Diluted p-Spin Glasses Overview

• Diluted p-spin glasses are spin-glass models defined on sparse random hypergraphs, where p-tuple interactions create complex energy landscapes.
• Researchers use replica and cavity methods to analyze transitions between replica symmetry, one-step RSB, and full RSB phases, crucial for understanding glass transitions.
• Numerical studies show FRSB-like behavior and disorder chaos, highlighting finite-size effects and the significant impact of dilution on phase structures.

Searching arXiv for relevant papers on diluted p-spin glasses and closely related results. arxiv_search.query({"^^^^^^^^3search_query3 p-spin\"^^^^^^^^3 OR ti:\3^^^^^^^^"diluted p-spin\"","start":^^^^3search_query3^^^^,"max_results":^^^^3all:\3search_query3 arxiv_search.search({"query":"diluted p-spin glass", "max_results": ^^^^3all:\3search_query3^^^^}) Diluted PRESERVED_PLACEHOLDER_^^^^3search_query3^^^^-spin glasses are spin-glass models in which the interaction set is not the complete graph of the fully connected mean-field theory, but a random hypergraph or a power-law diluted long-range network. In the cited literature, this class includes Poisson random PRESERVED_PLACEHOLDER_^^^^3all:\3^^^^-uniform hypergraphs with Hamiltonian

PRESERVED_PLACEHOLDER_^^^^3 OR ti:\3^^^^

general diluted models with random interaction functions $\theta_k$ on uniformly chosen $p$-tuples, mixed diluted $p$-spin systems on random hypergraphs $E=\bigcup_{p=2}^\Delta E_p$, and one-dimensional long-range proxies with fixed average coordination and power-law dilution (&&&^^^^3search_query3^^^^&&&, &&&^^^^3all:\3^^^^&&&, &&&^^^^3 OR ti:\3^^^^&&&, Gupta et al., 16 Apr 2026). The subject combines replica and cavity methods, asymptotic Gibbs-measure theory, recursive distributional equations, finite-size scaling, and spectral methods for disorder perturbations; its main themes are replica symmetry, one-step and full replica-symmetry breaking, and the existence or absence of thermodynamic glass transitions.

^^^^3all:\3^^^^. Model classes and Hamiltonian structure

A standard diluted $p$-spin model is defined on $N$ Ising spins $\sigma_i\in\{-1,+1\}$. In one formulation, a PRESERVED_PLACEHOLDER_^^^^3all:\3search_query3^^^^ number of PRESERVED_PLACEHOLDER_^^^^3all:\3all:\3^^^^-tuples is sampled i.i.d. uniformly from PRESERVED_PLACEHOLDER_^^^^3all:\3 OR ti:\3^^^^, each sampled tuple receives an independent standard Gaussian coupling, and the Hamiltonian is

PRESERVED_PLACEHOLDER_^^^^3all:\33^^^^

The free energy density is

PRESERVED_PLACEHOLDER_^^^^3all:\3^^^^4

and PRESERVED_PLACEHOLDER_^^^^3all:\3^^^^5 exists (&&&^^^^3search_query3^^^^&&&).

A broader formulation places the model on a random PRESERVED_PLACEHOLDER_^^^^3all:\3^^^^6-uniform hypergraph with sparsity parameter PRESERVED_PLACEHOLDER_^^^^3all:\3^^^^7. One draws PRESERVED_PLACEHOLDER_^^^^3all:\3^^^^8, chooses PRESERVED_PLACEHOLDER_^^^^3all:\3^^^^9 uniformly of size PRESERVED_PLACEHOLDER_^^^^3 OR ti:\3search_query3^^^^, assigns i.i.d. interaction functions PRESERVED_PLACEHOLDER_^^^^3 OR ti:\3all:\3^^^^, and writes

PRESERVED_PLACEHOLDER_^^^^3 OR ti:\3 OR ti:\3^^^^

This formulation accommodates Ising PRESERVED_PLACEHOLDER_^^^^3 OR ti:\33^^^^-spin, Potts, XY, and continuous hardcore examples within a common cavity framework (&&&^^^^3all:\3^^^^&&&).

A mixed diluted PRESERVED_PLACEHOLDER_^^^^3 OR ti:\3^^^^4-spin model allows hyperedges of several cardinalities. For each PRESERVED_PLACEHOLDER_^^^^3 OR ti:\3^^^^5, every PRESERVED_PLACEHOLDER_^^^^3 OR ti:\3^^^^6-subset PRESERVED_PLACEHOLDER_^^^^3 OR ti:\3^^^^7 is included independently with probability

PRESERVED_PLACEHOLDER_^^^^3 OR ti:\3^^^^8

and the Hamiltonian is

PRESERVED_PLACEHOLDER_^^^^3 OR ti:\3^^^^9

with $\theta_k$^^^^3search_query3^^^^ an odd measurable function and $\theta_k$^^^^3all:\3^^^^ independent standard Gaussians (&&&^^^^3 OR ti:\3^^^^&&&).

A different dilution mechanism appears in the balanced $\theta_k$^^^^3 OR ti:\3^^^^, $\theta_k$3 long-range proxy. There are $\theta_k$4 Ising spins $\theta_k$5 on each of $\theta_k$6 sites of a ring, and the Hamiltonian is

$\theta_k$7

In the diluted fixed-coordination-number $\theta_k$8 version, each rung is connected on average to $\theta_k$9 others, with

$p$^^^^3search_query3^^^^

where $p$^^^^3all:\3^^^^. In both the fully connected and diluted long-range constructions, the limit $p$^^^^3 OR ti:\3^^^^ recovers an infinite-range model, while larger $p$3 moves the system toward a short-range-like regime (Gupta et al., 16 Apr 2026).

^^^^3 OR ti:\3^^^^. Overlaps, order parameters, and mean-field organization

The basic order parameter is the overlap. For two configurations $p$4,

$p$5

In replicated formulations of $p$6-spin glasses, one also writes

$p$7

In the asymptotic ^^^^3all:\3^^^^-RSB setting, the overlap takes exactly two values $p$8: the self-overlap is $p$9, whereas for $p$^^^^3search_query3^^^^ one has $p$^^^^3all:\3^^^^ (&&&^^^^3search_query3^^^^&&&).

Mean-field $p$^^^^3 OR ti:\3^^^^-spin glasses with $p$3 exhibit a dynamical transition at $p$4, described as mode-coupling-like nonergodicity, and a static Kauzmann temperature $p$5 at which the configurational entropy vanishes. Their mean-field low-temperature phase is one-step replica-symmetry breaking: the order-parameter overlaps jump discontinuously at $p$6. At still lower temperature, a Gardner transition induces a full replica-symmetry-breaking hierarchy inside the ^^^^3all:\3^^^^-RSB phase (Gupta et al., 16 Apr 2026).

To distinguish weakly first-order ^^^^3all:\3^^^^-RSB behavior from a continuous FRSB transition, the replicated Gibbs free-energy expansion introduces

$p$7

where $p$8 and $p$9 are linear combinations of third-order cumulants of overlap fluctuations. For the balanced $E=\bigcup_{p=2}^\Delta E_p$^^^^3search_query3^^^^, $E=\bigcup_{p=2}^\Delta E_p$^^^^3all:\3^^^^ model, mean field gives $E=\bigcup_{p=2}^\Delta E_p$^^^^3 OR ti:\3^^^^. In that setting, $E=\bigcup_{p=2}^\Delta E_p$3 is associated with a discontinuous ^^^^3all:\3^^^^-RSB transition, whereas $E=\bigcup_{p=2}^\Delta E_p$4 yields a continuous FRSB transition (Gupta et al., 16 Apr 2026).

The overlap distribution $E=\bigcup_{p=2}^\Delta E_p$5 is correspondingly diagnostic. A two-peak structure is the characteristic mean-field signature of ^^^^3all:\3^^^^-RSB, while a broad continuous $E=\bigcup_{p=2}^\Delta E_p$6 extending from $E=\bigcup_{p=2}^\Delta E_p$7 to a maximum is characteristic of a direct transition to an FRSB-like phase in the numerical long-range proxy (Gupta et al., 16 Apr 2026). A frequent misconception is that dilution by itself enforces replica symmetry; the cited work does not support that conclusion. Rather, replica symmetry, ^^^^3all:\3^^^^-RSB, and FRSB appear as distinct regimes whose realization depends on temperature, sparsity, and the geometry of the underlying interaction structure.

3. Asymptotic Gibbs measures and the rigorous ^^^^3all:\3^^^^-RSB description

The rigorous theory of diluted $E=\bigcup_{p=2}^\Delta E_p$8-spin glasses studies infinite-volume Gibbs measures via exchangeability and cavity identities. In Panchenko’s framework, one draws $E=\bigcup_{p=2}^\Delta E_p$9, constructs a random probability measure $p$^^^^3search_query3^^^^ on

$p$^^^^3all:\3^^^^

and samples i.i.d. $p$^^^^3 OR ti:\3^^^^. The support of $p$3 then yields pure-state functions with weights $p$4, and independent uniform variables are inserted into these functions to generate spin variables in each pure state (&&&^^^^3search_query3^^^^&&&).

Under the ^^^^3all:\3^^^^-RSB hypothesis, the Mézard–Parisi ansatz predicts two key structures. First, the atomic masses $p$5 are independent of the array of spin magnetizations and have the Poisson–Dirichlet law $p$6, $p$7. Second, there is a measurable function $p$8 and an auxiliary uniform random variable $p$9 such that, conditionally on $N$^^^^3search_query3^^^^ and independent uniforms $N$^^^^3all:\3^^^^, $N$^^^^3 OR ti:\3^^^^, the magnetization of spin $N$3 in pure state $N$4 is

$N$5

This gives the factorized ^^^^3all:\3^^^^-RSB description of pure-state magnetizations in the diluted setting (&&&^^^^3search_query3^^^^&&&).

The associated cavity formalism produces a distributional fixed-point equation for cavity fields. In the ^^^^3all:\3^^^^-RSB setting, a single cavity field $N$6 satisfies in law

$N$7

with $N$8, $N$9, and

$\sigma_i\in\{-1,+1\}$^^^^3search_query3^^^^

This is the usual population-dynamics fixed-point equation in the ^^^^3all:\3^^^^-RSB cavity method (&&&^^^^3search_query3^^^^&&&).

The rigorous identification of the ansatz proceeds through cavity equations and hierarchical exchangeability. A hierarchical version of Aldous–Hoover shows that any such law factors through nested i.i.d. randomizations at each RSB level, while the cavity equations remove an extra Ghirlanda–Guerra coordinate and reduce the magnetization function from $\sigma_i\in\{-1,+1\}$^^^^3all:\3^^^^ to $\sigma_i\in\{-1,+1\}$^^^^3 OR ti:\3^^^^ (&&&^^^^3search_query3^^^^&&&).

The main theorem is conditional on being in the ^^^^3all:\3^^^^-RSB regime. If $\sigma_i\in\{-1,+1\}$3 and $\sigma_i\in\{-1,+1\}$4, any ^^^^3all:\3^^^^-RSB asymptotic Gibbs measure satisfies the Mézard–Parisi ansatz. If $\sigma_i\in\{-1,+1\}$5, then necessarily $\sigma_i\in\{-1,+1\}$6, and every ^^^^3all:\3^^^^-RSB Gibbs measure satisfies the full ^^^^3all:\3^^^^-RSB ansatz. The exceptional case $\sigma_i\in\{-1,+1\}$7 and $\sigma_i\in\{-1,+1\}$8 is only partially characterized: one proves a symmetry property of the conditional law of the spin-function, but not the full factorization formula (&&&^^^^3search_query3^^^^&&&). This establishes the rigorous structure of diluted ^^^^3all:\3^^^^-RSB Gibbs measures without asserting that every diluted $\sigma_i\in\{-1,+1\}$9-spin model actually realizes a ^^^^3all:\3^^^^-RSB phase.

4. Replica symmetry and recursive distributional equations

Replica symmetry in general diluted spin glasses can also be formulated as a fixed-point problem on laws of single-site log-densities. Let PRESERVED_PLACEHOLDER_^^^^3all:\3search_query3search_query3^^^^ represent the log-density of the single-site marginal. If PRESERVED_PLACEHOLDER_^^^^3all:\3search_query3all:\3^^^^ are i.i.d. with law PRESERVED_PLACEHOLDER_^^^^3all:\3search_query3 OR ti:\3^^^^, PRESERVED_PLACEHOLDER_^^^^3all:\3search_query33^^^^, and PRESERVED_PLACEHOLDER_^^^^3all:\3search_query3^^^^4 are i.i.d. interaction functions, then the cavity operator is

PRESERVED_PLACEHOLDER_^^^^3all:\3search_query3^^^^5

Replica symmetry means that the law PRESERVED_PLACEHOLDER_^^^^3all:\3search_query3^^^^6 of PRESERVED_PLACEHOLDER_^^^^3all:\3search_query3^^^^7 satisfies the recursive distributional equation

PRESERVED_PLACEHOLDER_^^^^3all:\3search_query3^^^^8

obtained by averaging over PRESERVED_PLACEHOLDER_^^^^3all:\3search_query3^^^^9 and the PRESERVED_PLACEHOLDER_^^^^3all:\3all:\3search_query3^^^^’s (&&&^^^^3all:\3^^^^&&&).

The corresponding RS free-energy functional is

PRESERVED_PLACEHOLDER_^^^^3all:\3all:\3all:\3^^^^

where

PRESERVED_PLACEHOLDER_^^^^3all:\3all:\3 OR ti:\3^^^^

Under the RS ansatz,

PRESERVED_PLACEHOLDER_^^^^3all:\3all:\33^^^^

and if the fixed point PRESERVED_PLACEHOLDER_^^^^3all:\3all:\3^^^^4 is unique then

PRESERVED_PLACEHOLDER_^^^^3all:\3all:\3^^^^5

The same framework admits a zero-temperature operator PRESERVED_PLACEHOLDER_^^^^3all:\3all:\3^^^^6 and a variational formula for the ground-state energy (&&&^^^^3all:\3^^^^&&&).

Two replica-symmetric regimes are isolated. The first is a high-temperature regime in which the cavity operator is contractive in PRESERVED_PLACEHOLDER_^^^^3all:\3all:\3^^^^7; one sufficient condition is

PRESERVED_PLACEHOLDER_^^^^3all:\3all:\3^^^^8

The second is the subcritical regime

PRESERVED_PLACEHOLDER_^^^^3all:\3all:\3^^^^9

where the underlying Galton–Watson tree of branching mean PRESERVED_PLACEHOLDER_^^^^3all:\3 OR ti:\3search_query3^^^^ is almost surely finite, giving uniqueness of the fixed point independently of temperature (&&&^^^^3all:\3^^^^&&&).

For the symmetric Ising PRESERVED_PLACEHOLDER_^^^^3all:\3 OR ti:\3all:\3^^^^-spin case with PRESERVED_PLACEHOLDER_^^^^3all:\3 OR ti:\3 OR ti:\3^^^^, PRESERVED_PLACEHOLDER_^^^^3all:\3 OR ti:\33^^^^, and uniform PRESERVED_PLACEHOLDER_^^^^3all:\3 OR ti:\3^^^^4 on PRESERVED_PLACEHOLDER_^^^^3all:\3 OR ti:\3^^^^5, the unique RS solution is PRESERVED_PLACEHOLDER_^^^^3all:\3 OR ti:\3^^^^6, and the explicit formulas are

PRESERVED_PLACEHOLDER_^^^^3all:\3 OR ti:\3^^^^7

These results delineate where dilution is compatible with a complete RS description and where more complicated Gibbs structures must be expected (&&&^^^^3all:\3^^^^&&&).

5. Numerical phase structure in a power-law diluted long-range proxy

A detailed numerical study of phase transitions was carried out for the balanced PRESERVED_PLACEHOLDER_^^^^3all:\3 OR ti:\3^^^^8, PRESERVED_PLACEHOLDER_^^^^3all:\3 OR ti:\3^^^^9 spin-glass model used as a one-dimensional long-range proxy for finite-dimensional short-range PRESERVED_PLACEHOLDER_^^^^3all:\33search_query3^^^^-spin glasses. The simulations employed Metropolis single-spin updates augmented by parallel tempering across PRESERVED_PLACEHOLDER_^^^^3all:\33all:\3^^^^ temperatures between PRESERVED_PLACEHOLDER_^^^^3all:\33 OR ti:\3^^^^ and PRESERVED_PLACEHOLDER_^^^^3all:\333^^^^. Four real replicas per disorder sample were simulated to compute overlap distributions and third-order cumulants, and equilibration was checked with the energy–link-overlap identity

PRESERVED_PLACEHOLDER_^^^^3all:\33^^^^4

where PRESERVED_PLACEHOLDER_^^^^3all:\33^^^^5 in the diluted model (Gupta et al., 16 Apr 2026).

The spin-glass susceptibility at zero momentum is

PRESERVED_PLACEHOLDER_^^^^3all:\33^^^^6

where PRESERVED_PLACEHOLDER_^^^^3all:\33^^^^7 runs over the 6 pairwise spin products on each rung, giving 36 channels in total. The finite-size scaling forms are known exactly in the long-range mean-field regime PRESERVED_PLACEHOLDER_^^^^3all:\33^^^^8,

PRESERVED_PLACEHOLDER_^^^^3all:\33^^^^9

and in the non-mean-field regime PRESERVED_PLACEHOLDER_^^^^3all:\3start3search_query3^^^^,

PRESERVED_PLACEHOLDER_^^^^3all:\3start3all:\3^^^^

with crossings drifting according to

PRESERVED_PLACEHOLDER_^^^^3all:\3start3 OR ti:\3^^^^

This supports extrapolation of PRESERVED_PLACEHOLDER_^^^^3all:\3^^^^43 from size-pair crossings (Gupta et al., 16 Apr 2026).

The extracted critical temperatures are as follows.

Model and regime	PRESERVED_PLACEHOLDER_^^^^3all:\3^^^^44	Critical observation
Fully connected (variance-scaled)	PRESERVED_PLACEHOLDER_^^^^3all:\3^^^^45	PRESERVED_PLACEHOLDER_^^^^3all:\3^^^^46
Fully connected (variance-scaled)	PRESERVED_PLACEHOLDER_^^^^3all:\3^^^^47	PRESERVED_PLACEHOLDER_^^^^3all:\3^^^^48
Diluted PRESERVED_PLACEHOLDER_^^^^3all:\3^^^^49	PRESERVED_PLACEHOLDER_^^^^3all:\3max_results3search_query3^^^^	PRESERVED_PLACEHOLDER_^^^^3all:\3max_results3all:\3^^^^
Diluted PRESERVED_PLACEHOLDER_^^^^3all:\3max_results3 OR ti:\3^^^^	PRESERVED_PLACEHOLDER_^^^^3all:\3^^^^53	PRESERVED_PLACEHOLDER_^^^^3all:\3^^^^54
Diluted PRESERVED_PLACEHOLDER_^^^^3all:\3^^^^55	PRESERVED_PLACEHOLDER_^^^^3all:\3^^^^56	PRESERVED_PLACEHOLDER_^^^^3all:\3^^^^57
Diluted non-MF	PRESERVED_PLACEHOLDER_^^^^3all:\3^^^^58	no crossing of PRESERVED_PLACEHOLDER_^^^^3all:\3^^^^59 no finite-PRESERVED_PLACEHOLDER_^^^^3all:\3query3search_query3^^^^

Below their respective PRESERVED_PLACEHOLDER_^^^^3all:\3query3all:\3^^^^, both the fully connected and diluted models exhibit broad, continuous PRESERVED_PLACEHOLDER_^^^^3all:\3query3 OR ti:\3^^^^ distributions extending from PRESERVED_PLACEHOLDER_^^^^3all:\3^^^^63 to a maximum, rather than the two-peak structure expected for a ^^^^3all:\3^^^^-RSB transition. The estimated PRESERVED_PLACEHOLDER_^^^^3all:\3^^^^64-parameter, obtained from three- and four-replica estimators, remains below unity near PRESERVED_PLACEHOLDER_^^^^3all:\3^^^^65 for all accessible sizes and all studied PRESERVED_PLACEHOLDER_^^^^3all:\3^^^^66. The numerical interpretation is therefore a direct transition from the paramagnetic state to an FRSB-like phase with renormalized PRESERVED_PLACEHOLDER_^^^^3all:\3^^^^67, despite the mean-field value PRESERVED_PLACEHOLDER_^^^^3all:\3^^^^68 for the balanced model (Gupta et al., 16 Apr 2026).

For PRESERVED_PLACEHOLDER_^^^^3all:\3^^^^69, which is described as roughly corresponding to a three-dimensional system, the results are stronger: PRESERVED_PLACEHOLDER_^^^^3all:\3diluted p-spin glass3search_query3^^^^ shows no bimodality, PRESERVED_PLACEHOLDER_^^^^3all:\3diluted p-spin glass3all:\3^^^^ throughout, and there are no signs of either a ^^^^3all:\3^^^^-RSB transition or a continuous FRSB transition at finite temperature. The authors argue that strong finite-size effects and closely spaced transition temperatures remove the expected ^^^^3all:\3^^^^-RSB transition for the accessible system sizes (Gupta et al., 16 Apr 2026). Taken together with the rigorous ^^^^3all:\3^^^^-RSB theory, this separates two questions that are often conflated: how a ^^^^3all:\3^^^^-RSB Gibbs measure is structured if such a phase occurs, and whether a specific diluted model actually realizes that phase.

6. Disorder chaos, dimensional implications, and open problems

Disorder chaos in diluted PRESERVED_PLACEHOLDER_^^^^3all:\3diluted p-spin glass3 OR ti:\3^^^^-spin glasses concerns the instability of equilibrium states under perturbations of the couplings. In the mixed diluted model, one compares the original disorder PRESERVED_PLACEHOLDER_^^^^3all:\3^^^^73 with a perturbed copy PRESERVED_PLACEHOLDER_^^^^3all:\3^^^^74, either through an Ornstein–Uhlenbeck interpolation,

PRESERVED_PLACEHOLDER_^^^^3all:\3^^^^75

or through independent replacement,

PRESERVED_PLACEHOLDER_^^^^3all:\3^^^^76

with PRESERVED_PLACEHOLDER_^^^^3all:\3^^^^77 i.i.d. standard Gaussians. Sampling PRESERVED_PLACEHOLDER_^^^^3all:\3^^^^78 and PRESERVED_PLACEHOLDER_^^^^3all:\3^^^^79, one defines the site overlap

PRESERVED_PLACEHOLDER_^^^^3all:\3max_results3search_query3^^^^

If

PRESERVED_PLACEHOLDER_^^^^3all:\3max_results3all:\3^^^^

then for either perturbation, every PRESERVED_PLACEHOLDER_^^^^3all:\3max_results3 OR ti:\3^^^^, and every PRESERVED_PLACEHOLDER_^^^^3all:\3^^^^83, there is a constant PRESERVED_PLACEHOLDER_^^^^3all:\3^^^^84 such that

PRESERVED_PLACEHOLDER_^^^^3all:\3^^^^85

For any fixed PRESERVED_PLACEHOLDER_^^^^3all:\3^^^^86, the squared overlap therefore vanishes at a polynomial rate as PRESERVED_PLACEHOLDER_^^^^3all:\3^^^^87 (&&&^^^^3 OR ti:\3^^^^&&&).

The proof uses the Hermite spectral method. Writing the two-spin correlation functions PRESERVED_PLACEHOLDER_^^^^3all:\3^^^^88 in the Hermite basis gives a semigroup identity under perturbation. A sign-flip lemma then eliminates low-frequency coefficients whenever a suitable sign vector can be chosen, and a local hypertree lemma shows that on tree-like balls PRESERVED_PLACEHOLDER_^^^^3all:\3^^^^89, coefficients with PRESERVED_PLACEHOLDER_^^^^3all:\3^^^^9^^^^3search_query3^^^^ must vanish. Combining this with an exploration process on the sparse hypergraph yields the overlap bound above (&&&^^^^3 OR ti:\3^^^^&&&). The result applies to any mixture, including odd PRESERVED_PLACEHOLDER_^^^^3all:\3^^^^9^^^^3all:\3 and removes the large-connectivity restriction of earlier work discussed in the same paper.

The dimensional implications of dilution remain contested. In the numerical long-range proxy, the mapping PRESERVED_PLACEHOLDER_^^^^3all:\3^^^^9^^^^3 OR ti:\3^^^^ to short-range PRESERVED_PLACEHOLDER_^^^^3all:\3^^^^93 is used to suggest the absence of any thermodynamic glass transition PRESERVED_PLACEHOLDER_^^^^3all:\3^^^^94 in three dimensions; the data are described as consistent with PRESERVED_PLACEHOLDER_^^^^3all:\3^^^^95, and the paper states that this supports the view that structural glasses do not undergo a finite-temperature Kauzmann transition in PRESERVED_PLACEHOLDER_^^^^3all:\3^^^^96 (Gupta et al., 16 Apr 2026). This is a numerical interpretation of a proxy model rather than a theorem, but it directly addresses a central question in the theory of structural glasses.

Several open problems are explicit in the rigorous literature. These include extension from ^^^^3all:\3^^^^-RSB to full PRESERVED_PLACEHOLDER_^^^^3all:\3^^^^97-RSB, treatment of the random PRESERVED_PLACEHOLDER_^^^^3all:\3^^^^98-SAT model, identification of the exact ^^^^3all:\3^^^^-RSB region, and proving absence of condensation transitions beyond one step (&&&^^^^3search_query3^^^^&&&). Another unresolved issue is how the rigorous RS regimes of subcritical or sufficiently high-temperature dilution interface with the non-RS phases observed numerically in long-range diluted proxies and with the universal disorder-chaos behavior on supercritical sparse hypergraphs. The current body of work shows that diluted PRESERVED_PLACEHOLDER_^^^^3all:\3^^^^99-spin glasses do not admit a single universal phase description: replica symmetry is exact in identifiable regimes, ^^^^3all:\3^^^^-RSB has a rigorous asymptotic structure when it occurs, FRSB-like behavior can dominate accessible numerical regimes, and sparse geometry produces strong sensitivity to perturbations of the disorder.