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The radius of a self-repelling star polymer

Carl Mueller, Eyal Neuman

TL;DR

The paper rigorously analyzes the radius of a weakly self-avoiding star polymer modeled by $N$ Brownian paths up to time $T$ with an exponential penalty for close visits. It develops a drift-change-of-measure framework to bound the partition function $Z_T$ and to control occupation measures of self- and cross-intersections, yielding dimension-dependent high-probability bounds on the radius $R_T$. In particular, it establishes a rigorous $d=2$ scaling $R_T$ proportional to $T^{3/4}$ up to logarithmic factors, aligning with the predicted two-dimensional SAW behavior, and provides complementary bounds in $d=1$ and $d=3$. The methods combine a careful occupation-measure analysis with large-deviation estimates and a time-dependent radial drift to balance competing energy and entropy terms, shedding light on SAW-like scaling in low dimensions for star polymers.

Abstract

We study the effective radius of weakly self-avoiding star polymers in one, two, and three dimensions. Our model includes $N$ Brownian motions up to time $T$, started at the origin and subject to exponential penalization based on the amount of time they spend close to each other, or close to themselves. The effective radius measures the typical distance from the origin. Our main result gives estimates for the effective radius where in two and three dimensions we impose the restriction that $T \leq N$. One of the highlights of our results is that in two dimensions, we find that the radius is proportional to $T^{3/4}$, up to logarithmic corrections. Our result may shed light on the well-known conjecture that for a single self-avoiding random walk in two dimensions, the end-to-end distance up to time $T$ is roughly $T^{3/4}$.

The radius of a self-repelling star polymer

TL;DR

The paper rigorously analyzes the radius of a weakly self-avoiding star polymer modeled by Brownian paths up to time with an exponential penalty for close visits. It develops a drift-change-of-measure framework to bound the partition function and to control occupation measures of self- and cross-intersections, yielding dimension-dependent high-probability bounds on the radius . In particular, it establishes a rigorous scaling proportional to up to logarithmic factors, aligning with the predicted two-dimensional SAW behavior, and provides complementary bounds in and . The methods combine a careful occupation-measure analysis with large-deviation estimates and a time-dependent radial drift to balance competing energy and entropy terms, shedding light on SAW-like scaling in low dimensions for star polymers.

Abstract

We study the effective radius of weakly self-avoiding star polymers in one, two, and three dimensions. Our model includes Brownian motions up to time , started at the origin and subject to exponential penalization based on the amount of time they spend close to each other, or close to themselves. The effective radius measures the typical distance from the origin. Our main result gives estimates for the effective radius where in two and three dimensions we impose the restriction that . One of the highlights of our results is that in two dimensions, we find that the radius is proportional to , up to logarithmic corrections. Our result may shed light on the well-known conjecture that for a single self-avoiding random walk in two dimensions, the end-to-end distance up to time is roughly .
Paper Structure (10 sections, 12 theorems, 153 equations)

This paper contains 10 sections, 12 theorems, 153 equations.

Table of Contents

  1. Introduction
  2. strategy
  3. Proofs of Theorems \ref{['thm-radius']} and \ref{['prop-z']}
  4. Proof of Proposition \ref{['prop-i-2']} for $d=2,3$
  5. Cross-intersections occupation measure for $d=2,3$
  6. Self-intersection occupation measure for $d=2,3$
  7. Proof of Proposition \ref{['prop-i-2']} for $d=1$
  8. Cross-intersection occupation measure
  9. Self-intersection occupation measure
  10. Proof of Lemma \ref{['lemma-ber']}

Key Result

Theorem 1.1

There exists positive constants $C_d, c_d$ and $C$ not depending on $N,T$ and $\beta$ such that,

Theorems & Definitions (27)

  • Theorem 1.1
  • Lemma 2.1
  • Theorem 2.2
  • proof : Proof of Theorem \ref{['prop-z']}
  • Remark 3.1
  • Proposition 3.2
  • proof : Proof of Theorem \ref{['thm-radius']}
  • proof : Proof of Proposition \ref{['prop-i-2']} for $d=2,3$
  • Proposition 4.1
  • Lemma 5.1
  • ...and 17 more