## The Dyson Equation with Linear Self-Energy: Spectral Bands, Edges and Cusps

##### Doc. Math. 25, 1421-1539 (2020)
DOI: 10.25537/dm.2020v25.1421-1539

### Summary

We study the unique solution $m$ of the Dyson equation $-m(z)^{-1} = z\1 - a + S[m(z)]$ on a von Neumann algebra $\mathcal{A}$ with the constraint $\text{Im}\, m\ge 0$. Here, $z$ lies in the complex upper half-plane, $a$ is a self-adjoint element of $\mathcal{A}$ and $S$ is a positivity-preserving linear operator on $\mathcal{A}$. We show that $m$ is the Stieltjes transform of a compactly supported $\mathcal{A}$-valued measure on $\mathbb{R}$. Under suitable assumptions, we establish that this measure has a uniformly $1/3$-Hölder continuous density with respect to the Lebesgue measure, which is supported on finitely many intervals, called bands. In fact, the density is analytic inside the bands with a square-root growth at the edges and internal cubic root cusps whenever the gap between two bands vanishes. The shape of these singularities is universal and no other singularity may occur. We give a precise asymptotic description of $m$ near the singular points. These asymptotics generalize the analysis at the regular edges given in the companion paper on the Tracy-Widom universality for the edge eigenvalue statistics for correlated random matrices [the first author et al., Ann. Probab. 48, No. 2, 963--1001 (2020; Zbl 1434.60017)] and they play a key role in the proof of the Pearcey universality at the cusp for Wigner-type matrices [\textit{G. Cipolloni} et al., Pure Appl. Anal. 1, No. 4, 615--707 (2019; Zbl 07142203); the second author et al., Commun. Math. Phys. 378, No. 2, 1203--1278 (2020; Zbl 07236118)]. We also extend the finite dimensional band mass formula from [the first author et al., loc. cit.] to the von Neumann algebra setting by showing that the spectral mass of the bands is topologically rigid under deformations and we conclude that these masses are quantized in some important cases.

### Mathematics Subject Classification

60B20, 46L54, 60F05

### Keywords/Phrases

Dyson equation, positive operator-valued measure, Stieltjes transform, band rigidity, eigenvalue distribution

### References

• 1. O. Ajanki, L. Erdős, and T. Krüger, Singularities of solutions to quadratic vector equations on the complex upper half-plane, Comm. Pure Appl. Math. 70 (2017), no. 9, 1672-1705. DOI 10.1002/cpa.21639; zbl 1419.15014; MR3684307; arxiv 1512.03703.
• 2. O. Ajanki, L. Erdős, and T. Krüger, Universality for general Wigner-type matrices, Probab. Theory Related Fields 169 (2017), no. 3-4, 667-727. DOI 10.1007/s00440-016-0740-2; zbl 1403.60010; MR3719056; arxiv 1506.05098.
• 3. O. Ajanki, L. Erdős, and T. Krüger, Quadratic vector equations on complex upper half-plane, Mem. Amer. Math. Soc. 261 (2019), no. 1261, v+133. DOI 10.1090/memo/1261; zbl 07161624; MR4031100.
• 4. O. Ajanki, L. Erdős, and T. Krüger, Stability of the matrix Dyson equation and random matrices with correlations, Probab. Theory Related Fields 173 (2019), no. 1-2, 293-373. DOI 10.1007/s00440-018-0835-z; zbl 07030873; MR3916109; arxiv 1604.08188.
• 5. A. B. Aleksandrov and V. V. Peller, Operator Lipschitz functions, Russian Math. Surveys 71 (2016), no. 4, 605-702. DOI 10.1070/RM9729; zbl 1356.26002; MR3588921; arxiv 1602.07994.
• 6. J. Alt, Singularities of the density of states of random Gram matrices, Electron. Commun. Probab. 22 (2017), paper no. 63, 13 pp. DOI 10.1214/17-ECP97; zbl 1378.60020; MR3734102; arxiv 1708.08442.
• 7. J. Alt, L. Erdős, T. Krüger, and Yu. Nemish, Location of the spectrum of Kronecker random matrices, Ann. Inst. Henri Poincaré Probab. Stat. 55 (2019), no. 2, 661-696. DOI 10.1214/18-AIHP894; zbl 07097327; MR3949949; arxiv 1706.08343.
• 8. J. Alt, L. Erdős, T. Krüger, and D. Schröder, Correlated random matrices: Band rigidity and edge universality, Ann. Probab. 48 (2020), no. 2, 963-1001. DOI 10.1214/19-AOP1379; zbl 1434.60017; MR4089499; arxiv 1804.07744.
• 9. J. Alt, L. Erdős, and T. Krüger, Local law for random Gram matrices, Electron. J. Probab. 22 (2017), paper no. 25, 41 pp. DOI 10.1214/17-EJP42; zbl 1376.60014; MR3622895; arxiv 1606.07353.
• 10. G. W. Anderson and O. Zeitouni, A CLT for a band matrix model, Probab. Theory Related Fields 134 (2006), no. 2, 283-338. DOI 10.1007/s00440-004-0422-3; zbl 1084.60014; MR2222385; arxiv math/0412040.
• 11. Z. D. Bai and J. W. Silverstein, Exact separation of eigenvalues of large-dimensional sample covariance matrices, Ann. Probab. 27 (1999), no. 3, 1536-1555. DOI 10.1214/aop/1022677458; zbl 0964.60041; MR1733159.
• 12. F. Bekerman, T. Leblé, and S. Serfaty, CLT for fluctuations of $\beta$-ensembles with general potential, Electron. J. Probab. 23 (2018), paper no. 115, 31 pp. DOI 10.1214/18-EJP209; zbl 1406.60036; MR3885548; arxiv 1706.09663.
• 13. F. A. Berezin, Some remarks on the Wigner distribution, Theoret. Math. Phys. 17 (1973), 1163-1171. DOI 10.1007/BF01037593; zbl 0291.60017; MR0465007.
• 14. R. Bhatia, Matrix analysis, Graduate Texts in Mathematics, vol. 169, Springer-Verlag, New York, 1997. zbl 0863.15001; MR1477662.
• 15. G. Cipolloni, L. Erdős, T. Krüger, and D. Schröder, Cusp universality for random matrices, II: The real symmetric case, Pure Appl. Anal. 1 (2019), no. 4, 615-707. DOI 10.2140/paa.2019.1.615; zbl 07142203; MR4026551; arxiv 1811.04055.
• 16. T. Claeys, I. Krasovsky, and A. Its, Higher-order analogues of the Tracy-Widom distribution and the Painlevé II hierarchy, Comm. Pure Appl. Math. 63, no. 3, 362-412. DOI 10.1002/cpa.20284; zbl 1198.34191; MR2599459; arxiv 0901.2473.
• 17. T. Claeys, A. B. J. Kuijlaars, K. Liechty, and D. Wang, Propagation of singular behavior for Gaussian perturbations of random matrices, Comm. Math. Phys. 362 (2018), no. 1, 1-54. DOI 10.1007/s00220-018-3195-8; zbl 1410.60011; MR3833603; arxiv 1608.05870.
• 18. P. Deift, T. Kriecherbauer, and K. T.-R McLaughlin, New results on the equilibrium measure for logarithmic potentials in the presence of an external field, J. Approx. Theory 95 (1998), no. 3, 388-475. DOI 10.1006/jath.1997.3229; zbl 0918.31001; MR1657691.
• 19. L. Erdős, T. Krüger, and D. Schröder, Cusp Universality for Random Matrices, I: Local Law and the Complex Hermitian Case, Commun. Math. Phys. 378 (2020), no. 2, 1203-1278. DOI 10.1007/s00220-019-03657-4; zbl 07236118; MR4134946; arxiv 1809.03971.
• 20. L. Erdős, T. Krüger, and D. Schröder, Random matrices with slow correlation decay, Forum Math. Sigma 7 (2019), paper no. e8, 89 pp. DOI 10.1017/fms.2019.2; zbl 1422.60014; MR3941370; arxiv 1705.10661.
• 21. L. Erdős and H.-T. Yau, A dynamical approach to random matrix theory, Courant Lecture Notes in Mathematics, vol. 28, Courant Institute of Mathematical Sciences, New York; American Mathematical Society, Providence, RI, 2017. DOI 10.1090/cln/028; zbl 1379.15003; MR3699468.
• 22. A. Garg, L. Gurvits, R. Mendes de Oliveira, and A. Wigderson, A deterministic polynomial time algorithm for non-commutative rational identity testing, 2016 IEEE 57th Annual Symposium on Foundations of Computer Science (FOCS) (2016), 109-117. MR3630971.
• 23. J. B. Garnett, Bounded analytic functions, first ed., Graduate Texts in Mathematics, vol. 236, Springer, New York, 2007. DOI 10.1007/0-387-49763-3; zbl 1106.30001; MR2261424.
• 24. V. L. Girko, Theory of stochastic canonical equations: Volumes I and II, Mathematics and Its Applications, Springer Netherlands, 2012. zbl 0996.60002(vol.1); zbl 0996.60003(vol.2); MR1887676(vol.2).
• 25. A. Guionnet, Large deviations upper bounds and central limit theorems for non-commutative functionals of Gaussian large random matrices, Ann. Inst. H. Poincaré Probab. Statist. 38 (2002), no. 3, 341-384. DOI 10.1016/S0246-0203(01)01093-7; zbl 0995.60028; MR1899457.
• 26. U. Haagerup, H. Schultz, and S. Thorbj\ornsen, A random matrix approach to the lack of projections in $C^*_{\text{red}}(\mathbb{F}_2)$, Adv. Math. 204 (2006), no. 1, 1-83. DOI 10.1016/j.aim.2005.05.008; zbl 1109.15020; MR2233126; arxiv math/0412545.
• 27. U. Haagerup and S. Thorbj\ornsen, A new application of random matrices: $\text{Ext}(C^*_{\text{red}} (\mathbb{F}_2))$ is not a group, Ann. of Math. (2) 162 (2005), no. 2, 711-775. DOI 10.4007/annals.2005.162.711; zbl 1103.46032; MR2183281; arxiv math/0212265.
• 28. Y. He, A. Knowles, and R. Rosenthal, Isotropic self-consistent equations for mean-field random matrices, Probab. Theory Relat. Fields 171 (2018), no. 1-2, 203-249. DOI 10.1007/s00440-017-0776-y; zbl 1392.15048; MR3800833; arxiv 1611.05364.
• 29. J. W. Helton, T. Mai, and R. Speicher, Applications of realizations (aka linearizations) to free probability, J. Funct. Anal. 274 (2018), no. 1, 1-79. DOI 10.1016/j.jfa.2017.10.003; zbl 1376.81026; MR3718048; arxiv 1511.05330.
• 30. J. W. Helton, R. Rashidi Far, and R. Speicher, Operator-valued semicircular elements: Solving a quadratic matrix equation with positivity constraints, Int. Math. Res. Not. IMRN (2007), no. 22, Art. ID rnm086. DOI 10.1093/imrn/rnm086; zbl 1139.15006; MR2376207; arxiv math/0703510.
• 31. A. M. Khorunzhy and L. A. Pastur, On the eigenvalue distribution of the deformed Wigner ensemble of random matrices, Spectral operator theory and related topics, Adv. Soviet Math., 19, Amer. Math. Soc., Providence, RI, 1994, pp. 97-127. zbl 0813.60036; MR1298444.
• 32. A. Knowles and J. Yin, Anisotropic local laws for random matrices, Probab. Theory Related Fields 169 (2017), no. 1-2, 257-352. DOI 10.1007/s00440-016-0730-4; zbl 1382.15051; MR3704770; arxiv 1410.3516.
• 33. S. G. Krantz and H. R. Parks, The Implicit Function Theorem - History, Theory, and Applications, Birkhäuser Basel, 2002. zbl 1012.58003; MR1894435.
• 34. T. Mai, R. Speicher, and M. Weber, Absence of algebraic relations and of zero divisors under the assumption of full non-microstates free entropy dimension, Adv. Math. 304 (2017), 1080-1107. DOI 10.1016/j.aim.2016.09.018; zbl 1368.46059; MR3558227; arxiv 1502.06357.
• 35. T. Mai, R. Speicher, and S. Yin, The free field: realization via unbounded operators and Atiyah property, 2019. arxiv 1905.08187.
• 36. J. A. Mingo and R. Speicher, Free probability and random matrices, Fields Institute Monographs, Springer New York, 2017. DOI 10.1007/978-1-4939-6942-5; zbl 1387.60005; MR3585560.
• 37. L. Pastur and M. Shcherbina, Eigenvalue distribution of large random matrices, Mathematical Surveys and Monographs, vol. 171, American Mathematical Society, Providence, RI, 2011. zbl 1244.15002; MR2808038.
• 38. V. I. Paulsen, Completely bounded maps and operator algebras, Cambridge Studies in Advanced Mathematics, vol. 78, Cambridge University Press, 2002. zbl 1029.47003; MR1976867.
• 39. D. Shlyakhtenko, Random Gaussian band matrices and freeness with amalgamation, Int. Math. Res. Not. IMRN (1996), no. 20, 1013-1025. DOI 10.1155/S1073792896000633; zbl 0872.15018; MR1422374.
• 40. D. Shlyakhtenko and P. Skoufranis, Freely independent random variables with non-atomic distributions, Trans. Amer. Math. Soc. 367 (2015), no. 9, 6267-6291. DOI 10.1090/S0002-9947-2015-06434-4; zbl 1338.46076; MR3356937; arxiv 1305.1920.
• 41. R. Speicher, Combinatorial theory of the free product with amalgamation and operator-valued free probability theory, Mem. Amer. Math. Soc. 132 (1998), no. 627. DOI 10.1090/memo/0627; zbl 0935.46056; MR1407898.
• 42. M. Takesaki, Theory of operator algebras I, Encyclopaedia of mathematical sciences, no. 124, Springer, 1979. zbl 0990.46034; MR1873025.
• 43. D. Voiculescu, The analogues of entropy and of Fisher's information measure in free probability theory. II, Invent. Math. 118 (1994), no.3, 411-440. DOI 10.1007/BF01231539; zbl 0820.60001; MR1296352.
• 44. D. Voiculescu, Operations on certain non-commutative operator-valued random variables, Astérisque (1995), no. 232, 243-275, Recent advances in operator algebras (Orléans, 1992). zbl 0839.46060; MR1372537.
• 45. D. Voiculescu, The analogues of entropy and of Fisher's information measure in free probability theory. V. Noncommutative Hilbert transforms, Invent. Math. 132 (1998), no. 1, 189-227. DOI 10.1007/s002220050222; zbl 0930.46053; MR1618636.
• 46. D. Voiculescu, The coalgebra of the free difference quotient and free probability, Int. Math. Res. Not. IMRN (2000), no. 2, 79-106. DOI 10.1155/S1073792800000064; zbl 0952.46038; MR1744647.
• 47. D. Voiculescu, Free entropy, Bull. Lond. Math. Soc. 34 (2002), no. 3, 257-278. DOI 10.1112/S0024609301008992; zbl 1036.46051; MR1887698; arxiv math/0103168.
• 48. F. J. Wegner, Disordered system with $n$ orbitals per site: $n =\infty$ limit, Physical Review B 19 (1979). DOI 10.1103/PhysRevB.19.783.

### Affiliation

Alt, Johannes
Section of Mathematics, University of Geneva, 2-4 Rue du Lièvre, CH-1211 Genève 4, Switzerland
Erdős, László
IST Austria, Am Campus 1, A-3400 Klosterneuburg, Austria
Krüger, Torben
Department of Mathematical Sciences, University of Copenhagen, Universitetsparken 5, DK-2100 København, Denmark