Table 3.

Analytical and simulation results for critical exponents ρ, ν_path and ν of annealed isolated self-avoiding lattice trees in a θ-solvent for different spatial dimensionalities d. $\frac{\delta \nu }{\delta {\nu }_{\text{ideal}}}$ is defined as in Table 1. Currently available exact values or best estimates (in boldface) are reported as symbols in Fig. 3, where predictions of Flory theory are shown as lines. To our knowledge, analytical or simulation data for quenched isolated branched polymers in θ-solutions are currently missing.

Annealed branching polymers in dilute solutions in a θ-solvent
d	ρ	νpath	ν	δν	$\frac{\delta \nu }{\delta {\nu }_{\text{ideal}}}$	Technique	Reference
2	3/4=0.75	5/6=0.833	5/8 = 0.625	0.0891	31%	Flory theory	This work
2	–	–	0.54±0.03	0.0041	1%	Monte Carlo	Madras & J. van Rensburg (1997)59
2	–	–	0.5359±0.0003	–	–	Monte Carlo	Hsu & Grassberger (2005)61
2	–	–	0.52±0.03	−0.0159	-6%	Renormalization group	Janssen & Stenull (2011)49
3	7/11=0.636	5/7=0.714	5/11 = 0.455	0.055	36%	Flory theory	This work
3	–	–	0.400 ±0.005	–	–	Monte Carlo	Madras & J. van Rensburg (1997)59
3	–	–	0.396 ±0.007	−0.004	-3%	Renormalization group	Janssen & Stenull (2011)49
4	4/7=0.571	5/8=0.625	5/14 = 0.357	0.028	35%	Flory theory	This work
4	–	–	0.329 ± 0.002	–	–	Renormalization group	Janssen & Stenull (2011)49
5	9/17=0.529	5/9=0.556	5/17 = 0.294	0.0091	26%	Flory theory	This work
5	–	–	0.2849 ± 0.0002	–	–	Renormalization group	Janssen & Stenull (2011)49
6	1/2=0.5	1/2=0.5	1/4 = 0.25	0	0%	Flory theory	This work
6	–	–	0.25	–	–	Renormalization group	Janssen & Stenull (2011)49