Non-Shattering at and Above the Dynamical Temperature in the Spherical Pure p-Spin Model

• URL: http://arxiv.org/abs/2608.14369v2
• Date: Fri, 21 Aug 2026 04:44:04 GMT
• Title: Non-Shattering at and Above the Dynamical Temperature in the Spherical Pure p-Spin Model
• Abstract summary: We consider the notion of shattering introduced by Ben Arous and Jagannath for spherical pure $p$-spin glasses with overlap $q$.<n>For every $pgeq 3$ and $0leq_mathrmsh(p)$, we rule out shattering whenever $qleq2-1/2$ or $q>sqrt(p-2)/(p-1)$.
• Score: 0.22843885788439797
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: We consider the notion of shattering introduced by Ben Arous and Jagannath for spherical pure $p$-spin glasses with overlap $q$. For every $p\geq 3$ and $0<β\leqβ_{\mathrm{sh}}(p)$, we rule out shattering whenever $q\leq2^{-1/2}$ or $q>\sqrt{(p-2)/(p-1)}$. The proof combines a deterministic $N+1$ bound for disjoint bands in the first range with a general-$p$ sign law showing that their total marked weight has subdominant free energy in the second. A spherical-code bound and Hölder's inequality give an additional $q$-dependent obstruction; in particular, they rule out every fixed overlap for $0<β\leq\sqrt{\log2}$. For $p=3$, the first two ranges already exhaust every fixed $q\in(0,1)$, so the landscape is not shattered at any $T\geq T_{\mathrm{sh}}$. For $p\geq4$, the cases not covered by our criteria are confined to $2^{-1/2}<q\leq\sqrt{(p-2)/(p-1)}$ and $\sqrt{\log2}<β\leqβ_{\mathrm{sh}}(p)$. In particular, this paper partially resolves Conjecture 1 of the paper above and also suggests new methods to show non-shattering.

Related papers

• Sharp Minimax Regret for Infinite-Memory Logistic Prediction [55.29259818039367]
  Lag $j$ can affect prediction by scale $r_j$ and enters only $n_T,j=T-j+1$ prediction rounds.<n>For every summable envelope, a localized mixture proves $cR_T(r)leq C_T(r)$.<n>For exponential and envelopes, under the stated finite-sample condition, a Toeplitz-design converse proves $cR_T(r)geq c_T(r)$.
  arXiv  Detail & Related papers  (2026-08-27T01:31:46Z)
• Isospectral majorization and isoperimetric inequalities for coherent states on the Bloch sphere [0.0]
  We prove an isospectral version of Lieb-Solovej inequality.<n>Spherical caps maximize every partial sum of the eigenvalues of Toeplitz operators with symbol $1_$.
  arXiv  Detail & Related papers  (2026-08-12T16:46:26Z)
• The Condition-Number Barrier in Sparse Least Squares [77.64108812086542]
  In [AS21], Axiotis and Sviridenko conjectured that the linear dependence on the restricted condition number in convex optimization cannot be improved by a sparse-time algorithm.<n>We establish their conjectured lower bound for least-squares objectives, conditional on the randomized exact-volume Small-Set Expansion Hypothesis.
  arXiv  Detail & Related papers  (2026-08-03T17:57:01Z)
• Random unitary circuits with constant spectral gap [0.34376560669160394]
  We prove constant lower bounds for the spectral gap of the following random walks on unitary groups $mathsfSU(2n)$ on $n$ qubits.<n> Importantly, the spectral gaps are independent of $n$ and apply for all finite dimensional unitary representations of $mathsfSU(2n)$ uniformly.
  arXiv  Detail & Related papers  (2026-07-23T04:56:09Z)
• Shallow ReLU$^s$ Networks in $L^p$-Type and Sobolev Spaces: Approximation and Path-Norm Controlled Generalization [12.871748162999062]
  In particular, when $_d$ is the uniform measure and $1le p2$, the approximation rate is $O!left(m-fracp(2s+2d+1)-2d2dpright)$ for $1le p*$ and $O!left(m-fracp(4s+3d-1)-2d+24dpright)$ for $p*p2$, where $p*=frac2d+2d+3$.<n>For non
  arXiv  Detail & Related papers  (2026-05-18T14:27:49Z)
• Tripartite information of free fermions: a universal entanglement coefficient from the sine kernel [51.56484100374058]
  We study the tripartite information I_3 of free fermions on two-dimensional lattices partitioned into three adjacent strips of width w.<n>We find that g(z) has a unique zero at z* = 1.329 +/- 0.001: modes with k_F w z* violate monogamy of mutual information.<n>Two exact cancellations -- of the z ln z area-law terms and of the z2 terms -- are intrinsic to the I_3 combination.
  arXiv  Detail & Related papers  (2026-03-03T15:39:35Z)
• Approximating the operator norm of local Hamiltonians via few quantum states [53.16156504455106]
  Consider a Hermitian operator $A$ acting on a complex Hilbert space of $2n$.<n>We show that when $A$ has small degree in the Pauli expansion, or in other words, $A$ is a local $n$-qubit Hamiltonian.<n>We show that whenever $A$ is $d$-local, textiti.e., $deg(A)le d$, we have the following discretization-type inequality.
  arXiv  Detail & Related papers  (2025-09-15T14:26:11Z)
• Dimension Independent Disentanglers from Unentanglement and Applications [55.86191108738564]
  We construct a dimension-independent k-partite disentangler (like) channel from bipartite unentangled input. We show that to capture NEXP, it suffices to have unentangled proofs of the form $| psi rangle = sqrta | sqrt1-a | psi_+ rangle where $| psi_+ rangle has non-negative amplitudes.
  arXiv  Detail & Related papers  (2024-02-23T12:22:03Z)
• Constrained HRT Surfaces and their Entropic Interpretation [0.0]
  We study the R'enyi entropies of subregion $A$ in a fixed-area state of subregion $B$. Our results are relevant to some of the issues associated with attempts to use standard random networks to describe time dependent geometries.
  arXiv  Detail & Related papers  (2023-11-30T06:50:40Z)
• Detection of Dense Subhypergraphs by Low-Degree Polynomials [72.4451045270967]
  Detection of a planted dense subgraph in a random graph is a fundamental statistical and computational problem. We consider detecting the presence of a planted $Gr(ngamma, n-alpha)$ subhypergraph in a $Gr(n, n-beta) hypergraph. Our results are already new in the graph case $r=2$, as we consider the subtle log-density regime where hardness based on average-case reductions is not known.
  arXiv  Detail & Related papers  (2023-04-17T10:38:08Z)
• The planted matching problem: Sharp threshold and infinite-order phase transition [25.41713098167692]
  We study the problem of reconstructing a perfect matching $M*$ hidden in a randomly weighted $ntimes n$ bipartite graph. We show that if $sqrtd B(mathcalP,mathcalQ) ge 1+epsilon$ for an arbitrarily small constant $epsilon>0$, the reconstruction error for any estimator is shown to be bounded away from $0$.
  arXiv  Detail & Related papers  (2021-03-17T00:59:33Z)
• Curse of Dimensionality on Randomized Smoothing for Certifiable Robustness [151.67113334248464]
  We show that extending the smoothing technique to defend against other attack models can be challenging. We present experimental results on CIFAR to validate our theory.
  arXiv  Detail & Related papers  (2020-02-08T22:02:14Z)

This list is automatically generated from the titles and abstracts of the papers in this site.

This site does not guarantee the quality of this site (including all information) and is not responsible for any consequences.