The phase transition for random loop models on trees Volker Betz TU Darmstadt Venice, 21 August 2019 Joint work with Johannes Ehlert, Benjamin Lees, Lukas Roth
The random loop model: intuition V. Betz (Darmstadt) Loops on trees
The random loop model: definition ◮ G = ( V, E ) a graph. Parameters β, u . ◮ ( X \ / e ) e ∈ E iid PPP with intensity u on [0 , β ) (’crosses’). ◮ ( X || e ) e ∈ E iid PPP with intensity 1 − u on [0 , β ) (’bars’). ◮ T β torus, X = { ( v, t ) : v ∈ V, t ∈ T β } . e : v ∈ e X \ / e ∪ X || ◮ The set � e separates { ( v, t ) : t ∈ T β } into disjoint open intervals. U ( v, t ) is the interval containing t . ◮ Connections: ( v, t ) ∼ ( v ′ , t ′ ) if ◮ v = v ′ and t ′ ∈ U ( v, t ) , or ◮ e := { v, v ′ } ∈ E , and there is precisely one element of X \ / e e ) between t and t ′ (considering periodicity) or (and none of X || ◮ e := { v, v ′ } ∈ E , U ( v, t ) ∩ U ( v ′ , t ′ ) � = ∅ and has at least one boundary point in X || e . ◮ Extend by transitivity. ◮ Percolation type model. Question: infinite cluster? V. Betz (Darmstadt) Loops on trees
The lack of monotonicity The main difficulty: adding connections can decrease the size of a connected component. Two mechanisms: 1. More than one connection per edge. V. Betz (Darmstadt) Loops on trees
The lack of monotonicity The main difficulty: adding connections can decrease the size of a connected component. Two mechanisms: 1. More than one connection per edge. 2. Loops in the underlying graph. V. Betz (Darmstadt) Loops on trees
The lack of monotonicity The main difficulty: adding connections can decrease the size of a connected component. Two mechanisms: 1. More than one connection per edge. 2. Loops in the underlying graph. We are only able to address problem 1. V. Betz (Darmstadt) Loops on trees
Random loop model and quantum theory ◮ Let P u,β be the joint measure of the PPP ( X \ e ) e ∈ E , ( X || / e ) e ∈ E . ◮ For θ > 0 , G finite let P θ,u,β ( A ) = E u,β ( θ L ) E u,β ( θ L 1 A ) , where 1 ◮ L ( ω ) is the total number of loops in the configuration produced by the X \ /, || e ( ω ) . ◮ Relevant quantum system has Hamiltonian � S (1) x S (1) + S (2) x S (2) + (2 u − 1) S (3) x S (3) H = − 2 y . y y { x,y }∈ E ◮ Heisenberg ferromagnet ( u = 1 ), anti-ferromagnet ( u = 0 ) or XY -model ( u = 1 / 2 ). ◮ Example for connection to random loop models: y � β ≡ tr(S (1) x S (1) y e − β H ) � S (1) x S (1) = P 2 ,u,β ( x ↔ y ) . tr e − β H V. Betz (Darmstadt) Loops on trees
History of the random loop model Case θ = 2 : ◮ Feynman 1953: basic idea to treat thermal states using functional integrals. ◮ Conlon and Solovej 1991: random walk representation for the ferromagnet. ◮ Toth 1993 improves this result using a random loop model. ◮ Aizenman and Nachtergaele 1994: extension to more general spin values and interactions. ◮ Ueltschi 2013: extension to general θ and all u . V. Betz (Darmstadt) Loops on trees
History of the random loop model Case θ = 2 : ◮ Feynman 1953: basic idea to treat thermal states using functional integrals. ◮ Conlon and Solovej 1991: random walk representation for the ferromagnet. ◮ Toth 1993 improves this result using a random loop model. ◮ Aizenman and Nachtergaele 1994: extension to more general spin values and interactions. ◮ Ueltschi 2013: extension to general θ and all u . Case θ = 1 : ◮ Harris 1972: random stirring model. ◮ Schramm 2005: emergence of infinite cycles for the complete graph. ◮ Berestycki (2011), Berestycki and Kozma (2015): extensions and simplifications of Schramms results. ◮ Kotecky, Milos, Uelschi (2016), results on the hypercube. V. Betz (Darmstadt) Loops on trees
Random loop models on trees ◮ Angel 2003: proof of existence of long loops for d � 4 , θ = 1 . ◮ Hammond 2013: sharp phase transition in β for d � 55 , for u = 1 , θ = 1 . ◮ Hammond 2015: strict bounds on β c for very high d . ◮ Hammond and Hedge 2018: improved those bounds to d � 56 and general u . ◮ Bj¨ ornberg, Ueltschi 2018, 2019: Asymptotics for large d , and for all θ, u : d + 1 − θu (1 − u ) − θ 2 (1 − u ) 2 / 6 β c θ = 1 + o ( d − 2 ) . d 2 ◮ Topic of this talk: proof of sharp phase transition for θ = 1 and all d � 3 , and (in principle) full asymptotic expansion of β c . V. Betz (Darmstadt) Loops on trees
Main Theorem Theorem: T the infinite d -regular, r its root, γ T the loop ( r, 0) . For all d � 3 , u ∈ [0 , 1] there exists β c > 0 and β + > β c such that 1. γ T is finite almost surely for all β � β c , 2. γ T is infinite with positive probability for all β ∈ ( β c , β + ) . d for all d � 3 and β + = ∞ for d � 16 . Moreover, β + � 1 √ V. Betz (Darmstadt) Loops on trees
Main Theorem Theorem: T the infinite d -regular, r its root, γ T the loop ( r, 0) . For all d � 3 , u ∈ [0 , 1] there exists β c > 0 and β + > β c such that 1. γ T is finite almost surely for all β � β c , 2. γ T is infinite with positive probability for all β ∈ ( β c , β + ) . d for all d � 3 and β + = ∞ for d � 16 . Moreover, β + � 1 √ We have the expansion n α k ( u ) � d k +1 + O ( d − n − 2 ) , β c = (1) k =0 where the α k are polynomials of order 2 k in u with recursively computable coefficients. V. Betz (Darmstadt) Loops on trees
Main idea of the proof: Galton Watson trees ◮ Let C 1 be the (random) maximal subtree of T containing the root and where each edge has at least two links. Percolation on trees: C is finite almost surely if β 2 � 1 /d . V. Betz (Darmstadt) Loops on trees
Main idea of the proof: Galton Watson trees ◮ Let C 1 be the (random) maximal subtree of T containing the root and where each edge has at least two links. Percolation on trees: C is finite almost surely if β 2 � 1 /d . ◮ Consider e = { x, y } with x ∈ C 1 , y / ∈ C 1 . ◮ Assume that γ T leaves C 1 through e . Then V. Betz (Darmstadt) Loops on trees
Main idea of the proof: Galton Watson trees ◮ Let C 1 be the (random) maximal subtree of T containing the root and where each edge has at least two links. Percolation on trees: C is finite almost surely if β 2 � 1 /d . ◮ Consider e = { x, y } with x ∈ C 1 , y / ∈ C 1 . ◮ Assume that γ T leaves C 1 through e . Then ◮ either it also leaves the next (iid) subtree C 2 attached at y ◮ or it returns by the same way into C 1 , i.e. using e in the opposite direction. γ T in both directions or not at all. ◮ So, e serves as a renewal edge, separating future and past. ◮ Let M i be the ’living’ renewal edges in the i -th generation. ( | M i | ) i ∈ N is a Galton-Watson process. V. Betz (Darmstadt) Loops on trees
Main idea of the proof: Galton Watson trees ◮ Let C 1 be the (random) maximal subtree of T containing the root and where each edge has at least two links. Percolation on trees: C is finite almost surely if β 2 � 1 /d . ◮ Consider e = { x, y } with x ∈ C 1 , y / ∈ C 1 . ◮ Assume that γ T leaves C 1 through e . Then ◮ either it also leaves the next (iid) subtree C 2 attached at y ◮ or it returns by the same way into C 1 , i.e. using e in the opposite direction. γ T in both directions or not at all. ◮ So, e serves as a renewal edge, separating future and past. ◮ Let M i be the ’living’ renewal edges in the i -th generation. ( | M i | ) i ∈ N is a Galton-Watson process. ◮ Therefore: γ T is finite almost surely if and only if E u,β ( M 1 ) � 1 . V. Betz (Darmstadt) Loops on trees
Computing E u,β ( M 1 ) Let C be the set of finite rooted subtrees of T , with edges e √ labelled by n e � 2 . In the case β < 1 / d we find � � � � E ( | M 1 | ) = | M 1 | � C 1 = C P ( C 1 = C ) . E C ∈C V. Betz (Darmstadt) Loops on trees
Computing E u,β ( M 1 ) Let C be the set of finite rooted subtrees of T , with edges e √ labelled by n e � 2 . In the case β < 1 / d we find � � � � E ( | M 1 | ) = | M 1 | � C 1 = C P ( C 1 = C ) . E C ∈C By independence, we have � � P ( C 1 = C ) = P ( | X e | = n e ) P ( | X e | � 1) e ∈ E ( C ) e ∈ ∂ + C β N ( C ) e ∈ E ( C ) n e ! e − βd | V ( C ) | (1 + β ) ( d − 1) | V ( C ) | +1 , = � β � � � and E | M 1 | � C 1 = C = � x ∈ V ( C ) ( d − d x ) p ( u, d ) where 1+ β p ( u, d ) is a polynomial in u . V. Betz (Darmstadt) Loops on trees
Uniqueness and sharpness of the phase transitions The end result is � f C ( βd, d − 1 ) g C ( d − 1 , u ) E β ( | M 1 | ) = C ∈C where (with N ( C ) the number of links on C ) α N ( C )+1 e − α (1 + αh ) 1 /h − 1 � | V ( C ) | ( N ( C ) + 1)! h N ( C ) − E ( C ) . � f C ( α, h ) = and g C ( h, u ) a polynomial, non-negative for all u and all h = 1 /d , d ∈ N . V. Betz (Darmstadt) Loops on trees
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