Different geometric routes can carry the same maximum flow through a holographic bottleneck, according to a new analytic preprint. The result is a precise example of max-flow nonuniqueness: one construction selects paths from endpoint-resolved entanglement data, while an independent construction follows geodesics normal to the bottleneck. In finite-cutoff Poincare AdS3, the two flows have the same bottleneck flux but different boundary densities, landing maps and pairings.
The study examines analytic geometries. It asks how exact endpoint-resolved PEE kernels generate large-scale bit-thread flows in planar BTZ and finite-cutoff AdS3, and how those flows compare with independent normal-geodesic representatives. In the paper's terminology, bit threads are the bulk flow lines generated by those endpoint data.
Turning endpoint data into a bulk flow
The authors begin with exact formulas for geodesic distance and differentiate them to define currents associated with individual endpoint pairs. They then superpose those source-resolved currents to obtain macroscopic fields that are divergenceless, meaning the flow has no net source or sink in the region being studied. The resulting fields are tested against the conditions for a max flow: they obey a pointwise norm bound, meet the RT bottleneck normally, and saturate it there.
The calculations cover asymptotic planar BTZ and finite-cutoff Poincare AdS3, global AdS3 and planar BTZ. A finite cutoff is introduced in the cutoff geometries. The work is an analytic set of exact geometric constructions within the modeled settings, rather than a statistical analysis.
A horizon sector tied to the two-sided state
In the asymptotic BTZ construction, the one-sided boundary-horizon measure is fixed by the Jacobian of the endpoint map that relates the two sides of the thermofield-double state. The relative factor of two in that measure comes from this geometric push-forward, rather than from a choice of orientation. The result ties the horizon contribution to the two-sided map used in the stated purification.
The physical BTZ endpoint chambers can be superposed into a divergenceless max flow that is normal to and saturated on the RT surface. In the selected flow, streamlines that end on the horizon cross an RT segment of length 2b/zh. The boundary and horizon contour densities also agree pointwise with the corresponding normal fluxes, providing a local check on how the flow is divided between its endpoints.
What changes when the boundary is brought inward
The finite-cutoff Poincare calculation changes the short-distance behavior of the endpoint weighting smoothly. Its two-point kernel suppresses arbitrarily short chords, goes to zero linearly when the endpoints coincide, and approaches the asymptotic conformal-field-theory density at large separation. The construction therefore does not impose a hard minimum chord length.
Those finite-cutoff PEE currents still produce a field that obeys the norm bound, meets the RT curve normally and saturates the bottleneck. But the individual streamlines need not be geodesics: away from the central case, the PEE-selected paths have nonzero geodesic curvature. The separate normal-geodesic flow reaches the same bottleneck flux while changing the boundary density, the map of where threads land and the boundary-to-boundary pairing.
The cutoff redistributes, but does not erase, the capacity
For finite-cutoff BTZ, the mixed cutoff-horizon kernel is exactly the push-forward of the two-sided cutoff-boundary measure under the stated thermofield-double purification. Again, the factor of two is supplied by the endpoint-map Jacobian. At fixed horizon scale zh, the integrated horizon contribution remains 2b/(4Gzh), independent of the cutoff position, while the explicit dependence on the cutoff lies in the cutoff-boundary sector.
The two endpoint sectors also satisfy an exact local capacity identity. In practical terms, changing the temperature reallocates a fixed local amount of PEE capacity between cutoff-cutoff and cutoff-horizon contributions. If the cutoff is moved toward the horizon while zh is held fixed, the cutoff-cutoff measure vanishes and the mixed kernel converges, in the distributional sense, to δ(r)/(4Gzh). Holding the physical cutoff temperature fixed is a different limiting procedure because zh then varies.
This split is not a universal property of the bottleneck itself. At finite cutoff, the PEE-selected BTZ flow and the normal-geodesic flow can assign different portions of the common RT flux to the horizon and boundary sectors. The total flux is fixed by maximality, but the endpoint decomposition depends on which max-flow representative is chosen.
A geometric result with a deliberately narrow reach
These are selected max-flow representatives built from a particular endpoint prescription, not a claim that the endpoint split is unique. Both the asymptotic and finite-cutoff horizon measures are conditional on the stated thermofield-double purification. At finite cutoff, the endpoint decomposition changes when the representative changes even though the total flux does not.
The conclusions are tied to the analytic settings actually constructed: asymptotic planar BTZ and finite-cutoff Poincare, global and planar BTZ geometries. The finite-cutoff results are exact statements about those modeled cases and their stated cutoff assumptions.
The manuscript is arXiv:2608.25782v1, dated 26 August 2026. The authors report support from NSFC Grant No. 12447108 and the Shing-Tung Yau Center of Southeast University.
Paper data and sources
Original title: PEE threads and bit-threads in BTZ black brane and finite-cutoff AdS$_3$
Authors: Debarshi Basu, Ashish Chandra, Qiang Wen
Journal/Repository: arXiv
Status: Preprint, not yet peer-reviewed
First online: 2026-08-26
DOI: Not available
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