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Talagrand's Conjecture 11.7.5 is resolved affirmatively for the critical Ising Sherrington-Kirkpatrick model: N2/3E⟨R1,22⟩ converges to a positive finite constant. The paper proves substantially more, showing that the entire quenched distribution of N1/3R1,2 converges to an explicit random probability measure defined from the reflected Airy1 point process. The same…
Closes both gaps left open by Bandeira and Maillard: exact fitting, and removal of the operator-norm constraint. The threshold turns out to be governed by the statistical dimension d(d+1)/4 of the PSD cone.
One variant case of Arnold's 1994 fast-dynamo problem, not the problem itself. Arnold asks for a single velocity field on T^3 that is smooth, divergence-free, autonomous and deterministic, fixed independently of the magnetic diffusivity, and that grows the magnetic field exponentially at every small enough diffusivity. The field constructed here is all of that except smooth: it is Lipschitz, not C^1. The sibling ent…
For every n>2, the paper constructs a bounded map U∈W1,n(Bn,Rn+2), smooth on Bn∖{0} but discontinuous at the origin, together with an antisymmetric potential
Ω∈Ln(Bn,so(n+2)⊗Rn)
such that
−Div(∣∇U∣n−2∇U)=Ω⋅∣∇U∣n−2∇Uin D′(Bn).
Moreover, the potential satisfies the sh…
For fixed δ∈(0,1) and all sufficiently large Δ depending only on δ, Glauber dynamics for proper q-colorings mixes rapidly on every graph of girth at least 5 whenever q≥(1+δ)Δ: spectral gap Ωδ(1/n) and tmix(ε)=Oδ(n2logq+nlog(1/ε)). An analogous theorem holds for the anti-ferromagnetic Potts model at…
Refuting log-concavity of the flat counts is weaker than refuting their unimodality, since log-concavity is the stronger property. A counterexample to unimodality followed three weeks later and is tracked separately as Rota's Unimodality Conjecture for Matroid Flats; this paper came first.
A dense case, not the conjecture: it remains open in general. The concrete gain is on the size of any counterexample - combined with the known minimum-outdegree results, this raises the best known lower bound on the order of a counterexample from 16 to 17, and to 19 conditional on the 2026 preprint of Sadhukhan, Sandeep and Sen. The novelty against the earlier dense-case work is that no structure is prescribed on th…
Constructed an explicit class of smooth random, time-dependent incompressible velocity fields on T^3, obtained by alternating smooth shear flows with iid random phases on finite time blocks. For every fixed sufficiently small resistivity, the magnetic field has an almost-sure exponential growth rate at least 1/2, together with a time-uniform lower bound whose random prefactor has a resistivity-uniform inverse-moment…
For every 0≤β≤21, Bondarenko and Seip construct a nonzero real-valued continuous even function
fβ∈L1(R)∩L2(R)
such that
fβ=fβ
and
fβ([log(e+n)]βn)=0(n≥0).
They normalize the construction by requiring fβ(1/2)=1, so the function is genuinely nontrivial.
For β=0, this gives…
VibeMathed recorded: The Kára–Pór–Wood Big-Line-Big-Clique Conjecture: Four Collinear Points or a Six-Clique. Consult the linked registry entry for the original report and scope.
For the Ising pure p-spin glass where p≥3, it was predicted by Gardner that there exists critical inverse temperatures 0<β1p<β2p<∞ such that: (1) When 0<β≤β1p, the Parisi measure is replica symmetric (RS); (2) When β1p<β≤β2p, the Parisi measure is one-step replica symmetry breaking (1-RSB); (3) When β>βp2, the Parisi measure is full rep…
VibeMathed recorded: The Optimal Approximation Ratio for Permanents of PSD Matrices. Consult the linked registry entry for the original report and scope.
If a bilinear form admits an (r,s)-sparse bound, its coordinate-wise extension to Cn-valued functions admits an (r,s)-convex body sparse bound, for 1≤r,s<∞ with r1+s1>1. It holds both in a fixed dyadic lattice (Theorem 2.6, constants independent of the ambient dimension) and for arbitrary cubes (Theorem 3.4), via a randomization of the good part of the form.
On scope, two thin…
The paper constructs a simple theory in which forking cannot always be witnessed by a stable formula. Precisely, what fails is: in a simple theory, if a↓Cb then there is φ(x,bˉ)∈tp(a/Cb) forking over C whose parameter-free form φ(x,y) is stable.
The counterexample is an infinite-dimensional vector space over the division ring of fractions of the quantum graph alg…
If Γ is a non-coHopfian property-(T) group and G is its ascending HNN extension, then the full group algebra C∗(G) is not finite. Indeed, Kazhdan projections pH<pΓ become unitarily equivalent under the stable letter, which cannot occur inside a finite C∗-algebra.
For
W=G/Γ⨁Z/2Z⋊G,
the paper proves more strongly that every homomorphism…
Let P⊆Qn be convex. Williams proves the stronger statement that for every k≥0,
w(P×Qk)≥w(Qn+k)∣P∣2−n,
where w denotes poset width.
Taking k=0 gives
w(P)≥∣P∣(⌊n/2⌋n)2−n,
which is exactly the Daykin-Frankl conjecture.
The proof proceeds by induction on n, reducing the step to a structural lemma for a convex subset of R×Q1 and…
The authors construct smooth Am and smooth solutions um in B2⊂R3 with one fixed ellipticity bound
I≤Am≤281I,
common boundary data and ∥um∥L∞(B2)≤1, but
∥Dum∥L1(B1)→∞.
Thus no interior W1,1 estimate can depend only on dimension and ellipticity. Consequently, no such W1,p estimate exists for any p≥1.
They further obtain a u…
For every neighborhood U of the origin in C×R and every ε>0, Curry constructs a smooth nonnegative perturbation ϕ, supported in U with ∥ϕ∥C2<ε, such that
Tϕ0,1=spanC{Lϕ}
is strongly pseudoconvex, its canonical bundle admits a nowhere-zero closed section, yet every C1 CR function near the origin satisfie…