Scaling Limit of the Prudent Walk

We describe the scaling limit of the nearest neighbour prudent walk on the square lattice, which performs steps uniformly in directions in which it does not see sites already visited. We show that the scaling limit is given by the process Z(u) = s_1 theta^+(3u/7) e_1 + s_2 theta^-(3u/7) e_2, where e_1, e_2 is the canonical basis, theta^+(t), resp. theta^-(t), is the time spent by a one-dimensional Brownian motion above, resp. below, 0 up to time t, and s_1, s_2 are two random signs. In particular, the asymptotic speed of the walk is well-defined in the L^1-norm and equals 3/7.Comment: Better exposition, stronger claim, simpler description of the limiting process; final version, to appear in Electr. Commun. Probab

Neoadjuvant capecitabine, radiotherapy, and bevacizumab (CRAB) in locally advanced rectal cancer: results of an open-label phase II study

<p>Abstract</p> <p>Background</p> <p>Preoperative capecitabine-based chemoradiation is a standard treatment for locally advanced rectal cancer (LARC). Here, we explored the safety and efficacy of the addition of bevacizumab to capecitabine and concurrent radiotherapy for LARC.</p> <p>Methods</p> <p>Patients with MRI-confirmed stage II/III rectal cancer received bevacizumab 5 mg/kg i.v. 2 weeks prior to neoadjuvant chemoradiotherapy followed by bevacizumab 5 mg/kg on Days 1, 15 and 29, capecitabine 825 mg/m²twice daily on Days 1-38, and concurrent radiotherapy 50.4 Gy (1.8 Gy/day, 5 days/week for 5 weeks + three 1.8 Gy/day), starting on Day 1. Total mesorectal excision was scheduled 6-8 weeks after completion of chemoradiotherapy. Tumour regression grades (TRG) were evaluated on surgical specimens according to Dworak. The primary endpoint was pathological complete response (pCR).</p> <p>Results</p> <p>61 patients were enrolled (median age 60 years [range 31-80], 64% male). Twelve patients (19.7%) had T3N0 tumours, 1 patient T2N1, 19 patients (31.1%) T3N1, 2 patients (3.3%) T2N2, 22 patients (36.1%) T3N2 and 5 patients (8.2%) T4N2. Median tumour distance from the anal verge was 6 cm (range 0-11). Grade 3 adverse events included dermatitis (n = 6, 9.8%), proteinuria (n = 4, 6.5%) and leucocytopenia (n = 3, 4.9%). Radical resection was achieved in 57 patients (95%), and 42 patients (70%) underwent sphincter-preserving surgery. TRG 4 (pCR) was recorded in 8 patients (13.3%) and TRG 3 in 9 patients (15.0%). T-, N- and overall downstaging rates were 45.2%, 73.8%, and 73.8%, respectively.</p> <p>Conclusions</p> <p>This study demonstrates the feasibility of preoperative chemoradiotherapy with bevacizumab and capecitabine. The observed adverse events of neoadjuvant treatment are comparable with those previously reported, but the pCR rate was lower.</p

On the Gibbs states of the noncritical Potts model on Z^2

We prove that all Gibbs states of the q-state nearest neighbor Potts model on Z^2 below the critical temperature are convex combinations of the q pure phases; in particular, they are all translation invariant. To achieve this goal, we consider such models in large finite boxes with arbitrary boundary condition, and prove that the center of the box lies deeply inside a pure phase with high probability. Our estimate of the finite-volume error term is of essentially optimal order, which stems from the Brownian scaling of fluctuating interfaces. The results hold at any supercritical value of the inverse temperature.Comment: Minor typos corrected after proofreading. Final version, to appear in Probab. Theory Relat. Field