Tube-Sector Probabilities Overview

• Tube-sector probabilities are a family of models that quantify the likelihood of tube-like structures occupying defined sectors in diverse fields like heavy-ion collisions, stochastic processes, and quantum systems.
• They employ methods ranging from binomial models for azimuthal occupancy and first-exit probability estimates to algebraic constructions in symmetry-resolved quantum distinguishability.
• Applications include hydrodynamic flow studies, diffusion survival analysis, transmission probabilities in molecular flows, and geometric measures in random tube networks.

Searching arXiv for the supplied topic and closely related usages to ground the article in the cited literature. “Tube-sector probabilities” is not a single standardized construction in the arXiv literature. The expression covers several technically distinct objects: inferred azimuthal sector-occupancy probabilities for localized peripheral tubes in fluctuating heavy-ion initial conditions; sojourn and first-exit probabilities for stochastic trajectories constrained to remain in finite-radius tubes around smooth paths; and canonical sector weights obtained from primitive central idempotents of a boundary tube algebra in generalized-symmetry-resolved quantum distinguishability [(Hama et al., 2012); (Kappler et al., 2020); (He, 18 Jun 2026)]. Related literatures use closely allied notions—transmission probabilities through tube-like vacuum components, overlap-dependent bond probabilities in random tube networks, spherical tube-volume probabilities for Gaussian random fields, and high-probability estimation-error tubes—while some papers use “tube” and “sector” in a wholly deterministic sense and explicitly do not introduce any probability theory [(Krause et al., 2018); (Szczygieł et al., 2015); (Kuriki et al., 2021); (Baranowski, 28 Jun 2026); (Destrade et al., 2013)].

1. Terminological scope and principal meanings

The arXiv record shows that “tube-sector probabilities” is best understood as a family resemblance term rather than a single formalism. In some papers, the “tube” is a geometric neighborhood around a path in configuration space; in some, it is a localized hydrodynamic hot spot distributed randomly in azimuth; in some, it is an algebraic object whose central idempotents define measurement sectors; and in some, it is merely part of a deterministic geometric description. This suggests that the phrase names different probabilistic constructions in different subfields rather than a universally accepted definition.

Domain	Tube object	Probability meaning
Heavy-ion hydrodynamics	Peripheral tube on a ring	Inferred azimuthal sector occupancy
Stochastic processes	Finite-radius moving tube around a path	Sojourn or first-exit probability
Generalized symmetries	Boundary tube algebra sector	Canonical readout probability
Molecular flow	Cylindrical tube or bellow	Transmission probability
Random tube networks	Tube overlap interval	Local bond probability
Geometric probability	Spherical tube around an index set	Tail probability via tube volume
Finite elasticity	Open sector closed into a tube	Not probabilistic

Two boundary cases are especially important. First, in the peripheral-tube model for ridge correlations, the paper does not develop a formal combinatorial theory of sector occupancy, but it does specify a stochastic construction from which such probabilities can be inferred (Hama et al., 2012). Second, in the finite-elasticity study of closing an annular sector into an intact tube, “tube-sector” language is entirely geometric and deterministic, and the paper explicitly does not define any probability notion (Destrade et al., 2013).

2. Azimuthal sector occupancy in the peripheral-tube model

In the heavy-ion setting, a tube is a localized high-energy-density bump superimposed on an otherwise smooth, isotropic background in the transverse plane. The multi-tube initial condition is

$$\epsilon=12\exp[-0.0004\, r^5]+\sum_{i=1}^n\frac{34}{0.845\pi} \exp\!\left[-\frac{|{\bf r}-{\bf r}_i|^2}{0.845}\right],$$

with $n=1,2,3,4$, and every tube center constrained by

$|{\bf r}_i|=r_0=5.4\ \mathrm{fm}.$

Thus the simplified stochastic geometry fixes tube multiplicity $n$, radius $r_0$, tube strength, and tube width, while randomizing only the azimuthal angles of the tube centers; the authors report that they computed “only 50 random events in each case” (Hama et al., 2012).

The paper states that the tubes are “distributed randomly in azimuth at a constant distance $r_0=5.4$ fm from the axis” and that “their azimuths are chosen randomly.” It does not write an explicit density $P(\phi)$, a joint law for the $n$ azimuths, or a multiplicity distribution $P(n)$. The natural interpretation, explicitly described in the technical summary as an inference rather than a theorem, is uniform sampling on $[0,2\pi)$, with independent and identically distributed azimuths for fixed $n=1,2,3,4$0. Under that interpretation, the probability that one tube lies in an azimuthal sector $n=1,2,3,4$1 of width $n=1,2,3,4$2 is

$n=1,2,3,4$3

the sector occupancy $n=1,2,3,4$4 is binomial,

$n=1,2,3,4$5

and

$n=1,2,3,4$6

These formulas are not written in the paper itself; they are the direct probabilistic reconstruction of the stated event construction if one adopts the implied uniform independent azimuthal sampling.

The dynamical significance of this reconstruction is constrained by the hydrodynamic observables. The paper studies $n=1,2,3,4$7, the Fourier harmonics $n=1,2,3,4$8, the event-plane differences $n=1,2,3,4$9 and $|{\bf r}_i|=r_0=5.4\ \mathrm{fm}.$0, and the two-particle correlation in $|{\bf r}_i|=r_0=5.4\ \mathrm{fm}.$1. For multi-tube events, the harmonic coefficients become broadly distributed and the symmetry-angle differences are “consistent with no-correlation,” but the two-particle correlation is “almost independent of the number of peripheral tubes.” The authors interpret this to mean that ridge production is local: each peripheral tube induces essentially the same characteristic flow deflection pattern, and random azimuthal averaging cancels interference from the global arrangement. In that sense, sector occupancy affects event-by-event harmonic content strongly, but affects the ensemble-averaged two-particle correlation only weakly (Hama et al., 2012).

3. Sojourn probabilities in finite-radius tubes around stochastic paths

In stochastic-process theory, the most literal use of a tube probability is the probability that a diffusion trajectory remains inside a moving neighborhood of a smooth reference path. For overdamped Langevin dynamics,

$|{\bf r}_i|=r_0=5.4\ \mathrm{fm}.$2

a smooth path $|{\bf r}_i|=r_0=5.4\ \mathrm{fm}.$3 and radius $|{\bf r}_i|=r_0=5.4\ \mathrm{fm}.$4 define the instantaneous tube

$|{\bf r}_i|=r_0=5.4\ \mathrm{fm}.$5

the tube event

$|{\bf r}_i|=r_0=5.4\ \mathrm{fm}.$6

and the sojourn probability

$|{\bf r}_i|=r_0=5.4\ \mathrm{fm}.$7

The associated measurable quantity is the first-exit hazard

$|{\bf r}_i|=r_0=5.4\ \mathrm{fm}.$8

For small tube radius,

$|{\bf r}_i|=r_0=5.4\ \mathrm{fm}.$9

so the Onsager–Machlup Lagrangian appears as the $n$0 path-dependent correction to a measurable exit rate rather than as an abstract path density (Kappler et al., 2020).

This construction was generalized to $n$1-dimensional Itô processes with state-dependent full-rank diffusion tensor,

$n$2

using Euclidean tubes of radius $n$3. The corresponding exit-rate expansion is

$n$4

A central difference from the additive-noise case is that for generic pairs of paths, the vanishing-radius ratio of sojourn probabilities diverges because the leading $n$5 term is path dependent when diffusivity is state dependent. By contrast, for a path and its time reversal the leading term cancels and the limit is finite, which the paper identifies as a pathwise irreversibility measure. The same paper emphasizes that Euclidean tubes are operationally natural because they correspond to neighborhoods one can define directly from data without first transforming to a diffusivity-adapted metric (Kappler et al., 2020).

Taken together, these papers make “tube probability” a measurable survival statistic for tubular ensembles of trajectories. The “sector” language is absent here, but a plausible implication is that any sector-like subdivision of configuration space would have to be subordinate to the first-exit framework rather than replacing it.

4. Tube sectors from boundary tube algebras in generalized-symmetry-resolved distinguishability

In a generalized-symmetry setting, “tube-sector probabilities” acquires an algebraic and operational meaning. Given global states $n$6, one reduces to subsystem $n$7,

$n$8

and forms the positive-overlap operator

$n$9

For a fusion category $r_0$0 and entangling-cut boundary module $r_0$1, the physically admissible commutative readout algebra is

$r_0$2

If $r_0$3 are the primitive central idempotents of this center and $r_0$4, then the tube-sector probabilities are

$r_0$5

These probabilities are not heuristic. They are defined as the classical output statistics of the tube POVM determined by the center of the boundary tube algebra (He, 18 Jun 2026).

The operational content comes from the admissible-instrument structure. The allowed measurements satisfy complete positivity, entanglement-cut locality, boundary-module covariance, and sequential stability. Under these constraints, Theorem 1 identifies $r_0$6 as the maximal physically admissible commutative readout algebra and shows that every admissible readout factors through the tube-center readout. In this framework, standard scalar overlap retains only $r_0$7, coarse symmetry resolution keeps only

$r_0$8

while the full tube-sector distribution retains the conditional structure inside each coarse fiber,

$r_0$9

The paper therefore treats tube-sector probabilities as a strict refinement of fidelity-based and symmetry-resolved diagnostics whenever a coarse sector contains more than one tube sector (He, 18 Jun 2026).

The doubled-Ising product-KW example shows this refinement explicitly. The trivial $r_0=5.4$0-character fiber splits as

$r_0=5.4$1

with lifted tube projectors $r_0=5.4$2 and $r_0=5.4$3. A coherent endpoint gives

$r_0=5.4$4

whereas an unread endpoint sign with probability $r_0=5.4$5 gives

$r_0=5.4$6

Scalar and coarse $r_0=5.4$7-resolved data can coincide in such cases, while the tube-sector distribution changes. In that sense, the “sector” is neither angular nor spatial; it is a primitive central block of the boundary tube algebra (He, 18 Jun 2026).

5. Transmission, occupancy, incidence, and spherical tube volume

In molecular-flow vacuum transport, the relevant probability is a through-probability. For a cylindrical tube or edge-welded bellow, the transmission probability is

$r_0=5.4$8

where $r_0=5.4$9 is the number of particles desorbed from the inlet facet and $P(\phi)$0 the number absorbed at the outlet facet. For a cylindrical tube of length $P(\phi)$1 and diameter $P(\phi)$2, the paper quotes the approximation

$P(\phi)$3

For mixed tube+bellow geometries, the transmission probability varies linearly with the bellow fraction $P(\phi)$4, and the paper derives an equivalent straight-tube length $P(\phi)$5 that preserves $P(\phi)$6. The long-component asymptotics are

$P(\phi)$7

with asymptotic ratio

$P(\phi)$8

Here “sector” is not angular, but the paper explicitly speaks of bellow-containing sectors and supplies a direct transport probability for them (Krause et al., 2018).

Geometric measure theory provides a different occupancy notion. A pair $P(\phi)$9 is called admissible if there exists a compact $n$0 with

$n$1

and

$n$2

for every tube $n$3 of width $n$4. The main theorem establishes that all pairs $n$5 are admissible for $n$6. If one normalizes $n$7 to a probability measure, this becomes a uniform tube-occupancy probability bound

$n$8

The paper is deterministic, but it immediately yields a probabilistic reading after normalization (Orponen, 2013).

For Gaussian random fields, the classical volume-of-tube method computes the probability content of spherical tube neighborhoods around an index manifold $n$9. In the inhomogeneous-variance extension, a field

$P(n)$0

leads to the spherical tube

$P(n)$1

and

$P(n)$2

The generalized critical radius controls self-overlap and the asymptotic tube-method error. This is a probability of a variable-width spherical tube or sector-like region on the sphere, not a discrete sector-occupancy law (Kuriki et al., 2021).

A discrete incidence analogue appears for $P(n)$3-tubes in $P(n)$4. Under a spacing condition—at most $P(n)$5 tubes in each $P(n)$6-tube and $P(n)$7-separated directions—the number of $P(n)$8-rich $P(n)$9-balls satisfies

$[0,2\pi)$0

The paper does not define probabilities, but normalized counts yield immediate sector-occupancy analogues for evenly spaced tube families (Fu et al., 2021).

6. Percolation, tube formation, and tube-constrained passage

In a discrete–continuous percolation model of parallel random tubes, geometry and connectivity are generated by two independent Poisson processes: one cuts parallel lines into tube segments, and the other generates bonds between overlapping adjacent tubes. If the overlap length is $[0,2\pi)$1, then the number of open bonds is Poisson with mean $[0,2\pi)$2,

$[0,2\pi)$3

so the probability that the two tubes are connected is

$[0,2\pi)$4

This is the clearest local “tube-sector” law in the overlap-based sense: a local overlap interval of length $[0,2\pi)$5 carries a direct connection probability. In two dimensions, duality yields $[0,2\pi)$6; in three dimensions, the reported threshold estimate is

$[0,2\pi)$7

The paper’s algorithmic contribution is an extension of Newman–Ziff to these inhomogeneous overlap-dependent bond probabilities (Szczygieł et al., 2015).

A higher-dimensional notion of tube formation appears in tube percolation by random $[0,2\pi)$8-plaquettes. The core event is that a $[0,2\pi)$9-sphere can escape to infinity through an open $n=1,2,3,4$00-tube. For width $n=1,2,3,4$01 and scale $n=1,2,3,4$02, the tubular one-arm probability is

$n=1,2,3,4$03

and the paper proves a sharp threshold: if $n=1,2,3,4$04, then

$n=1,2,3,4$05

while if $n=1,2,3,4$06, then

$n=1,2,3,4$07

For the box-crossing event $n=1,2,3,4$08, the crossing probability tends to $n=1,2,3,4$09 below criticality and to $n=1,2,3,4$10 above criticality for sufficiently large $n=1,2,3,4$11. The paper explicitly stresses that tube connectedness is not transitive, so there is no natural cluster notion analogous to ordinary bond percolation (Kanazawa et al., 14 May 2026).

Tube-constrained first-passage percolation supplies yet another probability of remaining in a tube. For $n=1,2,3,4$12, the minimal passage time between opposite sides of the periodic strip, the paper proves that for any $n=1,2,3,4$13 there exists $n=1,2,3,4$14 such that, with probability at least

$n=1,2,3,4$15

every geodesic for $n=1,2,3,4$16 lies in

$n=1,2,3,4$17

On this confinement event, $n=1,2,3,4$18 is exactly a minimum of cylinder passage times, which leads to fluctuations of at least order

$n=1,2,3,4$19

The paper explicitly states that it does not discuss sectors; its direct geometric object is a strip or cylinder rather than a wedge (Damron et al., 2022).

7. High-probability estimation tubes and deterministic nonprobabilistic limits

In continuous-time state estimation, deterministic ISS bounds can be reinterpreted as high-probability tube guarantees. If the estimation error satisfies

$n=1,2,3,4$20

and the aggregated disturbance satisfies

$n=1,2,3,4$21

then immediately

$n=1,2,3,4$22

With a quadratic Lyapunov function $n=1,2,3,4$23 satisfying

$n=1,2,3,4$24

the resulting tube is ellipsoidal in $n=1,2,3,4$25-coordinates and norm-ball-like after spectral bounds. The paper specializes this construction to positive and cooperative systems, but the controlled sets remain ISS tubes rather than sector decompositions (Baranowski, 28 Jun 2026).

By contrast, the finite-elasticity paper on deforming an open annular sector into an intact circular cylindrical tube contains no probability theory at all. Its central closure equation is

$n=1,2,3,4$26

and the main results are deterministic existence, uniqueness, and stability statements under strict convexity and the thickness restriction

$n=1,2,3,4$27

Stability is then analyzed by incremental deformation theory in Stroh form. This paper is important because it marks the limit of the phrase: “tube” and “sector” may coexist in a title and problem statement without generating any notion of probability whatsoever (Destrade et al., 2013).

Taken together, these usages suggest that “tube-sector probabilities” is an umbrella expression for several mathematically non-equivalent constructions. In heavy-ion hydrodynamics it denotes inferred occupancy of azimuthal sectors by randomly placed peripheral tubes; in stochastic analysis it denotes measurable survival or first-exit probabilities for path tubes; in generalized-symmetry quantum theory it denotes canonical probabilities of central tube-algebra sectors; and in several neighboring literatures it denotes transmission, occupancy, connection, or crossing probabilities for tube-like geometries. Where the phrase is used outside these probabilistic settings, the literature itself is explicit that the underlying problem is deterministic rather than stochastic.