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Random Circuits in the Black Hole Interior

Javier M. Magan, Martin Sasieta, Brian Swingle

arXiv:2412.08693v4hep-thquant-ph

TL;DR

The paper asks how microscopic randomness can encode the black hole interior, where an ab initio connection between circuit complexity and interior geometry remains open. It uses Brownian perturbations to prepare ER-caterpillar ensembles and finds a quantitative relation between k-design formation and wormhole length, with random-state indistinguishability at exponentially long times.

  • Problem

    An ab initio connection between quantum circuit complexity and black hole interior geometry remains unidentified.

    Source: §1 · p. 3

  • Method

    Brownian semiclassical sources perturb an AdS black hole, implementing a continuous random circuit whose states have ER-caterpillar wormhole duals.

    Source: §1.1 · pp. 5, 8

  • Results

    ER-caterpillar ensembles become approximate quantum state k-designs with circuit time scaling linearly in k, while their average wormhole length provides the corresponding randomness scale.

    Source: §1.1 · pp. 6–8

  • Takeaways & Limitations

    The results provide a quantitative randomness–geometry relation for interpreting ER-caterpillar geometry as a bulk random quantum circuit.

    Source: §1.1 · p. 8

    Matter backreaction limits how long the circuit can run before substantially changing the original black hole.

    Source: §2 · p. 10

Abstract

from arXiv · show

In this paper, we present a quantitative holographic relation between a microscopic measure of randomness and the geometric length of the wormhole in the black hole interior. To this end, we perturb an AdS black hole with Brownian semiclassical sources, implementing the continuous version of a random quantum circuit for the black hole. We use the random circuit to prepare ensembles of states of the black hole whose semiclassical duals contain Einstein-Rosen (ER) caterpillars: long cylindrical wormholes with large numbers of matter inhomogeneities, of linearly growing length with the circuit time. In this setup, we show semiclassically that the ensemble of ER caterpillars of average length $k\ell_{\Delta}$ and matter correlation scale $\ell_{\Delta}$ forms an approximate quantum state $k$-design of the black hole. At exponentially long circuit times, the ensemble of ER caterpillars becomes polynomial-copy indistinguishable from a collection of random states of the black hole. We comment on the implications of these results for holographic circuit complexity and for the holographic description of the black hole interior.

1 Introduction

This section introduces a Brownian random-circuit framework for describing black-hole interiors through holographic ER-caterpillar states, relating wormhole geometry to circuit-generated randomness. It also outlines implications for holographic circuit complexity, black-hole interiors, and firewalls, alongside the paper’s supporting technical appendices.

1 Introduction

  • Black-hole interiors remain microscopically elusive, motivating quantum complexity as a proposed measure of emergent interior space despite the absence of an ab initio complexity–geometry connection.The CV conjecture [19] provides the paper’s main circuit-picture motivation, but ambiguities in circuit complexity and realistic gravitational tensor networks leave the relation imprecise.Source: §1 · p. 3
  • The Brownian circuit is formed by independently randomizing time-dependent couplings to many Hermitian low-energy matter operators, producing an ensemble of black-hole evolution operators.Large K makes infinitesimal Trotter steps resemble a continuous random quantum circuit rather than evolution under the conserved Hamiltonian alone.Source: §1 · p. 4

1.1 Outline and summary of results

  • Brownian perturbations generate continuous random circuits whose holographic states contain ER caterpillars: cylindrical wormholes with many matter inhomogeneities and linearly growing average length.The matter correlation scale ℓ_Δ is set by the lightest bulk field’s conformal dimension, while cooling keeps the exterior equilibrated and controls backreaction.Source: §1.1 · pp. 4–5
  • The reduced states of the prepared two-sided black holes approach an equilibrium state whose thermodynamic-limit overlap with a canonical Gibbs state differs only through energy variance.The equilibrium state depends on perturbation details, including the operator set, K, and coupling J.Source: §1.1 · p. 6
  • The ensemble becomes an approximate k-design after a circuit time that grows linearly with k up to logarithmic factors, with slope controlled by E_gap.The same energy scale also controls the wormhole’s linear length growth, producing the randomness–length correspondence.Source: §1.1 · pp. 6–7
  • At late times, the design distance decays as approximately N_* k! k e^(-2tE_gap), governed by gapped fluctuations around a replica-symmetry-breaking wormhole saddle.The saddle’s ground states correspond to k copies of the two-boundary wormhole, while excitations determine the decay rate.Source: §1.1 · pp. 7–8
  • ER-caterpillars of average length and matter scale ℓ_Δ realize an approximate quantum state k-design, establishing a quantitative randomness–geometry relation.This formalizes the proposed circuit interpretation of the interior by relating microscopic randomness to wormhole geometry.Source: §1.1 · p. 8
  • At exponentially long circuit times, the ER-caterpillar ensemble becomes polynomial-copy indistinguishable from random equilibrium states of the two-sided black hole.The paper argues that microscopic randomness can continue growing linearly in this regime rather than saturating immediately.Source: §1.1 · p. 8
  • Section 5 discusses implications for holographic circuit complexity, the holographic description of the black hole interior, and firewalls.Source: §1.1 · p. 9
  • The appendices develop alternative low-backreaction circuit implementations, an explicit ER-caterpillar toy model, and the phase structure of the replica Hamiltonian Heff,k, including replica-symmetry breaking at high J.The toy model analyzes geometric properties in any dimension, while the replica analysis focuses on the ground and first excited states.Source: §1.1 · p. 9
  • Additional appendices analyze Heff,k in SYK, intrinsic randomness growth through unitary k-designs, and the relation between frame potentials and the black-hole spectral form factor.The SYK treatment derives Schwinger-Dyson equations and solves them at infinite temperature; the frame-potential analysis discusses gravitationally generated connected correlations.Source: §1.1 · p. 9
  • The final appendix gives asymptotic Haar-moment formulas with a combinatorial interpretation developed in the mathematical literature.Source: §1.1 · p. 9

2 Einstein-Rosen caterpillars

This section introduces ER-caterpillar states prepared by cooled Brownian random circuits and examines their averaged semiclassical geometry. It characterizes the resulting linearly growing cylindrical wormhole, including its trapped surfaces, future crunch, and near-extremal description in terms of two SYK models.

2 Einstein-Rosen caterpillars

  • Directly driving a thermofield-double state with a random circuit increases black-hole energy, preventing monitoring a fixed black hole for polynomially long circuit times.The alternative construction therefore injects perturbations into the interior while keeping the exterior effectively equilibrated.Source: §2 · p. 10, §2.1 · pp. 10–11

2.1 State preparation

  • The state-preparation protocol alternates Lorentzian Brownian-circuit evolution with Euclidean evolution under H0, gradually cooling the black hole along a Schwinger-Keldysh contour.Independent Gaussian Brownian couplings are used on infinitesimal contour segments, and symmetric states are prepared by sandwiching the contour with Euclidean TFD evolution.Source: §2.1 · pp. 11–12

2.2 Average geometry

  • Averaging over Brownian couplings replaces the Lorentzian contour by a Euclidean path integral governed by an effective Hamiltonian Heff,1 coupling the two replicas.The averaged norm is Z(t) = <TFD| exp(-tHeff,1) |TFD>, whose reflection-symmetric slice determines the ensemble’s average geometry.Source: §2.2 · p. 14
  • For large preparation time, approximate Euclidean time-translation symmetry prepares the effective Hamiltonian’s ground state and produces an ER-caterpillar wormhole whose length grows linearly with t.The argument assumes a unique semiclassical wormhole ground state; under suitable mixing and many-perturbation conditions, Heff,1 is gapped and its ground state substantially overlaps a TFD.Source: §2.2 · pp. 14–16
  • The averaged Lorentzian traversable-wormhole symmetry is an ensemble artifact, while the physically relevant continuation yields an anisotropic big-bang/big-crunch cosmology describing the black-hole interior.Individual realizations can have complex analytically continued metrics, so the averaged eternal-wormhole section is not the geometry occupied by the ensemble states.Source: §2.2 · p. 17
  • Trapped surfaces at t_L = 0 and constant τ, together with a future cosmological crunch, place this geometry inside the black hole under weak cosmic censorship.The paper states that this placement is explicitly demonstrated for a particular example in the next subsection.Source: §2.2 · p. 17
  • The average ER-caterpillar geometry is a cylindrical wormhole with length proportional to the preparation circuit time t.This reproduces the linear growth of the wormhole length with circuit time.Source: §2.2 · p. 18

2.3 Near-extremal caterpillars

  • For near-extremal black holes, the average ER-caterpillar geometry is related to preparation of the eternal traversable wormhole of Maldacena and Qi [61].Microscopically, the relevant ensemble consists of low-energy states in the Hilbert space of two SYK models.Source: §2.3 · p. 18

L b HSYK

This section develops the HSYK realization using Brownian Majorana-string drives and its JT-gravity description. It analyzes the resulting ER caterpillars, including their semiclassical growth, geometry within the two-sided black hole, and ensemble properties.

L b HSYK

  • HSYK implements the random circuit with fixed-even-length Majorana strings and Brownian couplings, yielding the Maldacena–Qi effective Hamiltonian.The coupling normalization is chosen so the perturbation is extensive in N.Source: L b HSYK · p. 19
  • The near-horizon theory reduces to JT gravity with matter and, after imposing PSL(2,R) constraints, to nonrelativistic particle mechanics in the renormalized wormhole-length variable.The relevant semiclassical regime is 1/N ! η ! 1, where the interaction produces substantial gravitational backreaction and a connected wormhole ground state.Source: L b HSYK · pp. 19–21
  • For 0 < ∆ < 1, the interacting Euclidean potential has a maximum at ℓmax = (∆^-1) log(η∆/2), enabling a long-lived bounce near equilibrium.The bounce contribution to the ensemble norm is evaluated semiclassically by the on-shell Euclidean action.Source: L b HSYK · pp. 21–22
  • The single-bounce solution ceases to exist for ˜β > ˜βc, so the controlled preparation tunes ˜β slightly above ˜βc to obtain the double-bounce solution.In this regime, 1 ! ˜βc ! N, and the relevant bounce in the interacting region is small but long-lived.Source: L b HSYK · p. 25, Figure 9 · p. 25
  • The analysis reproduces linear growth, while obtaining the onset time t* requires solving for the relevant energies at arbitrary times and substituting them into the matching relation.Source: L b HSYK · p. 26
  • The onset is estimated as t* = E_gap^-1 log N, after which the length grows as ℓ_ER(t) = E_gap(t − t*).The matter gap is set by the lowest-conformal-dimension excited matter field on the wormhole.Source: L b HSYK · p. 26
  • The family of near-extremal ER caterpillars exhibits linear growth, while the dilaton profile shows that the growing wormholes lie inside the two-sided black hole.The transverse sphere at the wormhole center is a maximal surface because the dilaton decreases away from it.Source: L b HSYK · pp. 26–28

2.4 Other properties

  • For relevant perturbations with sufficiently small η, the ground state has overlap close to one with the TFD, exactly reaching one in the large-q limit, while finite-q corrections can worsen with N.The overlap is stated for ∆ < 1/2 and receives corrections of the form e^{-#N/q^3+…} away from large q.Source: §2.4 · pp. 28–29
  • At sufficiently late times, the ensemble’s reduced states are equivalent to equilibrium states, and the bulk exterior has the geometry of a black hole at inverse temperature β(J).When the ground state overlaps strongly with the TFD, the equilibrium state approaches a canonical Gibbs state; equivalence holds for simple operators of O(N^0) energy in the large-N limit.Source: §2.4 · p. 29
  • The onset time satisfies E_gap t_* = O(N^0), parametrically earlier in N than the scrambling time for a single perturbation.For SYK, the corresponding scaling is E_gap t_* ∼ log N because the bulk contains N lightest fermion excitations of equal mass.Source: §2.4 · p. 30
  • The norm is self-averaging across ER caterpillars, strongly suggesting that coarse-grained geometric quantities, including wormhole length, are also self-averaging.A toy model in the appendix explicitly demonstrates self-averaging of the wormhole length.Source: §2.4 · pp. 30–31
  • Replica-symmetry breaking in the four-replica effective Hamiltonian makes two-replica wormholes dominant, reproducing products of the one-replica ground-space contributions at late times.The two-replica ground-state energy is twice the one-replica ground-state energy.Source: §2.4 · pp. 30–31

3 The growth of randomness in gravity

This section quantifies the growth of randomness in ER-caterpillar ensembles through moment-based diagnostics, k-design distances, and their microcanonical and semiclassical descriptions. It establishes the relation between circuit depth, wormhole geometry, and low-order equivalence to random-state ensembles, including the role of spectral data and fermion-parity superselection sectors.

3 The growth of randomness in gravity

  • The section defines approximate k-designs through moment-state distances: for p = 1, small distance means k-copy indistinguishability from the random-state ensemble.The paper focuses on the 2-norm because its design time is computable semiclassically and, in the relevant situations, implies the 1-norm criterion.Source: §3 · pp. 32–33

3.1 Microcanonical linear growth

  • In the microcanonical window, Brownian random circuits generate unitary evolution on a Hilbert space of dimension L = e^S, enabling comparison with Haar-random maximally entangled states.The comparison uses frame potentials, which control the 2-norm distance between ensemble moments; the reference ensemble differs from generic random states because every draw is maximally entangled.Source: §3.1 · pp. 33–34
  • Averaging Brownian couplings produces a time-independent 2k-replica effective Hamiltonian, and the frame potential is its thermal partition function at inverse temperature 2t.The invariant replica-pairing states are ground states of this Hamiltonian, so the infinite-time frame potential counts the ground-space dimension.Source: §3.1 · pp. 35–36, Figure 11 · p. 36
  • At infinite circuit time, the ER-caterpillar ensemble becomes a quantum state k-design for k ≤ L under generic drive-operator assumptions.Genericity excludes common global symmetries and disconnected operator-coupling clusters; the effective Hamiltonian’s ground space is then generated by k! invariant states.Source: §3.1 · pp. 36–37, Figure 12 · p. 36
  • The approach to a k-design is controlled by the effective Hamiltonian’s spectral gap and the number of first excited states.At sufficiently large circuit time, excited-state contributions to the frame potential are dominated by the first excited level.Source: §3.1 · p. 37
  • Linear design time follows when the effective-Hamiltonian gap is k-independent and the number of first excited states grows exponentially with k.These are identified as the two properties guaranteeing linear time to design.Source: §3.1 · p. 37
  • Up to logarithmic factors, the time to approximate a k-design grows linearly with k with slope (2E_gap)^-1, beginning at t* ~ (2E_gap)^-1 log N*.The 2-norm design distance is more restrictive than the trace-distance definition, so the corresponding times obey t_k,1 ≤ t_k,2.Source: §3.1 · pp. 38–39
  • Circuit randomness grows as the eigenphase distribution weakly converges to the circular unitary ensemble (CUE), whose nearby eigenphases repel.Frame potentials depend on eigenphase statistics, so CUE convergence controls approach to the Haar-random ensemble.Source: §3.1 · pp. 38–39, Figure 13 · p. 39

3.2 Semiclassical linear growth

  • At late times, the semiclassical moment calculation is dominated by replica-symmetry-breaking ground states, with non-diagonal replica overlaps suppressed by higher Rényi entropies.For k much smaller than the effective Hilbert-space dimension, corrections are exponentially suppressed by the equilibrium state's Rényi entropies.Source: §3.2 · pp. 41–43
  • For t,t' much larger than t*, the dominant semiclassical contribution to the moment two-point function is a connected Euclidean wormhole, while disconnected saddles decay as exp[-JK(t+t')].The wormhole contains two linearly growing, approximately static regions and receives late-time corrections from its lightest excitations.Source: §3.2 · pp. 43–44, Figure 16 · p. 44
  • At infinite time, only the saddle with k two-boundary wormholes survives, while other contributions decay with the number of perturbations, yielding approximate k-designs on a timescale linear in circuit time up to logarithmic factors.The late-time k-th moment is controlled by the ground space and first excitations, with kN* first excited states and a corresponding k-design approximation time.Source: §3.2 · pp. 45–46

3.3 Linear growth from the double trumpet

  • At large circuit times, the stabilized double-trumpet wormhole dominates the moment two-point function over the disconnected saddle and decays slowly because it has no classical entropy at large N.The double-trumpet solution describes the relevant two-boundary wormhole, whose one-loop correction is governed by the lightest excitations.Source: §3.3 · p. 50
  • Fermion parity is conserved for even Majorana-string perturbations, so ER-caterpillar ensembles become random only within each parity superselection sector, not across the total Hilbert space.For k>1, parity increases the numbers of ground and first excited states by a factor of 2^k and modifies the k-design time accordingly.Source: §3.3 · pp. 50–51

3.4 The random state ensemble

  • The random-state ensemble is defined as random purifications of the equilibrium state using Haar-random unitaries on a regulated finite-dimensional Hilbert space, with the regulator removed after computation.Its average norm is unity, while norm fluctuations and the average 2-norm distance of reduced states from equilibrium are exponentially suppressed under the stated entropy assumption.Source: §3.4 · pp. 51–52
  • Individual random-purification instances need not be close to the mean, despite the ensemble-average reduced state being exponentially close to equilibrium.The suppression concerns the average distance over realizations, not every state separately.Source: §3.4 · p. 52
  • The ER-caterpillar and random-state ensembles have coincident frame potentials and 2-norm-separated k-th moment states of O(k/L_eff), making them equivalent for k << L_eff.The frame-potential agreement follows from dominance of the diagonal Weingarten contribution up to corrections controlled by k/L_eff or higher Rényi entropies.Source: §3.4 · p. 53

3.5 Randomness/length relation

  • ER caterpillars form an ε-approximate quantum state k-design, with their length–randomness relation governed exactly by (3.97) and asymptotically by the logarithmic Lambert W-function.The relation is valid for k << L_eff; the paper defers the k ≳ L_eff regime to the next section.Source: §3.5 · p. 53
  • The ER-caterpillar geometry behaves as a bulk random quantum circuit, with ℓ corresponding to total elapsed time and ℓ_Δ to circuit time.This identifies the wormhole’s length and characteristic matter-correlation scale with the two relevant circuit-time scales.Source: §3.5 · p. 53

4 Towards random states at exponentially long circuit times

The analysis extends the ER-caterpillar ensemble to exponentially large copy numbers, finding linear growth of randomness while finite-resolution observers eventually see saturation and random-state indistinguishability.

Finite resolution and saturation

  • For finite Hilbert-space resolution ε0, the ER-caterpillar ensemble becomes indistinguishable from random states after k_max^ε0, with physically reasonable observers seeing k_max^ε0 ~ L^2.Here L^2 corresponds to the two-sided microcanonical Hilbert-space dimension.Source: Figure 22 · p. 55, §4 · p. 57

Moment scaling

  • For k > L, the frame-potential moments transition smoothly from k! to L^(2k) scaling near k ~ L^2.The transition is supported by asymptotic formulas and Fig. 22.Source: §4 · pp. 54–55

Design times

  • Under assumed effective-Hamiltonian gap structure, the ensemble becomes an ε-approximate quantum state k-design at times k log k - k + 2 log ε^-1 for k << L^2 and k log L + 2 log ε^-1 for k >> L^2.These assumptions are not established beyond numerics on small-L systems.Source: §4 · p. 56

Expected linear growth

  • The continuous random circuit therefore exhibits randomness that keeps growing linearly with circuit time rather than saturating in the idealized analysis.This conclusion contrasts with finite-resolution saturation in the original Hilbert space.Source: §4 · pp. 54, 56

No recurrences for randomness

  • Frame potentials are non-increasing in time, so ensemble-averaged randomness has no Poincaré recurrences.The monotonicity follows from the positive semidefiniteness of the effective Hamiltonian.Source: §4 · p. 57

5 Discussion

The discussion identifies wormhole length with randomness, explores a circuit-complexity interpretation, and highlights limits involving non-unitary preparation and indistinguishable long interiors.

5 Discussion

  • An ER-caterpillar ensemble with average length ℓ and matter scale ℓ_Δ forms an ε-approximate quantum state k-design when its length is related to k and ℓ_ε = ℓ_Δ log ε^-1.The wormhole geometry therefore parametrizes the randomness of the microscopic wavefunctions.Source: §5 · pp. 57–58

Holographic circuit complexity and volume

  • The proposed complexity bound gives a circuit-complexity relation with wormhole volume of CV form for the cylindrical ER-caterpillars.The geometric volume is Vol = 4G S ℓ and Vol_ε = 4G S ℓ_ε.Source: Holographic circuit complexity and… · p. 58
  • Because gradual cooling introduces Euclidean, non-unitary evolution, it remains unclear whether unitary circuit complexity scales linearly with entropy or whether wormhole volume only supplies a lower bound.The same uncertainty appears in the cylindrical tensor-network toy model.Source: Holographic circuit complexity and… · p. 58

Holography and the black hole interior

  • At exponentially long lengths, random states become k-copy indistinguishable from ER caterpillars even though the semiclassical interiors can be macroscopically different and contain no firewalls.The paper presents this as a tension for the holographic description of black-hole interiors.Source: Holography and the black hole interior · pp. 58–59
  • The indistinguishability claim depends on comparing ensembles in an effectively specified Hilbert space, while high-energy wavefunction tails may still distinguish different equilibrium states.The authors note that their ensembles live on infinite-dimensional Hilbert spaces and may have different equilibrium states.Source: Holography and the black hole interior · p. 59

Semiclassical disorder and Euclidean gravity

  • Explicit Brownian disorder produces replica interactions and Euclidean wormholes for ensemble-averaged quantities such as frame potentials without a factorization problem.The paper contrasts this visible semiclassical disorder with proposed microscopic pseudo-random disorder.Source: Semiclassical disorder and Euclidean… · pp. 59–60

A Backrection of the exterior random circuit

The exterior random-circuit construction is constrained by energy backreaction, motivating low-frequency perturbations and microcanonical regularization for long circuit times.

Backreaction

  • Directly applying the random circuit increases black-hole energy linearly with time, so matter backreaction limits the controlled exterior construction.The appendix models the growth as M(t) = M + Jt δE.Source: A Backrection of the exterior random… · p. 60

Low-frequency perturbations

  • The perturbations can be chosen as low-frequency bulk modes, allowing circuit times potentially exponential in entropy because the response peaks at exponentially small microscopic energy separations.The paper identifies modes with ω0 much smaller than the Hawking temperature as the relevant regime.Source: A Backrection of the exterior random… · p. 61

Microcanonical regularization

  • Microcanonical projection keeps perturbations within an energy band, but the required projectors are highly complex and may lack a semiclassical bulk interpretation.The resulting microcanonical unitaries remain useful in microscopic or tensor-network models but lose connection to the semiclassical bulk.Source: A Backrection of the exterior random… · p. 62

B A simple toy model of the states

The toy model replaces random-circuit steps with independently drawn thin-shell operators, yielding cylindrical ER caterpillars whose lengths self-average at large shell number.

Toy-model construction

  • Each shell operator is an independent draw, and the number of shells n plays the role of discrete circuit time.The shell ensemble is defined by a normally distributed mass or conformal dimension.Source: B A simple toy model of the states · p. 62

Semiclassical geometry

Length fluctuations

  • At large n, the wormhole length approaches a Gaussian variable whose relative fluctuations are suppressed by 1/sqrt(n), producing a self-averaging ensemble.This follows from the central limit theorem, independently of the original shell distribution.Source: B A simple toy model of the states · p. 64

Toy-model limitations

  • The toy model has limited microscopic control because fixed thin-shell operators may receive semiclassical corrections and remain compatible with additional pseudo-random microscopic ensembles.Its technical advantage is that precise geometries can be found in general dimensions.Source: B A simple toy model of the states · pp. 62, 64

C Comments on the effective Hamiltonian

The effective Hamiltonians are analyzed through replica symmetry, TFD-like ground states, and random-matrix numerics, with results depending on the coupling and microscopic drive operators.

Phase structure

  • The phase diagram of Heff,k depends on the single-replica Hamiltonian, drive operators, and coupling J, so the paper assumes a replica-symmetry-breaking pattern for all J > 0.The authors explicitly note that competing saddles have not been ruled out.Source: C Comments on the effective Hamiltonian · pp. 65, 67

Two-replica case

  • For k = 1, a sufficiently general perturbation set makes the infinite-temperature TFD the unique gapped ground state, while finite J can yield a TFD-like state at tunable β(J).The TFD-like state can differ from the canonical TFD through its energy variance.Source: C Comments on the effective Hamiltonian · pp. 65–66

Overlap

  • The optimized TFD overlap is predicted to remain nonzero at large system size when the energy variance is extensive, although it can vanish in models such as fixed-q SYK.The latter case is interpreted speculatively as a possible breakdown of the diagonal approximation.Source: C Comments on the effective Hamiltonian · p. 66

Multi-replica case

  • At large J, Heff,k is expected to have k! replica-symmetry-breaking ground states, each associated with a pairing of forward and backward replicas.For k less than the Hilbert-space dimension, these states are linearly independent, and a gap in Heff,1 is expected to persist in Heff,k.Source: C Comments on the effective Hamiltonian · pp. 66–67

Numerics

D SYK caterpillars

The SYK analysis formulates Brownian perturbations through replica effective actions and finds late-time gapped wormhole saddles whose frame-potential behavior approaches a plateau on a logarithmic timescale.

Replica effective theory

  • The large-N SYK treatment represents Brownian-circuit moments using 2k-replica collective fields and Schwinger-Dyson equations.The construction starts from a bosonic perturbation of q-SYK and integrates over the disorder to obtain an effective action.Source: D SYK caterpillars · pp. 68, 70

Infinite-temperature wormhole

  • At late times, the k = 1 correlators decay as e^(-Jp|s|), identifying an effective gap Egap = Jp in Heff,1.The effective Hamiltonian is described as a massive 1-1 Dirac fermion.Source: D SYK caterpillars · p. 71

Frame potential

  • The disconnected saddle decays exponentially, while wormhole saddles produce the late-time frame-potential plateau.The wormhole contribution becomes time-independent at late times.Source: D SYK caterpillars · p. 72

Timescale

  • The onset timescale is t* = J^-1 log N when Egap = 2Jp in the infinite-temperature case.The disconnected contribution becomes negligible before this onset scale.Source: D SYK caterpillars · p. 72

Multi-replica extension

  • For k > 1, the dominant late-time contribution comes from the replica-symmetry-breaking solution with n = k.The analysis supports the corresponding low-temperature SYK results but does not treat finite, non-small temperatures.Source: D SYK caterpillars · p. 72

E Unitary k-designs for Brownian systems

The paper maps Brownian-circuit moment superoperators to thermal evolution under replica effective Hamiltonians and uses their ground spaces to characterize unitary k-design formation.

Definitions

  • A unitary k-design reproduces the k-th Haar moment, while an ε-approximate design matches the first k moments within a specified trace-distance resolution.The paper notes that alternative diamond-distance definitions are equivalent up to dimension-dependent factors.Source: E Unitary k-designs for Brownian systems · p. 73

Physical interpretation

  • A 1-design models local thermalization on average, whereas a 2-design captures complete scrambling for observables measured by the four-point OTOC.These interpretations connect design order to progressively stronger randomization properties.Source: E Unitary k-designs for Brownian systems · p. 73

Infinite-time designs

  • At infinite time, Brownian circuits form unitary k-designs for any k when the invariant Haar states are the only ground states of Heff,k.The moment superoperator is Euclidean evolution generated by the replica effective Hamiltonian.Source: E Unitary k-designs for Brownian systems · p. 74

Haar comparison

  • For k ≤ L, Haar has k! independent invariant ground states, yielding the corresponding normalization of the moment superoperator.For k > L, the invariant-state representation is not unique.Source: E Unitary k-designs for Brownian systems · p. 74

Finite-time convergence

F Frame potential vs spectral form factor

The frame potential of Brownian circuits differs from the spectral form factor because time-dependent perturbations alter eigenphase correlations, removing the ramp and changing the plateau.

Effective-Hamiltonian interpretation

  • The first frame potential is represented as a real-time partition function of a generally non-Hermitian effective Hamiltonian with decaying excited states.Under generic perturbations, Heff,1 has a unique ground state.Source: F Frame potential vs spectral form… · p. 75

Main comparison

  • A small time-dependent perturbation removes the spectral-form-factor ramp from F1(t), leaving its intermediate-time approach controlled by connected correlations on O(log L) timescales.The perturbation suppresses disconnected contributions and defines the Thouless-like onset time t*.Source: F Frame potential vs spectral form… · pp. 75–76

Plateau behavior

  • The F1(t) plateau is suppressed by L^-1 relative to the SFF0 plateau because Brownian evolution develops CUE eigenphase repulsion while exp(-itH0) retains Poisson-distributed windings.Table 1 summarizes this eigenphase-correlation distinction.Source: F Frame potential vs spectral form… · p. 75, Table 1 · p. 77

Late-time semiclassics

  • The WKB description fails when time reaches the effective-Hamiltonian gap, requiring quantization of the double-cone moduli space.The resulting late-time structure includes a ground-state term and decaying excited-state contributions.Source: F Frame potential vs spectral form… · pp. 77–78

Gravitational interpretation

  • The late-time frame-potential plateau is expected to arise from a quantized double-cone wormhole for a few Brownian perturbations.With many perturbations, backreaction complicates the connection to an eternal traversable wormhole.Source: F Frame potential vs spectral form… · p. 78

Higher moments

  • Higher frame potentials probe finer spectral information than the SFF because Fk depends on the 2k-th moment of the eigenphase density.The SFF corresponds to the k = 1 case, while larger k accesses higher spectral moments.Source: F Frame potential vs spectral form… · pp. 78–79

G Distribution of the length of the longest increasing subsequences

This section derives the asymptotic behavior of f_k,L, the number of permutations in Sym(k) whose longest increasing subsequence has length at most L, across sub-exponential, intermediate, and exponential regimes.

Sub-exponential regime

  • For k ≤ L, f_k,L = k!, because no permutation has an increasing subsequence longer than L; this behavior must change before k greatly exceeds L^2.Here f_k,L is also the dimension of the invariant-state subspace within the L^(2k)-dimensional replica Hilbert space.Source: G Distribution of the length of the… · p. 79

Intermediate regime

  • For L < k ≪ 4L^2, f_k,L is identified with k! times the cumulative probability that a uniform permutation’s longest increasing subsequence has length at most L.The asymptotics are governed by the Tracy–Widom distribution in the intermediate regime.Source: G Distribution of the length of the… · pp. 79–80
  • For k = L^2/x^2 with x > 2, the cumulative probability approaches one, while for 1/2 ≲ x < 2 it approaches zero as L grows.The corresponding asymptotic forms include f_k,L ≈ k!e^(-kH(x)) in the latter regime and f_k,L ≈ k^k e^(-k(1+H(x))) for 1/2 ≲ x.Source: G Distribution of the length of the… · p. 80

Validity and limitations

  • The intermediate and exponential formulas have restricted validity: the exponential approximation requires k ≫ L^4, while one intermediate expression fails for x < 0.48789 because it would exceed the replica-space bound.These restrictions prevent extrapolating the displayed asymptotic formulas across all parameter ranges.Source: G Distribution of the length of the… · pp. 80, 82

Exponential regime

  • In the exponential regime k ≫ L^2/4, a generalized Stirling analysis yields f_k,L ≈ L^(2k)k^(-L^2/2) for k ≫ L^4.This approximation uses the generating function of the cumulative distribution and is explicitly stated to be valid only when k greatly exceeds L^4.Source: G Distribution of the length of the… · pp. 81–82
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