Holography on Biregular Trees

• The paper introduces a discrete holographic framework on semihomogeneous biregular trees, demonstrating vertex-dependent modifications in bulk propagators and boundary correlators.
• It employs a lattice scalar field approach and transfer-matrix techniques to reveal a two-step radial structure that alters the mass–dimension relation and near-boundary behavior.
• The work connects arithmetic realizations from Bruhat–Tits building theory with quantum graph methods, uncovering novel zeta function signatures in three-point functions.

to=arxiv_search 尚度json_code {"query":"Holography on biregular trees biregular tree holography Bruhat-Tits building", "max_results": 10} Holography on biregular trees is a discrete holographic framework in which the bulk geometry is a $(q_{+}+1,q_{-}+1)$-biregular tree $T_{q_{+},q_{-}}$, a bipartite tree with alternating vertex valences, and the boundary is the visual boundary $\partial T_{q_{+},q_{-}}$, equipped with an ultrametric defined from the Gromov product. In the formulation developed for scalar fields, the semihomogeneous nature of the tree—two vertex orbits rather than one—modifies both bulk propagators and boundary correlators relative to the regular Bruhat–Tits tree of $p$-adic AdS/CFT. The canonical example is the tree of type $(p^{3}+1,p+1)$, realized as the $1$-skeleton of the Bruhat–Tits building of a projective unitary group over an unramified quadratic extension, and interpreted as a discrete symmetric space associated with $SU(3)$-type $p$-adic geometry (Mondal et al., 28 Jul 2025). This setting combines features of negatively curved tree geometries with structures inherited from higher-rank buildings and their Euclidean apartments, thereby interpolating, at the level of discrete models, between familiar $p$-adic AdS/CFT on regular trees and more intricate non-Archimedean symmetric spaces (Marcolli, 2018, Mondal et al., 28 Jul 2025).

1. Biregular trees as discrete bulk geometries

A biregular tree is a locally finite bipartite tree with vertex set decomposed into two classes $V_{0}\sqcup V_{1}$, such that vertices in one class have one degree and vertices in the other class have another degree. In the standard notation, a $T_{q_{+},q_{-}}$0-biregular tree $T_{q_{+},q_{-}}$1 is a tree in which every “+” vertex has degree $T_{q_{+},q_{-}}$2, every “−” vertex has degree $T_{q_{+},q_{-}}$3, and adjacent vertices always have different degree (Thomas et al., 2010, Mondal et al., 28 Jul 2025). When $T_{q_{+},q_{-}}$4, the tree reduces to the usual $T_{q_{+},q_{-}}$5-regular Bruhat–Tits tree, and the semihomogeneous features disappear (Mondal et al., 28 Jul 2025).

The geometry is hyperbolic in the graph-theoretic sense. Infinite locally finite trees are Gromov hyperbolic, and their boundary at infinity is compact and totally disconnected; for locally finite trees of sufficiently large valence it is homeomorphic to a Cantor set (Thomas et al., 2010). In the biregular holographic construction, one fixes a base vertex $T_{q_{+},q_{-}}$6 and defines the Gromov product

$T_{q_{+},q_{-}}$7

where $T_{q_{+},q_{-}}$8 is graph distance. For boundary points $T_{q_{+},q_{-}}$9, the visual metric is defined by

$\partial T_{q_{+},q_{-}}$0

and is ultrametric (Mondal et al., 28 Jul 2025). This choice makes the boundary a totally disconnected ultrametric compact space, analogous to $\partial T_{q_{+},q_{-}}$1 in rank-one $\partial T_{q_{+},q_{-}}$2-adic holography, but now associated with a semihomogeneous bulk (Marcolli, 2018, Mondal et al., 28 Jul 2025).

The principal geometric novelty is the failure of vertex transitivity. The full automorphism group of a biregular tree has two vertex orbits, one for each homogeneity degree, and preserves parity along geodesics: even translations are allowed, odd ones are not (Mondal et al., 28 Jul 2025). This immediately implies that bulk observables can depend not only on distance but also on vertex type and parity, a feature absent in the homogeneous Bruhat–Tits tree of $\partial T_{q_{+},q_{-}}$3 (Marcolli, 2018, Mondal et al., 28 Jul 2025).

2. Arithmetic and building-theoretic origin

The most important arithmetic realization arises from unitary groups over an unramified quadratic extension. In the regular $\partial T_{q_{+},q_{-}}$4-adic AdS/CFT setting, the bulk is the Bruhat–Tits tree $\partial T_{q_{+},q_{-}}$5 of $\partial T_{q_{+},q_{-}}$6, whose vertices are homothety classes of $\partial T_{q_{+},q_{-}}$7-lattices in $\partial T_{q_{+},q_{-}}$8, whose edges encode one-step lattice refinements, and whose boundary is identified with $\partial T_{q_{+},q_{-}}$9 (Marcolli, 2018). Every vertex has valence $p$0, where $p$1 is the cardinality of the residue field $p$2, so the tree is homogeneous rather than biregular (Marcolli, 2018).

The biregular case described in the holographic model is instead tied to $p$3 and a projective unitary subgroup preserving a Hermitian form over the unramified quadratic extension $p$4. The corresponding Bruhat–Tits building of $p$5 is a $p$6-dimensional simplicial complex, while the building of the unitary subgroup appears as a subcomplex whose $p$7-skeleton is a biregular tree of type $p$8 (Mondal et al., 28 Jul 2025). Vertices fixed by the relevant involution are hyperspecial and have degree $p$9, whereas the remaining special vertices have degree $(p^{3}+1,p+1)$0 (Mondal et al., 28 Jul 2025).

This building-theoretic origin is significant for two reasons. First, it places biregular-tree holography within the general non-Archimedean dictionary relating arithmetic groups, buildings, and boundary projective geometries (Marcolli, 2018). Second, it explains why the model exhibits “features of both flat space and negatively curved space”: the biregular tree is a $(p^{3}+1,p+1)$1-dimensional slice of a higher-rank building whose apartments are Euclidean tilings of $(p^{3}+1,p+1)$2, while the tree itself still has exponential volume growth and a hyperbolic boundary structure (Mondal et al., 28 Jul 2025).

This suggests a broader hierarchy. Regular Bruhat–Tits trees provide rank-one bulk geometries for $(p^{3}+1,p+1)$3-adic AdS/CFT, higher-dimensional Bruhat–Tits buildings provide higher-rank analogues, and biregular trees occupy an intermediate position in which the bulk remains one-dimensional as a graph but inherits higher-rank arithmetic structure through its embedding in the building (Marcolli, 2018, Mondal et al., 28 Jul 2025).

3. Bulk scalar field theory and mass–dimension relation

The scalar bulk theory on a biregular tree is formulated as a nearest-neighbor lattice field theory on vertices. The action is

$(p^{3}+1,p+1)$4

with $(p^{3}+1,p+1)$5 over undirected edges (Mondal et al., 28 Jul 2025). The discrete Laplacian at $(p^{3}+1,p+1)$6 is

$(p^{3}+1,p+1)$7

so the equation of motion is

$(p^{3}+1,p+1)$8

Because the number of neighbors depends on the homogeneity degree $(p^{3}+1,p+1)$9, the operator is semihomogeneous rather than homogeneous (Mondal et al., 28 Jul 2025).

For the bulk-to-bulk Green’s function $1$0, spherical symmetry reduces the problem to a recurrence in the graph distance $1$1. However, since the coefficient $1$2 alternates with parity, the recursion splits into coupled even and odd sectors. Eliminating one sector yields a step-$1$3 constant-coefficient recurrence, whose decaying solution has the form

$1$4

rather than a pure exponential in $1$5 as on a regular tree (Mondal et al., 28 Jul 2025).

The resulting mass–dimension relation is

$1$6

with two roots $1$7 satisfying $1$8 (Mondal et al., 28 Jul 2025). The physical branch is $1$9, and the theory obeys a Breitenlohner–Freedman–type bound on $SU(3)$0 (Mondal et al., 28 Jul 2025). In the regular limit $SU(3)$1, this reduces to the familiar tree relation

$SU(3)$2

showing that biregularity modifies the mass spectrum by coupling the two vertex types (Mondal et al., 28 Jul 2025).

A related spectral perspective comes from quantum graph theory on biregular trees. For a metric biregular tree with Kirchhoff conditions, the quantum-graph Laplacian is controlled by transfer matrices and multipliers $SU(3)$3, whose modulus distinguishes spectral bands from decaying radial solutions (Carlson, 2023). This is not the same model as the discrete scalar theory on vertices, but it reinforces the same structural point: biregular trees require a two-step radial analysis, and their spectral theory is encoded by alternating local data rather than a single homogeneous branching number (Carlson, 2023).

4. Propagators and semihomogeneous radial structure

The bulk-to-bulk propagator on a biregular tree takes the form

$SU(3)$4

where

$SU(3)$5

Equivalently, an unnormalized propagator is

$SU(3)$6

and the relation

$SU(3)$7

encodes the alternation of vertex types (Mondal et al., 28 Jul 2025). The dependence on $SU(3)$8 and on the parity of $SU(3)$9 is the defining departure from the regular-tree propagator, which depends only on distance (Mondal et al., 28 Jul 2025).

The bulk-to-boundary propagator is built from a horospherical index

$p$0

where $p$1 is the unique bulk vertex common to the geodesics from $p$2, $p$3, and $p$4 (Mondal et al., 28 Jul 2025). The solution of the bulk field equation with a boundary insertion at $p$5 is

$p$6

again with an explicit type-dependent prefactor (Mondal et al., 28 Jul 2025). In the regular-tree limit the $p$7-ratio becomes $p$8, recovering the standard $p$9-adic bulk-to-boundary kernel (Mondal et al., 28 Jul 2025).

A number of multiplicative identities survive from the homogeneous theory, but now with care required for the vertex-type factors. For $p$0 on the geodesic segment $p$1,

$p$2

and if $p$3 lies on the geodesic from $p$4 to $p$5,

$p$6

These splitting identities are the fundamental combinatorial tools for evaluating contact and exchange diagrams (Mondal et al., 28 Jul 2025).

The same radial-multiplicative viewpoint appears in the spectral geometry of biregular quantum graphs, where the resolvent kernel on the tree is expressed through distinguished solutions $p$7, a Wronskian $p$8, and multipliers $p$9 controlling growth or decay along rays (Carlson, 2023). This suggests a broader common mechanism: in both discrete scalar theory and quantum-graph formulations, radial propagation on a biregular tree is governed by alternating local coefficients whose two-step transfer structure replaces the simpler one-step analysis of regular trees.

5. Boundary geometry, conformal data, and two-point correlators

The boundary theory is defined on the ultrametric compact space $V_{0}\sqcup V_{1}$0. Bulk solutions sourced by boundary data $V_{0}\sqcup V_{1}$1 are written using a Patterson–Sullivan measure $V_{0}\sqcup V_{1}$2 as

$V_{0}\sqcup V_{1}$3

with normalization chosen so that the leading near-boundary behavior matches the source (Mondal et al., 28 Jul 2025). The measure itself depends on the parity of the shell relative to $V_{0}\sqcup V_{1}$4, reflecting semihomogeneity (Mondal et al., 28 Jul 2025).

Near the boundary, the scalar field exhibits the analogue of the usual AdS asymptotics. For appropriate sequences of vertices approaching a boundary point, one identifies $V_{0}\sqcup V_{1}$5 with an ultrametric boundary scale $V_{0}\sqcup V_{1}$6, and the dominant falloff behaves as $V_{0}\sqcup V_{1}$7 (Mondal et al., 28 Jul 2025). This suggests the same boundary scaling dimension $V_{0}\sqcup V_{1}$8 that already appeared in the bulk mass–dimension relation.

The two-point correlator of the boundary operator $V_{0}\sqcup V_{1}$9 has the standard conformal-distance dependence

$T_{q_{+},q_{-}}$00

up to an overall normalization that the construction does not fully fix without a fuller harmonic analysis on $T_{q_{+},q_{-}}$01 (Mondal et al., 28 Jul 2025). The key point is that semihomogeneity does not alter the ultrametric power law itself; rather, it complicates the normalization and becomes visible more sharply in higher-point functions (Mondal et al., 28 Jul 2025).

The general background for boundary holography on $T_{q_{+},q_{-}}$02-adic trees is provided by the regular Bruhat–Tits case, where $T_{q_{+},q_{-}}$03, and classical or quantum holographic codes map bulk data on the tree to boundary data on the projective line (Marcolli, 2018). In that setting the local geometry is regular, and the code architecture only introduces effective asymmetry through distinguished root and incoming directions (Marcolli, 2018). By contrast, biregular-tree holography incorporates asymmetry directly into the bulk geometry itself (Mondal et al., 28 Jul 2025).

6. Three-point functions, discrete tensor structure, and zeta factors

The most distinctive result of holography on biregular trees is the form of the three-point correlator. Given three distinct boundary points $T_{q_{+},q_{-}}$04, the three geodesics joining them meet at a unique bulk vertex

$T_{q_{+},q_{-}}$05

On a regular tree or in continuum AdS, this bulk point carries no invariant extra datum beyond its role in the geodesic tripod. On a biregular tree, however, the vertex type $T_{q_{+},q_{-}}$06 is invariant under the isometry group, because the group cannot exchange the two homogeneity classes by an odd translation (Mondal et al., 28 Jul 2025). This produces a nontrivial discrete “tensor structure.”

For a cubic interaction, the tree-level contact diagram is

$T_{q_{+},q_{-}}$07

and the sum can be decomposed into the three legs of the boundary tripod and the finite central region near $T_{q_{+},q_{-}}$08, using the bulk-to-boundary splitting identity (Mondal et al., 28 Jul 2025). The result has the conformal-distance factor familiar from CFT,

$T_{q_{+},q_{-}}$09

but multiplied by a vertex-type-dependent structure (Mondal et al., 28 Jul 2025).

In the compact form given in the construction,

$T_{q_{+},q_{-}}$10

where $T_{q_{+},q_{-}}$11 distinguishes which homogeneity class the join point belongs to, and $T_{q_{+},q_{-}}$12 are explicit functions built from the $T_{q_{+},q_{-}}$13-factors (Mondal et al., 28 Jul 2025). This is the precise sense in which the three-point function possesses a nontrivial “tensor structure”: it carries a discrete internal label attached to the geometry of the bulk tripod (Mondal et al., 28 Jul 2025).

The coefficient

$T_{q_{+},q_{-}}$14

shows that the OPE data are expressed through local zeta functions (Mondal et al., 28 Jul 2025). The same analysis indicates that the coefficients can be rewritten in terms of zeta functions at both $T_{q_{+},q_{-}}$15 and $T_{q_{+},q_{-}}$16, corresponding to the trivial and nontrivial multiplicative sign characters of the unramified quadratic extension underlying the arithmetic construction (Mondal et al., 28 Jul 2025). This suggests that the three-point coefficients carry genuinely number-theoretic information absent from the basic regular-tree model.

In the regular limit $T_{q_{+},q_{-}}$17, the distinction between the two tensor structures collapses and the usual $T_{q_{+},q_{-}}$18-adic result is recovered (Mondal et al., 28 Jul 2025). Thus semihomogeneity is invisible in the two-point power law but explicit in the three-point OPE data.

7. Relations to lattices, spectral quotients, and code-based holography

Biregular trees arise not only as symmetric spaces for unitary groups but also as factors in lattice actions and quotient constructions. In the general theory of products of locally finite biregular trees, the ambient automorphism group is a locally compact group

$T_{q_{+},q_{-}}$19

and irreducible lattices $T_{q_{+},q_{-}}$20 are discrete subgroups of finite covolume whose projections to the factors are dense or non-discrete in the appropriate sense (Thomas et al., 2010). A key theorem states that an irreducible lattice acting on a product of two or more locally finite, biregular trees is finitely generated (Thomas et al., 2010). This implies that finite-covolume discrete symmetry groups for tree-based bulk spaces are controlled by finite combinatorial data, which is relevant for quotient geometries, boundary actions, and possible holographic orbifolds (Thomas et al., 2010).

A spectral version of quotient holography appears in the quantum-graph approach to biregular trees. There the infinite biregular tree $T_{q_{+},q_{-}}$21 serves as the universal cover of a finite biregular graph $T_{q_{+},q_{-}}$22, and the resolvent on the quotient is obtained by summing the tree resolvent over lifts. The trace of the resolvent is related to the generating function $T_{q_{+},q_{-}}$23 for non-backtracking closed walks by

$T_{q_{+},q_{-}}$24

with $T_{q_{+},q_{-}}$25 the decaying multiplier and $T_{q_{+},q_{-}}$26 a Wronskian (Carlson, 2023). Ratios of resolvent traces for different graphs with the same local biregularity equal ratios of their walk-generating functions (Carlson, 2023). This suggests a plausible holographic interpretation in which bulk spectral data on the tree control loop observables or zeta-type functions on finite quotients.

The code-theoretic perspective developed for regular Bruhat–Tits trees also illuminates biregular settings indirectly. Classical and quantum holographic codes on $T_{q_{+},q_{-}}$27 are built from Reed–Solomon and generalized Reed–Solomon codes placed at vertices, with bulk logical data encoded into boundary configurations on $T_{q_{+},q_{-}}$28 (Marcolli, 2018). Although the underlying graph is regular, the constructions already distinguish root and non-root roles, incoming and outgoing legs, and in unramified quadratic extensions distinguish subsets of “preferred” directions inside a higher-valence tree (Marcolli, 2018). A plausible implication is that biregular-tree holography may admit analogous code-based realizations in which the two vertex types of the geometry itself replace or reinforce the effective asymmetries introduced by encoding protocols.

The contrast with forest constructions inside hyperbolic tessellations is also instructive. In regular hyperbolic honeycombs $T_{q_{+},q_{-}}$29, one can build a forest of embedded trees whose shell populations obey a $T_{q_{+},q_{-}}$30 transfer matrix

$T_{q_{+},q_{-}}$31

with dominant eigenvalue

$T_{q_{+},q_{-}}$32

governing exponential radial growth (Németh, 2015). These trees are not biregular in the strict graph-theoretic sense, but they exhibit a two-type radial structure whose transfer matrix plays a role similar to the alternation between the two homogeneity classes in biregular-tree holography (Németh, 2015). This suggests a broader discrete-holographic theme: semihomogeneous or multi-type radial recursions encode curvature and scaling data through low-dimensional transfer operators.

8. Conceptual significance and directions of extension

Holography on biregular trees generalizes $T_{q_{+},q_{-}}$33-adic AdS/CFT by replacing the homogeneous Bruhat–Tits tree of $T_{q_{+},q_{-}}$34 with a semihomogeneous tree associated with a unitary group over an unramified quadratic extension (Marcolli, 2018, Mondal et al., 28 Jul 2025). The principal consequences are structural rather than merely technical. Bulk propagators depend on vertex type as well as distance, boundary three-point functions acquire a discrete tensor structure labeled by the homogeneity class of the join point, and OPE coefficients involve zeta functions reflecting the arithmetic of the quadratic extension (Mondal et al., 28 Jul 2025).

This places biregular-tree holography at the intersection of several domains. From geometric group theory, it draws on the theory of biregular trees, their automorphism groups, and lattices on products of trees (Thomas et al., 2010). From non-Archimedean geometry, it inherits the Bruhat–Tits and Drinfeld framework in which discrete trees and continuous $T_{q_{+},q_{-}}$35-adic symmetric spaces share a common boundary and admit bulk–boundary maps (Marcolli, 2018). From spectral graph and quantum graph theory, it gains explicit transfer-matrix methods, resolvent kernels, and quotient trace identities (Carlson, 2023). From combinatorial hyperbolic models, it echoes the use of radial transfer matrices and shell growth rates as discrete curvature data (Németh, 2015).

Several open directions are explicit in the existing literature. One is extension from biregular trees to higher-rank buildings, where the boundary is a spherical building rather than a rank-one ultrametric projective line, and where one expects more intricate analogues of conformal blocks and tensor structures (Marcolli, 2018, Mondal et al., 28 Jul 2025). Another is the systematic development of boundary field theories on the hermitian flag complex or related ultrametric fractals, potentially yielding direct derivations of nonlocal kinetic operators from bulk integration (Mondal et al., 28 Jul 2025). A further direction is the study of quotients by discrete subgroups, including Mumford-curve or black-hole-type geometries in the non-Archimedean setting (Marcolli, 2018).

A plausible implication is that biregular trees provide a minimal setting in which a holographic theory can remain graphically one-dimensional while already exhibiting genuinely higher-rank, arithmetic, and representation-theoretic phenomena. In that sense, they occupy a distinctive niche between homogeneous tree holography and the still less developed holography of full Bruhat–Tits buildings (Marcolli, 2018, Mondal et al., 28 Jul 2025).