGENPHASES ?N SO THAT THEY HAVE UNIT MEAN SPACING, ?N = ?N N 2? , (12) THE TWO-POINT CORRELATION FUNCTION FOR A GIVEN MATRIX A MAY BE DE?NED AS R2(A; X) = ? K=?? ?(X + KN ??N + ?M), (13) SO THAT N N,M F(?N ??M) = N R2(A; X)F(X)DX (14) R2(A; X) IS CLEARLY PERIODIC IN X, SO CAN BE EXPRESSED AS A FOURIER SERIES: R2(A; X) = ? K=?? |TRAK|2E2?IKX/N (15) THE CUE CORRESPONDS TAKING MATRICES FROM U(N) WITH A PROBABILITY MEA- SURE GIVEN BY THE NORMALIZED HAAR MEASURE ON THE GROUP (I E THE UNIQUE MEA- SURE THAT IS INVARIANT UNDER ALL UNITARY TRANSFORMATIONS) IT FOLLOWS FROM (15) THAT THE CUE AVERAGE OF R2(A; X) MAY BE EVALUATED BY COMPUTING THE CORRE- SPONDING AVERAGE OF THE FOURIER COEF?CIENTS |TRAK|2 THIS WAS DONE BY DYSON [14]: U(N) |TRAK|2DµHAAR(A) = ? ? ? N K = |K| |K| ?N N |K| > N (16) THERE ARE SEVERAL METHODS FOR PROVING THIS ONE REASONABLY ELEMENTARY PROOF INVOLVES USING HEINE’S IDENTITY U(N) FC(?1, , ?N)DµHAAR(A) = (2?)N 2? FC(?1, , ?N) DET(EI?N(N?M))D?1 D?N (17) FOR CLASS FUNCTIONS FC(A) = FC(?1, ?2, , ?N) (I E FUNCTIONS FC THAT ARE SYM- METRIC IN ALL OF THEIR VARIABLES) TO GIVE U(N) |TRAK|2DµHAAR(A) = J L EIK(?J??L) × EI(N?1)?N EI(N?2)?N D?1 D?N (18) THE NET CONTRIBUTION FROM THE DIAGONAL (J = L) TERMS IN THE DOUBLE SUM IS N, BECAUSE THE MEASURE IS NORMALIZED AND THERE ARE N DIAGONAL TERMS USING THE FACT THAT 2? 2? EIN?D? = N = N ?= 0 , (19) IF K ?N THEN THE INTEGRAL OF THE OFF-DIAGONAL TERMS IS ZERO, BECAUSE, FOR EXAMPLE, WHEN THE DETERMINANT IS EXPANDED OUT AND MULTIPLIED BY THE PREFACTOR THERE IS NO POSSIBILITY OF ?1 CANCELLING IN THE EXPONENT IF K = N ?S, S = 1, , N ?1, THEN THE OFF-DIAGONAL TERMS CONTRIBUTE ?S; FOR EXAMPLE, WHEN S = 1 ONLY ONE NON-ZERO TERM SURVIVES WHEN THE DETERMINANT IS EXPANDED OUT, MULTIPLIED BY THE PREFACTOR, AND INTEGRATED TERM-BY-TERM – THIS IS THE TERM COMING FROM MULTIPLYING THE BOTTOM-LEFT ENTRY BY THE TOP-RIGHT ENTRY AND ALL OF THE DIAGONAL ENTRIES ON THE OTHER ROWS THUS THE COMBINED DIAGONAL AND OFF-DIAGONAL TERMS ADD UP TO GIVE THE EXPRESSION IN (16), BEARING IN MIND THAT WHEN K = 0 THE TOTAL IS JUST N 2, THE NUMBER OF TERMS IN THE SUM OVER J AND L RANDOM MATRICES AND NUMBER THEORY APPLICATIONS OF RANDOM MATRICES IN PHYSICS HEINE’S IDENTITY ITSELF MAY BE PROVED USING THE WEYL INTEGRATION FORMULA [35] U(N) FC(A)DµHAAR(A) = (2?)N N! 2? FC(?1, , ?N) × 1?J<K?N |EI?J ?EI?K|2D?1 D?N (20) FOR CLASS FUNCTIONS FC(A), THE VANDERMONDE IDENTITY 1?J<K?N |EI?J ?EI?K|2 = DET MM † (21) WHERE M = ? ? ? ? ? EI?N ? ? ?U †, (24) WHERE U IS AN N × N UNITARY MATRIX THEREFORE THE INTEGRAL OVER A CAN BE WRITTEN AS AN INTEGRAL OVER THE MATRIX ELEMENTS OF U AND THE EIGENPHASES ?N BECAUSE THE MEASURE IS INVARIANT UNDER UNITARY TRANSFORMATIONS, THE INTEGRAL OVER THE MATRIX ELEMENTS OF U CAN BE EVALUATED STRAIGHTFORWARDLY, LEAVING THE INTEGRAL OVER THE EIGENPHASES (20) HENCEFORTH, TO SIMPLIFY THE NOTATION, I SHALL DROP THE SUBSCRIPT ON THE MEA- SURE Dµ(A) – IN ALL INTEGRALS OVER COMPACT GROUPS THE MEASURE MAY BE TAKEN TO BE THE HAAR MEASURE ON THE GROUP IT FOLLOWS FROM DYSON’S THEOREM (16) THAT U(N) R2(A; X)Dµ(A) = ? K=?? E2?IKX/N ? ? ? N DX (27) FOR EXAMPLE, LIM N?? U(N) DX (28) THE KEY POINT IS NOW THAT THE RIGHT-HAND SIDES OF (27) AND (28) COINCIDE PRECISELY WITH THOSE OF (10) AND (9) RESPECTIVELY THAT IS, THE PAIR CORRELATION OF THE RIEMANN ZEROS, IN THE LIMIT AS THE HEIGHT UP THE CRITICAL LINE TENDS TO IN?NITY, IS, CONJECTURALLY, THE SAME AS THAT OF THE EIGENPHASES OF RANDOM UNITARY (OR HERMITIAN) MATRICES IN THE LIMIT AS THE MATRIX SIZE TENDS TO IN?NITY IT IS IMPORTANT TO NOTE THAT THE PROOF OF MONTGOMERY’S THEOREM DOES NOT IN- VOLVE ANY OF THE STEPS IN THE DERIVATION OF THE CUE PAIR CORRELATION FUNCTION IT IS INSTEAD BASED ENTIRELY ON THE CONNECTION BETWEEN THE RIEMANN ZEROS AND THE PRIMES IN OUTLINE, THE PROOF INVOLVES COMPUTING THE PAIR CORRELATION FUNCTION OF THE DERIVATIVE OF N(T ) USING THE EXPLICIT FORMULA (5), THIS PAIR CORRELATION FUNCTION CAN BE EXPRESSED AS A SUM OVER PAIRS OF PRIMES, P AND Q THE DIAGO- NAL TERMS, FOR WHICH P = Q, OBVIOUSLY INVOLVE ONLY SINGLE PRIMES THEIR SUM CAN THEN BE EVALUATED USING THE PRIME NUMBER THEOREM, WHICH GOVERNS THE ASYMPTOTIC DENSITY OF PRIMES (ROUGHLY SPEAKING, THE PRIME NUMBER THEO- REM GUARANTEES THAT PRIME SUMS P F(P) MAY, FOR APPROPRIATE FUNCTIONS F, BE APPROXIMATED BY F(X)/ LOG XDX ) THE OFF-DIAGONAL TERMS (P ?= Q) CANNOT BE SUMMED RIGOROUSLY HOWEVER, IT CAN BE SHOWN THAT THESE TERMS DO NOT CON- TRIBUTE TO THE LIMITING FORM OF THE PAIR CORRELATION FUNCTION FOR TEST FUNCTIONS F(X) IN (10) WHOSE FOURIER TRANSFORMS HAVE SUPPORT IN (-1, 1) THIS FOLLOWS FROM THE FACT THAT THE SEPARATION BETWEEN THE PRIMES IS BOUNDED FROM BELOW RANDOM MATRICES AND NUMBER THEORY