Matching Tag: noise-stability
Let n ≥ 2 n\geq2 n ≥ 2 ρ ∈ [ − 1 , 1 ] \rho\in[-1,1] ρ ∈ [ − 1 , 1 ] 3 ≤ k ≤ n + 1 3\leq k\leq n+1 3 ≤ k ≤ n + 1 { A i } i = 1 k \{A_i\}_{i=1}^k { A i } i = 1 k R n \mathbb{R}^n R n ⋃ i = 1 k A i = R n \bigcup_{i=1}^k A_i=\mathbb{R}^n ⋃ i = 1 k A i = R n
Let m ≥ 3 m\geq 3 m ≥ 3 0 < ρ < 1 0<\rho<1 0 < ρ < 1 Ω 1 , … , Ω m \Omega_1,\ldots,\Omega_m Ω 1 , … , Ω m Ω 1 ′ , … , Ω m ′ \Omega'_1,\ldots,\Omega'_m Ω 1 ′ , … , Ω m ′ R n + 1 \mathbb{R}^{n+1} R n + 1
Let m ≥ 2 m\geq 2 m ≥ 2 ρ ∈ [ 0 , 1 ] \rho\in[0,1] ρ ∈ [ 0 , 1 ] ε > 0 \varepsilon>0 ε > 0 f : { 1 , … , m } n → Δ m f:\{1,\ldots,m\}^n\to\Delta_m f : { 1 , … , m } n → Δ m Inf i ( f j ) \operatorname{Inf}_i(f_j) Inf i ( f j ) i i i
Plurality is Stablest conjecture. If ρ ≥ 0 \rho\geq0 ρ ≥ 0 E f = 1 k ∑ j = 1 k e j \mathbb{E}f=\frac1k\sum_{j=1}^{k}e_j E f = k 1 ∑ j = 1 k e j
Let 0 ≤ ρ ≤ 1 0\leq\rho\leq1 0 ≤ ρ ≤ 1 q ≥ 2 q\geq2 q ≥ 2 f : [ q ] n → [ 0 , 1 ] f:[q]^n\to[0,1] f : [ q ] n → [ 0 , 1 ] S ρ ( f ) \mathbb{S}_\rho(f) S ρ ( f ) I n f i ( f ) \mathrm{Inf}_i(f) Inf i ( f )
For q ≥ 1 q\geq 1 q ≥ 1 ln q : ( 0 , + ∞ ) → R \ln_q:(0,+\infty)\to\mathbb{R} ln q : ( 0 , + ∞ ) → R q q q Φ q s y m ( t ) = t ln q ( t ) + ( 1 − t ) ln q ( 1 − t ) \Phi_q^{\mathrm{sym}}(t)=t\ln_q(t)+(1-t)\ln_q(1-t) Φ q sym ( t ) = t ln q ( t ) + ( 1 − t ) ln q ( 1 − t ) q q q
Vector-valued Borell inequality. If 0 < ρ < 1 0<\rho<1 0 < ρ < 1
Let ρ ∈ ( 0 , 1 ) \rho\in(0,1) ρ ∈ ( 0 , 1 ) n ≥ k ! + k − 1 n\geq k!+k-1 n ≥ k ! + k − 1 Ω 1 , … , Ω k \Omega_1,\ldots,\Omega_k Ω 1 , … , Ω k R n \mathbb{R}^n R n γ n ( Ω i ) = 1 / k \gamma_n(\Omega_i)=1/k γ n ( Ω i ) = 1/ k
Let S k S_k S k k k k f : S k n → Δ k f\colon S_k^n\to\Delta_k f : S k n → Δ k Δ k \Delta_k Δ k
Let n n n k ≥ 3 k\geq 3 k ≥ 3
Let C n = { − 1 , 1 } n \mathcal{C}_n=\{-1,1\}^n C n = { − 1 , 1 } n X X X C n \mathcal{C}_n C n N X , ρ \mathcal{N}_{X,\rho} N X , ρ
Let ρ \rho ρ 0 ≤ ρ ≤ 1 0\leq\rho\leq1 0 ≤ ρ ≤ 1 b 1 ∈ [ 1 , 9 ] b1\in[1,9] b 1 ∈ [ 1 , 9 ] M a x S t a b α ( 1 / 2 ) \mathbf{MaxStab}_{\alpha}(1/2) MaxStab α ( 1/2 ) b 1 b1 b 1 1 / 2 1/2 1/2
Let m ≥ 3 m\geq3 m ≥ 3 0 < ρ < 1 0<\rho<1 0 < ρ < 1 ( Ω i ) i = 1 m (\Omega_i)_{i=1}^m ( Ω i ) i = 1 m ( Ω i ′ ) i = 1 m (\Omega_i')_{i=1}^m ( Ω i ′ ) i = 1 m
Let f : { − 1 , 1 } n → { − 1 , 1 } f\colon\{-1,1\}^{n}\rightarrow\{-1,1\} f : { − 1 , 1 } n → { − 1 , 1 } n n n ρ ∈ [ 0 , 1 ] \rho\in[0,1] ρ ∈ [ 0 , 1 ] Stab ρ [ f ] \operatorname{Stab}_{\rho}[f] Stab ρ [ f ] f f f
Quadratic Symmetric Gaussian Problem. Either B ( 0 , r a ) B(0,r_a) B ( 0 , r a ) B ( 0 , r a ′ ) c B(0,r_a')^c B ( 0 , r a ′ ) c
Symmetric Gaussian Problem. If ρ > 0 \rho>0 ρ > 0 ( B ( 0 , r a ) , B ( 0 , r b ) c ) (B(0,r_a),B(0,r_b)^c) ( B ( 0 , r a ) , B ( 0 , r b ) c ) ( B ( 0 , r a ′ ) c , B ( 0 , r b ′ ) ) (B(0,r_a')^c,B(0,r_b')) ( B ( 0 , r a ′ ) c , B ( 0 , r b ′ ))
The perturbed-metric conjecture. For every 0 < s < 1 0<s<1 0 < s < 1 C s , c s > 0 C_s,c_s>0 C s , c s > 0 ε ( A ) < e − 1 / ρ \varepsilon(A)<e^{-1/\rho} ε ( A ) < e − 1/ ρ
