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A Solvable P-adic Random Tensor Cascade with an Exact Bulk–Boundary Dictionary: Holographic Features, Perturbation Experiments, and the Weingarten-to-CFT Chain

Kevin Donahue

Published Mar 19, 2026• 0 citations
Publisher

Abstract

Abstract We construct a solvable p-adic random tensor model exhibiting an exact bulk–boundary correspondence with holographic features. A binary tree of Haar-random quantum operations on the Bruhat-Tits tree T₂ produces a boundary quantum state whose entanglement structure is consistent with a p-adic conformal field theory. Key CFT data — including the scaling dimension, bulk scalar mass, and leading non-Gaussian coupling — are derived from the Weingarten moments of the random tensors with no free parameters. We derive a bulk–boundary dictionary for two-point observables and modular Hamiltonian structure: the mean mutual information is consistent with the boundary two-point function of a free scalar with Δ=log₂(5/2) and m²=3/10; the modular Hamiltonian admits a decomposition H_A = H_Aᵐᵉʳᵉᵉ + g·δH_A where g = η₄/η₂² − 1 = 9/56 is derived from the fourth Weingarten moment; and the weight function w(v) = K(v,A) = (η₂/2)^{depth(v)} is the unique quasi-local weight in the 13/28 eigenspace of the transfer operator. We test the holographic dictionary under two bulk perturbations. Experiment A (ER=EPR): severing the root edge destroys cross-half mutual information exactly while releasing monogamy to saturate intra-half entanglement at the Page entropy. Experiment D (RT locality): a mid-tree defect reduces mutual information to zero for all boundary pairs whose minimal geodesic crosses the defect, while leaving remote pairs unchanged to within 1%. Both effects are predicted from the Weingarten structure alone. The entire structure — exact results, derivations, and experimental observations — follows from two numbers: η₂ = 2/5 and η₄ = 13/70. The p-adic cascade is structurally distinct from finite-D U(D) Haar cascades; the latter approach the p-adic fixed point at large D but do not realise it exactly. Numerical validation is performed using large-D U(D) cascades, which approximate but do not realise the p-adic fixed point exactly. Keywords: p-adic AdS/CFT, Bruhat-Tits tree, random tensor network, Weingarten calculus, modular Hamiltonian, ER=EPR, Ryu-Takayanagi, holographic entanglement

DOI
10.5281/zenodo.19110041

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Tensor networks provide a natural framework for exploring holographic duality because they obey entanglement area laws. They have been used to construct explicit toy models realizing many of the interesting structural features of the AdS/CFT correspondence, including the non-uniqueness of bulk operator reconstruction in the boundary theory. In this article, we explore the holographic properties of networks of random tensors. We find that our models naturally incorporate many features that are analogous to those of the AdS/CFT correspondence. When the bond dimension of the tensors is large, we show that the entanglement entropy of all boundary regions, whether connected or not, obey the Ryu-Takayanagi entropy formula, a fact closely related to known properties of the multipartite entanglement of assistance. We also discuss the behavior of Rényi entropies in our models and contrast it with AdS/CFT. Moreover, we find that each boundary region faithfully encodes the physics of the entire bulk entanglement wedge, i.e., the bulk region enclosed by the boundary region and the minimal surface. Our method is to interpret the average over random tensors as the partition function of a classical ferromagnetic Ising model, so that the minimal surfaces of Ryu-Takayanagi appear as domain walls. Upon including the analog of a bulk field, we find that our model reproduces the expected corrections to the Ryu-Takayanagi formula: the bulk minimal surface is displaced and the entropy is augmented by the entanglement of the bulk field. Increasing the entanglement of the bulk field ultimately changes the minimal surface behavior topologically, in a way similar to the effect of creating a black hole. Extrapolating bulk correlation functions to the boundary permits the calculation of the scaling dimensions of boundary operators, which exhibit a large gap between a small number of low-dimension operators and the rest. While we are primarily motivated by the AdS/CFT duality, the main results of the article define a more general form of bulk-boundary correspondence which could be useful for extending holography to other spacetimes.
A bstract The AdS/CFT correspondence realises the holographic principle where information in the bulk of a space is encoded at its border. We are yet a long way from a full mathematical construction of AdS/CFT, but toy models in the form of holographic quantum error correcting codes (HQECC) have replicated some interesting features of the correspondence. In this work we construct new HQECCs built from random stabilizer tensors that describe a duality between models encompassing local Hamiltonians whilst exactly obeying the Ryu-Takayanagi entropy formula for all boundary regions. We also obtain complementary recovery of local bulk operators for any boundary bipartition. Existing HQECCs have been shown to exhibit these properties individually, whereas our mathematically rigorous toy models capture these features of AdS/CFT simultaneously, advancing further towards a complete construction of holographic duality.
We propose a family of exactly solvable toy models for the AdS/CFT correspondence based on a novel construction of quantum error-correcting codes with a tensor network structure. Our building block is a special type of tensor with maximal entanglement along any bipartition, which gives rise to an isometry from the bulk Hilbert space to the boundary Hilbert space. The entire tensor network is an encoder for a quantum error-correcting code, where the bulk and boundary degrees of freedom may be identified as logical and physical degrees of freedom respectively. These models capture key features of entanglement in the AdS/CFT correspondence; in particular, the Ryu-Takayanagi formula and the negativity of tripartite information are obeyed exactly in many cases. That bulk logical operators can be represented on multiple boundary regions mimics the Rindlerwedge reconstruction of boundary operators from bulk operators, realizing explicitly the quantum error-correcting features of AdS/CFT recently proposed in [1].
A bstract We present a general procedure for constructing tensor networks that accurately reproduce holographic states in conformal field theories (CFTs). Given a state in a large- N CFT with a static, semiclassical gravitational dual, we build a tensor network by an iterative series of approximations that eliminate redundant degrees of freedom and minimize the bond dimensions of the resulting network. We argue that the bond dimensions of the tensor network will match the areas of the corresponding bulk surfaces. For “tree” tensor networks (i.e., those that are constructed by discretizing spacetime with non­ intersecting Ryu-Takayanagi surfaces), our arguments can be made rigorous using a version of one-shot entanglement distillation in the CFT. Using the known quantum error correcting properties of AdS/CFT, we show that bulk legs can be added to the tensor networks to create holographic quantum error correcting codes. These codes behave similarly to previous holographic tensor network toy models, but describe actual bulk excitations in continuum AdS/CFT. By assuming some natural generalizations of the “holographic entanglement of purification” conjecture, we are able to construct tensor networks for more general bulk discretizations, leading to finer-grained networks that partition the information content of a Ryu-Takayanagi surface into tensor-factorized subregions. While the granularity of such a tensor network must be set larger than the string/Planck scales, we expect that it can be chosen to lie well below the AdS scale. However, we also prove a no-go theorem which shows that the bulk-to-boundary maps cannot all be isometries in a tensor network with intersecting Ryu-Takayanagi surfaces.
We present a general procedure for constructing tensor networks that accurately reproduce holographic states in conformal field theories (CFTs). Given a state in a large-N CFT with a static, semiclassical gravitational dual, we build a tensor network by an iterative series of approximations that eliminate redundant degrees of freedom and minimize the bond dimensions of the resulting network. We argue that the bond dimensions of the tensor network will match the areas of the corresponding bulk surfaces. For “tree” tensor networks (i.e., those that are constructed by discretizing spacetime with nonintersecting Ryu-Takayanagi surfaces), our arguments can be made rigorous using a version of one-shot entanglement distillation in the CFT. Using the known quantum error correcting properties of AdS/CFT, we show that bulk legs can be added to the tensor networks to create holographic quantum error correcting codes. These codes behave similarly to previous holographic tensor network toy models, but describe actual bulk excitations in continuum AdS/CFT. By assuming some natural generalizations of the “holographic entanglement of purification” conjecture, we are able to construct tensor networks for more general bulk discretizations, leading to finer-grained networks that partition the information content of a Ryu-Takayanagi surface into tensor-factorized subregions. While the granularity of such a tensor network must be set larger than the string/Planck scales, we expect that it can be chosen to lie well below the AdS scale. However, we also prove a no-go theorem which shows that the bulk-to-boundary maps cannot all be isometries in a tensor network with intersecting Ryu-Takayanagi surfaces.
We introduce a branching quantum circuit on a binary (Bruhat–Tits) tree in which each internal node applies an independent Haar-random two-qubit unitary, and show that entanglement organizes along genealogical distance. For two particles separated by k generations, the expected von Neumann entropy of an interval of size 2k grows linearly: S_{VN}(k) = (c_{VN}/3)·k, with exact coefficient c_{VN} = m²·c_{RT} = (9/10)·log₂(5/2) ≈ 1.1897 bits. Both inputs are derived from first principles: a single Haar average yields η₂ = 2/5, from which we obtain the boundary scaling dimension Δ = log₂(5/2), the bulk mass m² = 3/10, and a cascade propagator that is an eigenmode of the p-adic Laplacian, satisfying a discrete Klein–Gordon equation. This matches the spectral data of a p-adic bulk field. Invoking the p-adic AdS/CFT entropy formula of Gubser–Heydeman–Parra-Martinez–Parikh (2016) then fixes the entropy coefficient, giving c_{VN} = m²·c_{RT}. The result is confirmed numerically up to depth D = 50 at 10.34σ. A fully self-contained proof is reduced to a single analytic continuation problem (Conjecture~1), formulated in terms of Selberg integrals and spherical functions on SU(4)/S(U(2)×U(2)). Together with a numerical verification of the entanglement first law in the same model, these results provide a concrete setting in which entanglement scaling, ultrametric geometry, and a Klein–Gordon spectrum emerge from a single random-unitary construction.