Weakly self-avoiding walk in dimension four Gordon Slade University - PowerPoint PPT Presentation

Weakly self-avoiding walk in dimension four Gordon Slade University of British Columbia Mathematical Statistical Physics Kyoto: July 29, 2013 Abstract We report on recent and ongoing work on the continuous-time weakly self- avoiding walk on the 4-dimensional integer lattice, with focus on a proof that the susceptibility diverges at the critical point with a logarithmic correction to mean- field scaling. The proof is based on a rigorous renormalisation group analysis of a supersymmetric field theory representation of the weakly self-avoiding walk. The talk is based on collaborations with David Brydges, and with Roland Bauerschmidt and David Brydges. Research supported in part by NSERC.

Self-avoiding walk Discrete-time model: Let S n ( x ) be the set of ω : { 0 , 1 , . . . , n } → Z d with: ω (0) = 0 , ω ( n ) = x , | ω ( i + 1) − ω ( i ) | = 1 , and ω ( i ) ̸ = ω ( j ) for all i ̸ = j . Let S n = ∪ x ∈ Z d S n ( x ) . x c n ( x ) = |S n | . Easy: c 1 /n Let c n ( x ) = |S n ( x ) | . Let c n = ∑ → µ . n Declare all walks in S n to be equally likely: each has probability c − 1 n . Two-point function: G z ( x ) = ∑ ∞ n =0 c n ( x ) z n , radius of convergence z c = µ − 1 . Predicted asymptotic behaviour: E n | ω ( n ) | 2 ∼ Dn 2 ν , c n ∼ Aµ n n γ − 1 , G zc ( x ) ∼ C | x | − ( d − 2+ η ) , with universal critical exponents γ, ν, η obeying γ = (2 − η ) ν .

Dimensions other than d = 4 Theorem. (Brydges–Spencer (1985); Hara–Slade (1992); Hara (2008)...) For d ≥ 5 , 1 E n | ω ( n ) | 2 ∼ Dn, c n ∼ Aµ n , G zc ( x ) ∼ c | x | − ( d − 2) , √ ω ( ⌊ nt ⌋ ) ⇒ B t . Dn Proof uses lace expansion, requires d > 4 . d = 2 . Prediction: γ = 43 32 , ν = 3 5 4 , η = 24 , Nienhuis (1982); Lawler–Schramm-Werner (2004) — connection with SLE 8 / 3 . d = 3 . Numerical: γ ≈ 1 . 16 , ν ≈ 0 . 588 , η ≈ 0 . 031 . E.g., Clisby (2011): ν = 0 . 587597(7) . Theorem. (lower: Madras 2012, upper: Duminil-Copin–Hammond 2012) 6 n 4 / 3 d ≤ E n | ω ( n ) | 2 ≤ o ( n 2 ) , 1 so ν ≥ 2 / (3 d ) . Not proved for d = 2 , 3 , 4 : E n | ω ( n ) | 2 ≤ O ( n 2 − ϵ ) , i.e., that ν < 1 .

Predictions for d = 4 Prediction is that upper critical dimension is 4 , and asymptotic behaviour for Z 4 has log corrections (e.g., Br´ ezin, Le Guillou, Zinn-Justin 1973): E n | ω ( n ) | 2 ∼ Dn (log n ) 1 / 4 , c n ∼ Aµ n (log n ) 1 / 4 , G zc ( x ) ∼ c | x | − 2 . The susceptibility and correlation length are defined by: ∞ 1 1 ∑ c n z n , χ ( z ) = ξ ( z ) = − lim n log G z ( ne 1 ) . n →∞ n =0 For these the prediction is: χ ( z ) ∼ A ′ | log(1 − z/z c ) | 1 / 4 ξ ( z ) ∼ D ′ | log(1 − z/z c ) | 1 / 8 , as z ↑ z c . (1 − z/z c ) 1 / 2 1 − z/z c Universality hypothesis.

Continuous-time weakly self-avoiding walk A.k.a. discrete Edwards model. Let E 0 denote the expectation for continuous-time nearest-neighbour simple random walk X ( t ) on Z d started from 0 (steps at events of rate- 2 d Poisson process). ∫ T Let L u,T = 0 1 X ( s )= u ds and ∫ T ∫ T L 2 ∑ I ( T ) = 1 X ( s )= X ( t ) ds dt = u,T . 0 0 u ∈ Z d Let g ∈ (0 , ∞ ) , ν ∈ ( −∞ , ∞ ) . The two-point function is ∫ ∞ ( e − gI ( T ) 1 X ( T )= x ) e − νT dT G g,ν ( x ) = E 0 0 n c n ( x ) z n ). (compare ∑ Subadditivity ⇒ ∃ ν c ( g ) s.t. susceptibility χ g ( ν ) = ∑ x ∈ Z d G g,ν ( x ) obeys χ g ( ν ) < ∞ ( ν > ν c ( g )) , χ g ( ν ) = ∞ ( ν < ν c ( g )) .

Main results Theorem 1 (Bauerschmidt–Brydges–Slade 2013+). Let d = 4 . There exists g 0 > 0 such that for 0 < g ≤ g 0 , as t ↓ 0 , χ g ( ν c (1 + t )) ∼ A (log | t | ) 1 / 4 . t Theorem 2 (Brydges–Slade 2011, 2013+). Let d ≥ 4 . There exists g 0 > 0 such that for 0 < g ≤ g 0 , as | x | → ∞ , c G g,νc ( x ) ∼ | x | d − 2 . Related results: • weakly SAW on 4-dimensional hierarchical lattice (replacement of Z 4 by a recursive structure well-suited to RG): Brydges–Evans–Imbrie (1992); Brydges–Imbrie (2003); and with different RG method Ohno (2013+). • 4-dimensional ϕ 4 field theory: Gaw¸ edzki–Kupiainen (1985), Feldman–Magnen– Rivasseau–S´ en´ eor (1987), Hara–Tasaki (1987).

Bubble diagram and role of d = 4 Let ∆ denote the discrete Laplacian on Z d , i.e., ∆ ϕ x = ∑ y : | y − x | =1 ( ϕ y − ϕ x ) . Let ∫ ∞ E 0 ( 1 X ( T )= x ) e − m 2 T dT = ( − ∆ + m 2 ) − 1 C m 2 ( x ) = 0 x . 0 Let X, Y be independent continuous-time simple random walks started from 0 ∈ Z d . The simple random walk bubble diagram is ∫ ∞ E 0 , 0 ( 1 X ( T )= Y ( S ) ) e − m 2 S e − m 2 T dSdT, ( C m 2 ( x )) 2 = ∑ B m 2 = 0 x ∈ Z d and the expected mutual intersection time is ∫ ∞ B 0 = E 0 , 0 ( 1 X ( T )= Y ( S ) ) dSdT. 0 Direct calculation shows d = 4 is critical: as m 2 ↓ 0 ,  cm − ( d − 4) d < 4    B m 2 ∼ c | log m | d = 4  c d > 4 .  

Bubble diagram and role of d = 4 For d ≥ 5 and use of the lace expansion an essential feature is B 0 < ∞ . For d = 4 , the logarithmic divergence B m 2 ∼ c | log m | is the source of the logarithmic corrections to scaling for the 4-d SAW.

Comparison of WSAW and SRW Our strategy is to determine an effective approximation of the WSAW two-point function by the two-point function of a renormalised SRW: with m 2 ↓ 0 as ν ↓ ν c . G g,ν ( x ) ≈ (1 + z 0 ) G 0 ,m 2 ( x ) In physics terminology: • m is the renormalised mass (or physical mass), • 1 + z 0 is the field strength renormalisation. We use a rigorous RG method to construct z 0 = z 0 ( g, ν ) and m 2 = m 2 ( g, ν ) such that χ g ( ν ) = (1 + z 0 ) χ 0 ( m 2 ) = (1 + z 0 ) m − 2 with, as t ↓ 0 , t m 2 ( g, ν c (1 + t )) ∼ const z 0 ( g, ν c (1 + t )) → const , | log t | 1 / 4 .

Finite-volume approximation Fix g > 0 . Given a (large) positive integer L , let Λ N be the torus Z d /L N Z d . Finite-volume two-point function is defined by ∫ ∞ ( ) E N e − gI ( T ) 1 X ( T )= x e − νT dT, G N,ν ( x ) = 0 0 with E N 0 the expectation for the continuous-time simple random walk on Λ N . Let χ N ( ν ) = ∑ x ∈ Λ N G N,ν ( x ) denote the susceptibility on Λ N . Easy: N →∞ χ N ( ν ) = χ ( ν ) ∈ [0 , ∞ ] lim ( ν ∈ R ) , N →∞ χ ′ N ( ν ) = χ ′ ( ν ) lim ( ν > ν c ) . We work in finite volume, maintaining sufficient control to take the limit.

Gaussian expectation and super-expectation Let ϕ : Λ → C , with complex conjugate ¯ ϕ , and let C = ( − ∆ + m 2 ) − 1 . The standard Gaussian expectation is ∫ ϕC − 1 ϕ F ( ¯ C Λ e − ¯ E C F ( ¯ ϕ, ϕ ) = Z − 1 ϕ, ϕ ) d ¯ ϕdϕ. C The super-expectation is (differentials anti-commute) ∫ ϕC − 1 ϕ − 1 ϕC − 1 dϕ F ( ¯ C Λ e − ¯ 2 πid ¯ E C F ( ¯ ϕ, ϕ, d ¯ ϕ, ϕ, d ¯ ϕ, dϕ ) = ϕ, dϕ ) . Then E C F ( ¯ ϕ, ϕ ) = E C F ( ¯ so in particular E C ¯ ϕ 0 ϕ x = E C ¯ ϕ, ϕ ) , ϕ 0 ϕ x = C 0 x . Much of the standard theory of Gaussian integration carries over to this setting, with beautiful properties, e.g., for a function of τ = ( τ x ) with τ x = ¯ 2 πi d ¯ 1 ϕ x ϕ x + ϕ x dϕ x , E C F ( τ ) = F (0) .

Functional integral representation Let τ x = ϕ x ¯ 2 πi dϕ x d ¯ 1 ϕ x + ϕ x , τ ∆ ,x = 1 ( ) ϕ x ( − ∆ ¯ 2 πi dϕ x ( − ∆ d ¯ 1 ϕ ) x + ϕ ) x + c.c. , 2 Theorem. ∫ ∞ ( ) E N e − gI ( T ) 1 X ( T )= x e − νT dT G N,ν ( x ) = 0 0 ∫ u ∈ Λ( gτ 2 u + ντu + τ ∆ ,u ) ¯ e − ∑ = ϕ 0 ϕ x . C Λ N RHS is the two-point function of a supersymmetric field theory with boson field ( ϕ, ¯ ϕ ) and fermion field ( dϕ, d ¯ ϕ ) . (Parisi–Sourlas ’80; McKane ’80; Dynkin ’83; Le Jan ’87; Brydges–Imbrie ’03; Brydges–Imbrie–Slade ’09).

Renormalised parameters and Gaussian approximation Change of variables ϕ x �→ √ 1 + z 0 ϕ x in the integral Let z 0 > − 1 and m 2 > 0 . representation gives G g,ν ( x ) = (1 + z 0 ) E C ( e − V 0 ¯ ϕ 0 ϕ x ) where E C denotes Gaussian super-expectation with covariance C = ( − ∆ + m 2 ) − 1 , and ∑ ( g 0 τ 2 V 0 = u + ν 0 τ u + z 0 τ ∆ ,u ) u ∈ Λ g 0 = g (1 + z 0 ) 2 , ν 0 = (1 + z 0 ) ν − m 2 . Thus the two-point function is the two-point function of a perturbation (by e − V 0 ) of a supersymmetric Gaussian field. Now we study E C ( e − V 0 ¯ ϕ 0 ϕ x ) and forget about the walks.

Objective Given m 2 , g 0 , ν 0 , z 0 , define C = ( − ∆+ m 2 ) − 1 , V 0 = ∑ u ∈ Λ ( g 0 τ 2 u + ν 0 τ u + z 0 τ ∆ ,u ) , E C ( e − V 0 ¯ χ N ( g 0 , ν 0 , z 0 , m 2 ) = ∑ χ N = ˆ ˆ ϕ 0 ϕ x ) , χ = lim ˆ N →∞ ˆ χ N . x ∈ Λ Objective: choose z 0 , ν 0 depending on g 0 , m 2 such that 1 ∂ ˆ χ 1 1 χ = ˆ m 2 , ∼ − c g 0 . B 1 / 4 m 4 ∂ν 0 m 2 This suffices because after some implicit function theory it allows ν c ( g ) to be identified and gives ∂χ ∂ν ∼ − C g χ 2 (log χ ) 1 / 4 ( ν ↓ ν c ) which implies that χ ( ν c (1 + t )) ∼ ct − 1 ( | log t | ) 1 / 4 . So our focus now is on ˆ χ N .