Critical values in an anisotropic percolation model Hao Xue EPF - - PowerPoint PPT Presentation

Critical values in an anisotropic percolation model Hao Xue EPF - Lausanne (Based on joint work with T. Mountford and M.E. Vares) 23rd EBP - July 2019 1 / 11

Overview Basic model on Z 1+1 Supercritical/ higher dimensions Z d +1 (on going) 2 / 11

Related works A model investigated by Fontes, Marchetti, Merola, Presutti, Vares: ± 1 spins σ ( x , i ) , x ∈ Z , i ∈ Z s.t. 3 / 11

Related works A model investigated by Fontes, Marchetti, Merola, Presutti, Vares: ± 1 spins σ ( x , i ) , x ∈ Z , i ∈ Z s.t. ◮ Horizontal interaction with strength γ and range 1 /γ : On i -th horizontal level, it follows a Kac potential − 1 � 2 J γ ( x , y ) σ ( x , i ) σ ( y , i ) , J γ ( x , y ) = 1 , y � = x where J γ = c γ γ J ( γ ( x − y )) and J ( r ) , r ∈ R is smooth and symmetric � J ( r ) dr = 1, c γ is the with support in [ − 1 , 1] , J (0) > 0 , normalization constant that tends to 1 as γ → 0. 3 / 11

Related works A model investigated by Fontes, Marchetti, Merola, Presutti, Vares: ± 1 spins σ ( x , i ) , x ∈ Z , i ∈ Z s.t. ◮ Horizontal interaction with strength γ and range 1 /γ : On i -th horizontal level, it follows a Kac potential − 1 � 2 J γ ( x , y ) σ ( x , i ) σ ( y , i ) , J γ ( x , y ) = 1 , y � = x where J γ = c γ γ J ( γ ( x − y )) and J ( r ) , r ∈ R is smooth and symmetric � J ( r ) dr = 1, c γ is the with support in [ − 1 , 1] , J (0) > 0 , normalization constant that tends to 1 as γ → 0. ◮ Vertical interaction between nearest neighbours with strength ǫ ( γ ). − ǫσ ( x , i ) σ ( x , i + 1) . 3 / 11

Related works (cont’d) Question : How small ǫ is to still observe a phase transition for the anisotropic Ising model (for all γ small)? 4 / 11

Related works (cont’d) Question : How small ǫ is to still observe a phase transition for the anisotropic Ising model (for all γ small)? Conjecture ǫ ( γ ) = κγ 2 / 3 . 4 / 11

Related works (cont’d) Question : How small ǫ is to still observe a phase transition for the anisotropic Ising model (for all γ small)? Conjecture ǫ ( γ ) = κγ 2 / 3 . Ising ⇒ FK percolation with q = 2: p ( � v 1 , v 2 � ) = 1 − e − J γ ( x , y ) 1 � v 1 , v 2 �∈ E h − e − 2 ǫ ( γ ) 1 � v 1 , v 2 �∈ E v . FK measure: 1 Z ( G , p , q ) p | ω | (1 − p ) | E \ ω | q k ( ω ) . P ( ω ) = 4 / 11

Basic model Graph Z 2 = ( V , E ), V = { ( x , i ) : x ∈ Z , i ∈ Z } , E = E h ∪ E v , s.t. 5 / 11

Basic model Graph Z 2 = ( V , E ), V = { ( x , i ) : x ∈ Z , i ∈ Z } , E = E h ∪ E v , s.t. ◮ Horizontal edges E h = { e = � v 1 , v 2 � : 1 ≤ | x 1 − x 2 | ≤ N , i 1 = i 2 } . λ Open probability 2 N ; critical case: λ = 1. 5 / 11

Basic model Graph Z 2 = ( V , E ), V = { ( x , i ) : x ∈ Z , i ∈ Z } , E = E h ∪ E v , s.t. ◮ Horizontal edges E h = { e = � v 1 , v 2 � : 1 ≤ | x 1 − x 2 | ≤ N , i 1 = i 2 } . λ Open probability 2 N ; critical case: λ = 1. ◮ Vertical edges E v = { e = � v 1 , v 2 � : x 1 = x 2 , | i 1 − i 2 | = 1 } . Open probability ǫ ( N ) = κ N − b . 5 / 11

Basic model Graph Z 2 = ( V , E ), V = { ( x , i ) : x ∈ Z , i ∈ Z } , E = E h ∪ E v , s.t. ◮ Horizontal edges E h = { e = � v 1 , v 2 � : 1 ≤ | x 1 − x 2 | ≤ N , i 1 = i 2 } . λ Open probability 2 N ; critical case: λ = 1. ◮ Vertical edges E v = { e = � v 1 , v 2 � : x 1 = x 2 , | i 1 − i 2 | = 1 } . Open probability ǫ ( N ) = κ N − b . ◮ Initial open sites at layer 0: 2 N 2 α on {− N 1+ α , · · · , 0 , · · · , N 1+ α } with distance N 1 − α 5 / 11

Picutre Figure: Anisotropic percolation on Z 2 6 / 11

Horizontal Rescale the horizontal space by N 1+ α and its horizontal time step by N 2 α 7 / 11

Horizontal Rescale the horizontal space by N 1+ α and its horizontal time step by N 2 α ξ n ( · ) : Z / N 1+ α → { 0 , 1 } . ◮ ˆ 7 / 11

Horizontal Rescale the horizontal space by N 1+ α and its horizontal time step by N 2 α ξ n ( · ) : Z / N 1+ α → { 0 , 1 } . ◮ ˆ ◮ � ξ j ( x ) = 0 and � N k ( x ) if � j ≤ k ˆ w =1 η w 1 k +1 ≥ 1 ˆ ξ k +1 ( x ) = 0 otherwise , i.i.d. where N k ( x ) = � y ∼ x ˆ ξ k ( y ) , η w ∼ Bernoulli(1 / 2 N ). k +1 7 / 11

Horizontal Rescale the horizontal space by N 1+ α and its horizontal time step by N 2 α ξ n ( · ) : Z / N 1+ α → { 0 , 1 } . ◮ ˆ ◮ � ξ j ( x ) = 0 and � N k ( x ) if � j ≤ k ˆ w =1 η w 1 k +1 ≥ 1 ˆ ξ k +1 ( x ) = 0 otherwise , i.i.d. where N k ( x ) = � y ∼ x ˆ ξ k ( y ) , η w ∼ Bernoulli(1 / 2 N ). k +1 � ◮ Approximation of density ( A ˆ 1 y ∼ x ˆ ξ )( x ) = ξ ( y ). 2 N α 7 / 11

Horizontal Rescale the horizontal space by N 1+ α and its horizontal time step by N 2 α ξ n ( · ) : Z / N 1+ α → { 0 , 1 } . ◮ ˆ ◮ � ξ j ( x ) = 0 and � N k ( x ) if � j ≤ k ˆ w =1 η w 1 k +1 ≥ 1 ˆ ξ k +1 ( x ) = 0 otherwise , i.i.d. where N k ( x ) = � y ∼ x ˆ ξ k ( y ) , η w ∼ Bernoulli(1 / 2 N ). k +1 � ◮ Approximation of density ( A ˆ 1 y ∼ x ˆ ξ )( x ) = ξ ( y ). 2 N α ◮ ˆ ξ k +1 ( x ) = ξ k ( x ) + Laplacian + martingale − 1 � � ˆ ˆ ˆ ξ k ( y ) ξ j ( x ) + Error . 2 N y ∼ x j ≤ k � �� � converges when α =1 / 5 7 / 11

Heuristics ξ n ( · ) : Z / N 1+ α → { 0 , 1 } , ( A ˆ ◮ ˆ 1 � y ∼ x ˆ ξ )( x ) = ξ ( y ) 2 N α 8 / 11

Heuristics ξ n ( · ) : Z / N 1+ α → { 0 , 1 } , ( A ˆ ◮ ˆ 1 � y ∼ x ˆ ξ )( x ) = ξ ( y ) 2 N α ξ 0 ) = 1 [ − 1 , 1] , N α − 1 on each site and total number is N 2 α ◮ Initially, A (ˆ 8 / 11

Heuristics ξ n ( · ) : Z / N 1+ α → { 0 , 1 } , ( A ˆ ◮ ˆ 1 � y ∼ x ˆ ξ )( x ) = ξ ( y ) 2 N α ξ 0 ) = 1 [ − 1 , 1] , N α − 1 on each site and total number is N 2 α ◮ Initially, A (ˆ ◮ After N 2 α steps, dying chance is N 3 α − 1 8 / 11

Heuristics ξ n ( · ) : Z / N 1+ α → { 0 , 1 } , ( A ˆ ◮ ˆ 1 � y ∼ x ˆ ξ )( x ) = ξ ( y ) 2 N α ξ 0 ) = 1 [ − 1 , 1] , N α − 1 on each site and total number is N 2 α ◮ Initially, A (ˆ ◮ After N 2 α steps, dying chance is N 3 α − 1 ◮ Total attrition is N 5 α − 1 8 / 11

Main result ◮ C i : cluster at layer i 9 / 11

Main result ◮ C i : cluster at layer i ◮ |C i | ≈ N 2 α 9 / 11

Main result ◮ C i : cluster at layer i ◮ |C i | ≈ N 2 α Theorem 1 (Mountford, Vares, X.). The critical value is b = 2 / 5 ( ǫ ( N ) = κ N − b ): there exist positive constants C 1 and C 2 such that for κ < C 1 , there is no percolation and for κ > C 2 , the percolation appears. 9 / 11

Supercritical/ higher dimensions cases (on going) ◮ When λ > 1 , d = 1 (horizontal open probability is λ/ 2 N ): the critical vertical interaction is ǫ ( N ) = e − κ N . 10 / 11

Supercritical/ higher dimensions cases (on going) ◮ When λ > 1 , d = 1 (horizontal open probability is λ/ 2 N ): the critical vertical interaction is ǫ ( N ) = e − κ N . ◮ When λ > 1 , d > 1, there always exists a percolation. ◮ What is the probability of a large but finite size cluster? 10 / 11

Thanks! 11 / 11