Exact (symbolic, no sampling) Weingarten calculus for networks of Haar-random SO(m) matrices, via two structurally independent computation engines (a "cut and glue" symbolic surgery engine, and a "direct index sum" engine using union-find and bucket elimination). This is the reusable computational core extracted from a research project studying the Weingarten calculus underlying Nissim's SO(3) deconfinement mechanism and its generalizations (see the related technical notes below). It computes exact expectation values of products of traces of Haar-random SO(m) matrices whenever every edge in the network occurs exactly 2, 4, or m times -- the three cases with a known closed-form Weingarten identity. Implemented and tested for SO(m), m >= 3 (numerically exercised for m = 3, 5, 7); SU(N) is not implemented. Ships with a known-answer test suite (classical Haar moments, two real regression cases from development, and a cross-check that both engines agree). MIT licensed. Deliberately does not include the tube-specific geometry that originally used this engine -- this is the generic calculus layer, independent of any one application. See the included README.md for full documentation, scope, and a worked example.
Martín Baca· Zenodo (CERN European Organi...· 0 citations
Exact (symbolic, no sampling) Weingarten calculus for networks of Haar-random SO(m) matrices, via two structurally independent computation engines (a "cut and glue" symbolic surgery engine, and a "direct index sum" engine using union-find and bucket elimination). This is the reusable computational core extracted from a research project studying the Weingarten calculus underlying Nissim's SO(3) deconfinement mechanism and its generalizations (see the related technical notes below). It computes exact expectation values of products of traces of Haar-random SO(m) matrices whenever every edge in the network occurs exactly 2, 4, or m times -- the three cases with a known closed-form Weingarten identity. Implemented and tested for SO(m), m >= 3 (numerically exercised for m = 3, 5, 7); SU(N) is not implemented. Ships with a known-answer test suite (classical Haar moments, two real regression cases from development, and a cross-check that both engines agree). MIT licensed. Deliberately does not include the tube-specific geometry that originally used this engine -- this is the generic calculus layer, independent of any one application. See the included README.md for full documentation, scope, and a worked example.
Martín Baca· Zenodo (CERN European Organi...· 0 citations
Nissim (arXiv:2605.16162) proved that SO(3) lattice Yang-Mills theory at strong coupling fails Wilson's confinement criterion, via an explicit "tube" cluster in the cluster expansion whose leading contribution is computed exactly using third-moment identities for Haar(SO(3)). A natural question, raised explicitly in Nissim's paper (Section 5), is whether the same mechanism extends to SO(2N+1) for N>1, which would be a step toward deconfinement results for general odd orthogonal gauge groups. We record two results toward this question. First, we give an independent derivation of the exact m-th moment formula for Haar(SO(m)) (a corollary of Lehrer-Zhang's invariant theory computation) in the form needed here. Second, we prove that Nissim's exact tube construction, transported verbatim to SO(2N+1), gives an identically zero contribution for every N>1 — not a numerically small one — because every loop edge in that construction only reaches total multiplicity 3, which is below the threshold m=2N+1 required for a nonvanishing odd moment. This yields an explicit necessary condition any candidate replacement construction must satisfy. We also report a negative computational finding obtained while testing the simplest such replacement, which we record to save a future attempt the same work: a "boosted-degree" variant of the tube for SO(7) that reaches the correct total multiplicity does not show a statistically significant nonzero signal up to N=1.5×10^7 Monte Carlo samples. The extension of the deconfinement mechanism to SO(2N+1), N>1, remains open. (A separate, unrelated discrepancy we encountered while reconstructing Nissim's own SO(3) tube combinatorially is reported in a companion note, https://doi.org/10.5281/zenodo.22778193.)
Martín Baca· Zenodo (CERN European Organi...· 0 citations
Nissim (arXiv:2605.16162) proved that SO(3) lattice Yang-Mills theory at strong coupling fails Wilson's confinement criterion, via an explicit "tube" cluster in the cluster expansion whose leading contribution is computed exactly using third-moment identities for Haar(SO(3)). A natural question, raised explicitly in Nissim's paper (Section 5), is whether the same mechanism extends to SO(2N+1) for N>1, which would be a step toward deconfinement results for general odd orthogonal gauge groups. We record two results toward this question. First, we give an independent derivation of the exact m-th moment formula for Haar(SO(m)) (a corollary of Lehrer-Zhang's invariant theory computation) in the form needed here. Second, we prove that Nissim's exact tube construction, transported verbatim to SO(2N+1), gives an identically zero contribution for every N>1 — not a numerically small one — because every loop edge in that construction only reaches total multiplicity 3, which is below the threshold m=2N+1 required for a nonvanishing odd moment. This yields an explicit necessary condition any candidate replacement construction must satisfy. We also report a negative computational finding obtained while testing the simplest such replacement, which we record to save a future attempt the same work: a "boosted-degree" variant of the tube for SO(7) that reaches the correct total multiplicity does not show a statistically significant nonzero signal up to N=1.5×10^7 Monte Carlo samples. The extension of the deconfinement mechanism to SO(2N+1), N>1, remains open. (A separate, unrelated discrepancy we encountered while reconstructing Nissim's own SO(3) tube combinatorially is reported in a companion note, https://doi.org/10.5281/zenodo.22778193.)
Martín Baca· Zenodo (CERN European Organi...· 0 citations
We use cookies to run the site and, with your consent, for analytics and to show ads.
See our Cookie Policy.
