Matching Tag: random-polymers
Derrida–Griffiths–Higgs conjecture. The following assertions hold: (i) the specific heat, namely the derivative of the finite-volume free energy, converges P \mathbb P P
Fix t > 0 t>0 t > 0 Δ > 0 \Delta>0 Δ > 0 [ X , X ′ ] [X,X'] [ X , X ′ ] [ Y , Y ′ ] [Y,Y'] [ Y , Y ′ ] R \mathbb R R ( x 1 , … , x n ) (x_1,\dots,x_n) ( x 1 , … , x n ) ( y 1 , … , y n ) (y_1,\dots,y_n) ( y 1 , … , y n ) ( x ~ 1 , … , x ~ k ) (\tilde x_1,\dots,\tilde x_k) ( x ~ 1 , … , x ~ k )
Fix k ≥ 1 k\geq 1 k ≥ 1 c ∈ ( 0 , ∞ ) c\in(0,\infty) c ∈ ( 0 , ∞ ) α 1 , … , α k \alpha_1,\ldots,\alpha_k α 1 , … , α k F ( z 1 , … , z k ) F(z_1,\ldots,z_k) F ( z 1 , … , z k ) α k = 0 \alpha_k=0 α k = 0 α j > α j + 1 + c \alpha_j>\alpha_{j+1}+c α j > α j + 1 + c 1 ≤ j ≤ k − 1 1\leq j\leq k-1 1 ≤ j ≤ k − 1
Let α ∈ ( 0 , ∞ ) \alpha\in(0,\infty) α ∈ ( 0 , ∞ ) χ : = ∑ n ∈ N [ P ( S n = 0 ) ] 2 \chi:=\sum_{n\in\mathbb{N}}[\textbf{P}(S_n=0)]^2 χ := ∑ n ∈ N [ P ( S n = 0 ) ] 2 β c \beta_c β c
